def main(): # Get the path for the mesh to load, either from the program argument if # one was given, or a dialog otherwise if(len(sys.argv) > 1): filename = sys.argv[1] else: print("ERROR: No file name specified. Proper syntax is 'python testview.py path/to/your/mesh.obj'.") exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename)) # Create a viewer window winName = 'meshview -- ' + os.path.basename(filename) meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) meshDisplay.startMainLoop()
def main(): # Get the path for the mesh to load, either from the program argument if # one was given, or a dialog otherwise if (len(sys.argv) > 1): filename = sys.argv[1] else: print( "ERROR: No file name specified. Proper syntax is 'python testview.py path/to/your/mesh.obj'." ) exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename)) # Create a viewer window winName = 'meshview -- ' + os.path.basename(filename) meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) meshDisplay.startMainLoop()
def main(): # Get the path for the mesh to load from the program argument if(len(sys.argv) == 3): partString = sys.argv[1] if partString not in ['part1','part2','part3']: print("ERROR part specifier not recognized. Should be one of 'part1', 'part2', or 'part3'") exit() filename = sys.argv[2] else: print("ERROR: Incorrect call syntax. Proper syntax is 'python Assignment3.py partN path/to/your/mesh.obj'.") exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename)) # Create a viewer object winName = 'DDG Assignment3 ' + partString + '-- ' + os.path.basename(filename) meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) ###################### BEGIN YOUR CODE # implement the body of each of these functions ############################ # Part 0: Helper functions # ############################ # Implement a few useful functions that you will want in the remainder of # the assignment. @property @cacheGeometry def cotanWeight(self): """ Return the cotangent weight for an edge. Since this gets called on an edge, 'self' will be a reference to an edge. This will be useful in the problems below. Don't forget, everything you implemented for the last homework is now available as part of the library (normals, areas, etc). (Moving forward, Vertex.normal will mean area-weighted normals, unless otherwise specified) """ return 0.0 # placeholder value @property @cacheGeometry def dualArea(self): """ Return the dual area associated with a vertex. Since this gets called on a vertex, 'self' will be a reference to a vertex. Recall that the dual area can be defined as 1/3 the area of the surrounding faces. """ return 0.0 # placeholder value def enumerateVertices(mesh): """ Assign a unique index from 0 to (N-1) to each vertex in the mesh. Should return a dictionary containing mappings {vertex ==> index}. You will want to use this function in your solutions below. """ return None # placeholder value ################################# # Part 1: Dense Poisson Problem # ################################# # Solve a Poisson problem on the mesh. The primary function here # is solvePoissonProblem_dense(), it will get called when you run # python Assignment3.py part1 /path/to/your/mesh.obj # and specify density values with the mouse (the press space to solve). # # Note that this code will be VERY slow on large meshes, because it uses # dense matrices. def buildLaplaceMatrix_dense(mesh, index): """ Build a Laplace operator for the mesh, with a dense representation 'index' is a dictionary mapping {vertex ==> index} Returns the resulting matrix. """ return None # placeholder value def buildMassMatrix_dense(mesh, index): """ Build a mass matrix for the mesh. Returns the resulting matrix. """ return None # placeholder value def solvePoissonProblem_dense(mesh, densityValues): """ Solve a Poisson problem on the mesh. The results should be stored on the vertices in a variable named 'solutionVal'. You will want to make use of your buildLaplaceMatrix_dense() function from above. densityValues is a dictionary mapping {vertex ==> value} that specifies densities. The density is implicitly zero at every vertex not in this dictionary. When you run this program with 'python Assignment3.py part1 path/to/your/mesh.obj', you will get to click on vertices to specify density conditions. See the assignment document for more details. """ pass # remove this line once you have implemented the method ################################## # Part 2: Sparse Poisson Problem # ################################## # Solve a Poisson problem on the mesh. The primary function here # is solvePoissonProblem_sparse(), it will get called when you run # python Assignment3.py part2 /path/to/your/mesh.obj # and specify density values with the mouse (the press space to solve). # # This will be very similar to the previous part. Be sure to see the wiki # for notes about the nuances of sparse matrix computation. Now, your code # should scale well to larger meshes! def buildLaplaceMatrix_sparse(mesh, index): """ Build a laplace operator for the mesh, with a sparse representation. This will be nearly identical to the dense method. 'index' is a dictionary mapping {vertex ==> index} Returns the resulting sparse matrix. """ return None # placeholder value def buildMassMatrix_sparse(mesh, index): """ Build a sparse mass matrix for the system. Returns the resulting sparse matrix. """ return None # placeholder value def solvePoissonProblem_sparse(mesh, densityValues): """ Solve a Poisson problem on the mesh, using sparse matrix operations. This will be nearly identical to the dense method. The results should be stored on the vertices in a variable named 'solutionVal'. densityValues is a dictionary mapping {vertex ==> value} that specifies any densities. The density is implicitly zero at every vertex not in this dictionary. Note: Be sure to look at the notes on the github wiki about sparse matrix computation in Python. When you run this program with 'python Assignment3.py part2 path/to/your/mesh.obj', you will get to click on vertices to specify density conditions. See the assignment document for more details. """ pass # remove this line once you have implemented the method ############################### # Part 3: Mean Curvature Flow # ############################### # Perform mean curvature flow on the mesh. The primary function here # is meanCurvatureFlow(), which will get called when you run # python Assignment3.py part3 /path/to/your/mesh.obj # You can adjust the step size with the 'z' and 'x' keys, and press space # to perform one step of flow. # # Of course, you will want to use sparse matrices here, so your code # scales well to larger meshes. def buildMeanCurvatureFlowOperator(mesh, index, h): """ Construct the (sparse) mean curvature operator matrix for the mesh. It might be helpful to use your buildLaplaceMatrix_sparse() and buildMassMatrix_sparse() methods from before. Returns the resulting matrix. """ return None # placeholder value def meanCurvatureFlow(mesh, h): """ Perform mean curvature flow on the mesh. The result of this operation is updated positions for the vertices; you should conclude by modifying the position variables for the mesh vertices. h is the step size for the backwards euler integration. When you run this program with 'python Assignment3.py part3 path/to/your/mesh.obj', you can press the space bar to perform this operation and z/x to change the step size. Recall that before you modify the positions of the mesh, you will need to set mesh.staticGeometry = False, which disables caching optimizations but allows you to modfiy the geometry. After you are done modfiying positions, you should set mesh.staticGeometry = True to re-enable these optimizations. You should probably have mesh.staticGeometry = True while you assemble your operator, or it will be very slow. """ pass # remove this line once you have implemented the method ###################### END YOUR CODE Edge.cotanWeight = cotanWeight Vertex.dualArea = dualArea # A pick function for choosing density conditions densityValues = dict() def pickVertBoundary(vert): value = 1.0 if pickVertBoundary.isHigh else -1.0 print(" Selected vertex at position:" + printVec3(vert.position)) print(" as a density with value = " + str(value)) densityValues[vert] = value pickVertBoundary.isHigh = not pickVertBoundary.isHigh pickVertBoundary.isHigh = True # Run in part1 mode if partString == 'part1': print("\n\n === Executing assignment 2 part 1") print(""" Please click on vertices of the mesh to specify density conditions. Alternating clicks will specify high-value (= 1.0) and low-value (= -1.0) density conditions. You may select as many density vertices as you want, but >= 2 are necessary to yield an interesting solution. When you are done, press the space bar to execute your solver and view the results. """) meshDisplay.pickVertexCallback = pickVertBoundary meshDisplay.drawVertices = True def executePart1Callback(): print("\n=== Solving Poisson problem with your dense solver\n") # Print and check the density values print("Density values:") for key in densityValues: print(" " + str(key) + " = " + str(densityValues[key])) if len(densityValues) < 2: print("Aborting solve, not enough density vertices specified") return # Call the solver print("\nSolving problem...") t0 = time.time() solvePoissonProblem_dense(mesh, densityValues) tSolve = time.time() - t0 print("...solution completed.") print("Solution took {:.5f} seconds.".format(tSolve)) print("Visualizing results...") # Error out intelligently if nothing is stored on vert.solutionVal for vert in mesh.verts: if not hasattr(vert, 'solutionVal'): print("ERROR: At least one vertex does not have the attribute solutionVal defined.") exit() if not isinstance(vert.solutionVal, float): print("ERROR: The data stored at vertex.solutionVal is not of type float.") print(" The data has type=" + str(type(vert.solutionVal))) print(" The data looks like vert.solutionVal="+str(vert.solutionVal)) exit() # Visualize the result # meshDisplay.setShapeColorFromScalar("solutionVal", definedOn='vertex', cmapName="seismic", vMinMax=[-1.0,1.0]) meshDisplay.setShapeColorFromScalar("solutionVal", definedOn='vertex', cmapName="seismic") meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback(' ', executePart1Callback, docstring="Solve the Poisson problem and view the results") # Start the GUI meshDisplay.startMainLoop() # Run in part2 mode elif partString == 'part2': print("\n\n === Executing assignment 2 part 2") print(""" Please click on vertices of the mesh to specify density conditions. Alternating clicks will specify high-value (= 1.0) and low-value (= -1.0) density conditions. You may select as many density vertices as you want, but >= 2 are necessary to yield an interesting solution. When you are done, press the space bar to execute your solver and view the results. """) meshDisplay.pickVertexCallback = pickVertBoundary meshDisplay.drawVertices = True def executePart2Callback(): print("\n=== Solving Poisson problem with your sparse solver\n") # Print and check the density values print("Density values:") for key in densityValues: print(" " + str(key) + " = " + str(densityValues[key])) if len(densityValues) < 2: print("Aborting solve, not enough density vertices specified") return # Call the solver print("\nSolving problem...") t0 = time.time() solvePoissonProblem_sparse(mesh, densityValues) tSolve = time.time() - t0 print("...solution completed.") print("Solution took {:.5f} seconds.".format(tSolve)) print("Visualizing results...") # Error out intelligently if nothing is stored on vert.solutionVal for vert in mesh.verts: if not hasattr(vert, 'solutionVal'): print("ERROR: At least one vertex does not have the attribute solutionVal defined.") exit() if not isinstance(vert.solutionVal, float): print("ERROR: The data stored at vertex.solutionVal is not of type float.") print(" The data has type=" + str(type(vert.solutionVal))) print(" The data looks like vert.solutionVal="+str(vert.solutionVal)) exit() # Visualize the result # meshDisplay.setShapeColorFromScalar("solutionVal", definedOn='vertex', cmapName="seismic", vMinMax=[-1.0,1.0]) meshDisplay.setShapeColorFromScalar("solutionVal", definedOn='vertex', cmapName="seismic") meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback(' ', executePart2Callback, docstring="Solve the Poisson problem and view the results") # Start the GUI meshDisplay.startMainLoop() # Run in part3 mode elif partString == 'part3': print("\n\n === Executing assignment 2 part 3") print(""" Press the space bar to perform one step of mean curvature flow smoothing, using your solver. Pressing the 'z' and 'x' keys will decrease and increase the step size (h), respectively. """) stepSize = [0.01] def increaseStepsize(): stepSize[0] += 0.001 print("Increasing step size. New size h="+str(stepSize[0])) def decreaseStepsize(): stepSize[0] -= 0.001 print("Decreasing step size. New size h="+str(stepSize[0])) meshDisplay.registerKeyCallback('z', decreaseStepsize, docstring="Increase the value of the step size (h) by 0.1") meshDisplay.registerKeyCallback('x', increaseStepsize, docstring="Decrease the value of the step size (h) by 0.1") def smoothingStep(): print("\n=== Performing mean curvature smoothing step\n") print(" Step size h="+str(stepSize[0])) # Call the solver print(" Solving problem...") t0 = time.time() meanCurvatureFlow(mesh, stepSize[0]) tSolve = time.time() - t0 print(" ...solution completed.") print(" Solution took {:.5f} seconds.".format(tSolve)) print("Updating display...") meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback(' ', smoothingStep, docstring="Perform one step of your mean curvature flow on the mesh") # Start the GUI meshDisplay.startMainLoop()
def main(inputfile, show=False, StaticGeometry=False): # Get the path for the mesh to load, either from the program argument if # one was given, or a dialog otherwise if (len(sys.argv) > 1): filename = sys.argv[1] elif inputfile is not None: filename = inputfile else: string1 = "ERROR: No file name specified. " string2 = "Proper syntax is 'python Assignment2.py path/to/your/mesh.obj'." print(string1 + string2) exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename), staticGeometry=StaticGeometry) # Create a viewer object winName = 'DDG Assignment2 -- ' + os.path.basename(filename) if show: meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) ###################### BEGIN YOUR CODE # implement the body of each of these functions # # # def buildLaplaceMatrix_dense(mesh, index): # """ # Build a Laplace operator for the mesh, with a dense representation # # 'index' is a dictionary mapping {vertex ==> index} # # Returns the resulting matrix. # """ # #index_map = mesh.enumerateVertices() # index_map = enumerateVertices(mesh) # # return Laplacian @property @cacheGeometry def faceArea(self): """ Compute the area of a face. Though not directly requested, this will be useful when computing face-area weighted normals below. This method gets called on a face, so 'self' is a reference to the face at which we will compute the area. """ v = list(self.adjacentVerts()) a = 0.5 * norm( cross(v[1].position - v[0].position, v[2].position - v[0].position)) return a @property @cacheGeometry def faceNormal(self): """ Compute normal at a face of the mesh. Unlike at vertices, there is one very obvious way to do this, since a face uniquely defines a plane. This method gets called on a face, so 'self' is a reference to the face at which we will compute the normal. """ v = list(self.adjacentVerts()) n = normalize( cross(v[1].position - v[0].position, v[2].position - v[0].position)) return n @property @cacheGeometry def vertexNormal_EquallyWeighted(self): """ Compute a vertex normal using the 'equally weighted' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. http://brickisland.net/cs177/?p=217 Perhaps the simplest way to get vertex normals is to just add up the neighboring face normals: """ normalSum = np.array([0.0, 0.0, 0.0]) for face in self.adjacentFaces(): normalSum += face.normal n = normalize(normalSum) #issue: # two different tessellations of the same geometry # can produce very different vertex normals return n @property @cacheGeometry def vertexNormal_AreaWeighted(self): """ Compute a vertex normal using the 'face area weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. The area-weighted normal vector for this vertex""" normalSum = np.array([0.0, 0.0, 0.0]) for face in self.adjacentFaces(): normalSum += face.normal * face.area n = normalize(normalSum) #print 'computed vertexNormal_AreaWeighted n = ',n return n @property @cacheGeometry def vertexNormal_AngleWeighted(self): """ element type : vertex Compute a vertex normal using the 'Tip-Angle Weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. A simple way to reduce dependence on the tessellation is to weigh face normals by their corresponding tip angles theta, i.e., the interior angles incident on the vertex of interest: """ normalSum = np.array([0.0, 0.0, 0.0]) for face in self.adjacentFaces(): vl = list(face.adjacentVerts()) vl.remove(self) v1 = vl[0].position - self.position v2 = vl[1].position - self.position # norm ->no need for check: # it doesn not matter what the sign is? #area = norm(cross(v1, v2)) ##if area < 0.0000000001*max((norm(v1),norm(v2))): #if area < 0.: # area *= -1. alpha = np.arctan2(norm(cross(v1, v2)), dot(v1, v2)) #print v1 #print v2 #print alpha #print '' normalSum += face.normal * alpha n = normalize(normalSum) return n @property @cacheGeometry def cotan(self): """ element type : halfedge Compute the cotangent of the angle OPPOSITE this halfedge. This is not directly required, but will be useful when computing the mean curvature normals below. This method gets called on a halfedge, so 'self' is a reference to the halfedge at which we will compute the cotangent. https://math.stackexchange.com/questions/2041099/ angle-between-vectors-given-cross-and-dot-product see half edge here: Users/lukemcculloch/Documents/Coding/Python/ DifferentialGeometry/course-master/libddg_userguide.pdf """ if self.isReal: # Relevant vectors A = -self.next.vector B = self.next.next.vector # Nifty vector equivalent of cot(theta) val = np.dot(A, B) / norm(cross(A, B)) return val else: return 0.0 @property @cacheGeometry def vertex_Laplace(self): """ element type : vertex Compute a vertex normal using the 'mean curvature' method. del del phi = 2NH -picked up negative sign due to cross products pointing into the page? -no they are normalized. -picked up a negative sign due to the cotan(s) being defined for pj, instead of pi. But how did it change anything? SwissArmyLaplacian.pdf, page 147 Applying 'L' to a column bector u implements the cotan formula M = [square diagonal] """ hl = list(self.adjacentHalfEdges()) pi = self.position sumj = 0. ot = 1. / 3. for hlfedge in hl: pj = hlfedge.vertex.position ct1 = hlfedge.cotan ct2 = hlfedge.twin.cotan sumj += (ct1 + ct2) * (pj - pi) #laplace = .5*sumj return normalize(.5 * sumj) @property @cacheGeometry def vertexNormal_MeanCurvature(self): """ element type : vertex Compute a vertex normal using the 'mean curvature' method. Be sure to understand the relationship between this method and the area gradient method. aka, http://brickisland.net/cs177/?p=217: (the remarkable fact is that the most straightforward discretization of laplacian leads us right back to the cotan formula! I n other words, the vertex normals we get from the mean curvature vector are precisely the same as the ones we get from the area gradient.) p 60 siggraph2013 del del phi = 2NH This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. http://brickisland.net/cs177/?p=309 For the dual area of a vertex you can simply use one-third the area of the incident faces hl[0].next.next.next is hl[0] >>> True hl[0].twin.twin is hl[0] >>> True """ hl = list(self.adjacentHalfEdges()) # lenhl = len(hl) # # for hlfedge in self.adjacentHalfEdges: # pass. pi = self.position sumj = 0. ot = 1. / 3. for hlfedge in hl: pj = hlfedge.vertex.position #ct1 = hlfedge.next.cotan ct2 = hlfedge.cotan #ct2 = hlfedge.twin.next.cotan ct1 = hlfedge.twin.cotan #dual_area = -ot*hlfedge.face.area #wtf sumj += (ct2 + ct1) * (pj - pi) #/dual_area laplace = .5 * sumj """ Picked up a sign because? -picked up negative sign due to cross products pointing into the page? -no they are normalized. -picked up a negative sign due to the cotan(s) being defined for pj, instead of pi. But how did it change anything? """ return normalize(laplace) #return normalize(laplace*(.5/self.angleDefect)) @property @cacheGeometry def vertexNormal_SphereInscribed(self): """ element type : vertex Compute a vertex normal using the 'inscribed sphere' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. normal at a vertex pi can be expressed purely in terms of the edge vectors ej = pj-pi where pj are the immediate neighbors of pi """ vl = list(self.adjacentVerts()) lenvl = len(vl) vl.append(vl[0]) # Ns = Vector3D(0.0,0.0,0.0) # for i in range(lenvl): # v1 = vl[i].position # v2 = vl[i+1].position # e1 = v1 - self.position # e2 = v2 - self.position # Ns += cross(e1,e2)/((norm(e1)**2)* # (norm(e2)**2)) hl = list(self.adjacentHalfEdges()) lenhl = len(hl) hl.append(hl[0]) Ns = Vector3D(0.0, 0.0, 0.0) for i in range(lenhl): e1 = hl[i].vector e2 = hl[i + 1].vector #Ns += cross(e1,e2)/(sum(abs(e1)**2)* # sum(abs(e2)**2)) Ns += cross(e1, e2) / ((norm(e1)**2) * (norm(e2)**2)) return normalize(-Ns) #return Vector3D(0.0,0.0,0.0) # placeholder value @property @cacheGeometry def angleDefect(self): """ angleDefect <=> local Gaussian Curvature element type : vertex Compute the angle defect of a vertex, d(v) (see Assignment 1 Exercise 8). This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the angle defect. """ """ el = list(self.adjacentEdges()) evpl = list(self.adjacentEdgeVertexPairs()) fl = list(self.adjacentFaces()) vl = list(self.adjacentVerts()) https://scicomp.stackexchange.com/questions/27689/ numerically-stable-way-of-computing-angles-between-vectors #""" hl = list(self.adjacentHalfEdges()) lenhl = len(hl) hl.append(hl[0]) alpha = 0. for i in range(lenhl): v1 = hl[i].vector v2 = hl[i + 1].vector alpha += np.arctan2(norm(cross(v1, v2)), dot(v1, v2)) #dv = 2.*np.pi - alpha return 2. * np.pi - alpha def totalGaussianCurvature(): """ Compute the total Gaussian curvature in the mesh, meaning the sum of Gaussian curvature at each vertex. Note that you can access the mesh with the 'mesh' variable. """ tot = 0. for vel in mesh.verts: tot += vel.angleDefect return tot def gaussianCurvatureFromGaussBonnet(): """ Compute the total Gaussian curvature that the mesh should have, given that the Gauss-Bonnet theorem holds (see Assignment 1 Exercise 9). Note that you can access the mesh with the 'mesh' variable. The mesh includes members like 'mesh.verts' and 'mesh.faces', which are sets of the vertices (resp. faces) in the mesh. """ V = len(mesh.verts) E = len(mesh.edges) F = len(mesh.faces) EulerChar = V - E + F return 2. * np.pi * EulerChar ###################### END YOUR CODE # Set these newly-defined methods # as the methods to use in the classes Face.normal = faceNormal Face.area = faceArea Vertex.normal = vertexNormal_AreaWeighted Vertex.vertexNormal_EquallyWeighted = vertexNormal_EquallyWeighted Vertex.vertexNormal_AreaWeighted = vertexNormal_AreaWeighted Vertex.vertexNormal_AngleWeighted = vertexNormal_AngleWeighted Vertex.vertexNormal_MeanCurvature = vertexNormal_MeanCurvature # Vertex.vertex_Laplace = vertex_Laplace # Vertex.vertexNormal_SphereInscribed = vertexNormal_SphereInscribed Vertex.angleDefect = angleDefect HalfEdge.cotan = cotan if show: ## Functions which will be called # by keypresses to visualize these definitions def toggleFaceVectors(): print("\nToggling vertex vector display") if toggleFaceVectors.val: toggleFaceVectors.val = False meshDisplay.setVectors(None) else: toggleFaceVectors.val = True meshDisplay.setVectors('normal', vectorDefinedAt='face') meshDisplay.generateVectorData() toggleFaceVectors.val = False # ridiculous Python scoping hack meshDisplay.registerKeyCallback( '1', toggleFaceVectors, docstring="Toggle drawing face normal vectors") def toggleVertexVectors(): print("\nToggling vertex vector display") if toggleVertexVectors.val: toggleVertexVectors.val = False meshDisplay.setVectors(None) else: toggleVertexVectors.val = True meshDisplay.setVectors('normal', vectorDefinedAt='vertex') meshDisplay.generateVectorData() toggleVertexVectors.val = False # ridiculous Python scoping hack meshDisplay.registerKeyCallback( '2', toggleVertexVectors, docstring="Toggle drawing vertex normal vectors") def toggleDefect(): print("\nToggling angle defect display") if toggleDefect.val: toggleDefect.val = False meshDisplay.setShapeColorToDefault() else: toggleDefect.val = True meshDisplay.setShapeColorFromScalar("angleDefect", cmapName="seismic") # vMinMax=[-pi/8,pi/8]) meshDisplay.generateFaceData() toggleDefect.val = False meshDisplay.registerKeyCallback( '3', toggleDefect, docstring="Toggle drawing angle defect coloring") def useEquallyWeightedNormals(): mesh.staticGeometry = False print("\nUsing equally-weighted normals") Vertex.normal = vertexNormal_EquallyWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '4', useEquallyWeightedNormals, docstring="Use equally-weighted normal computation") def useAreaWeightedNormals(): mesh.staticGeometry = False print("\nUsing area-weighted normals") Vertex.normal = vertexNormal_AreaWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '5', useAreaWeightedNormals, docstring="Use area-weighted normal computation") def useAngleWeightedNormals(): mesh.staticGeometry = False print("\nUsing angle-weighted normals") Vertex.normal = vertexNormal_AngleWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '6', useAngleWeightedNormals, docstring="Use angle-weighted normal computation") def useMeanCurvatureNormals(): mesh.staticGeometry = False print("\nUsing mean curvature normals") Vertex.normal = vertexNormal_MeanCurvature mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '7', useMeanCurvatureNormals, docstring="Use mean curvature normal computation") def useSphereInscribedNormals(): mesh.staticGeometry = False print("\nUsing sphere-inscribed normals") Vertex.normal = vertexNormal_SphereInscribed mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '8', useSphereInscribedNormals, docstring="Use sphere-inscribed normal computation") def computeDiscreteGaussBonnet(): print("\nComputing total curvature:") computed = totalGaussianCurvature() predicted = gaussianCurvatureFromGaussBonnet() print(" Total computed curvature: " + str(computed)) print(" Predicted value from Gauss-Bonnet is: " + str(predicted)) print(" Error is: " + str(abs(computed - predicted))) meshDisplay.registerKeyCallback('z', computeDiscreteGaussBonnet, docstring="Compute total curvature") def deformShape(): print("\nDeforming shape") mesh.staticGeometry = False # Get the center and scale of the shape center = meshDisplay.dataCenter scale = meshDisplay.scaleFactor # Rotate according to swirly function ax = eu.Vector3(-1.0, .75, 0.5) for v in mesh.verts: vec = v.position - center theta = 0.8 * norm(vec) / scale newVec = np.array(eu.Vector3(*vec).rotate_around(ax, theta)) v.position = center + newVec mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( 'x', deformShape, docstring="Apply a swirly deformation to the shape") ## Register pick functions that output useful information on click def pickVert(vert): print(" Position:" + printVec3(vert.position)) print(" Angle defect: {:.5f}".format(vert.angleDefect)) print(" Normal (equally weighted): " + printVec3(vert.vertexNormal_EquallyWeighted)) print(" Normal (area weighted): " + printVec3(vert.vertexNormal_AreaWeighted)) print(" Normal (angle weighted): " + printVec3(vert.vertexNormal_AngleWeighted)) print(" Normal (sphere-inscribed): " + printVec3(vert.vertexNormal_SphereInscribed)) print(" Normal (mean curvature): " + printVec3(vert.vertexNormal_MeanCurvature)) meshDisplay.pickVertexCallback = pickVert def pickFace(face): print(" Face area: {:.5f}".format(face.area)) print(" Normal: " + printVec3(face.normal)) print(" Vertex positions: ") for (i, vert) in enumerate(face.adjacentVerts()): print(" v{}: {}".format((i + 1), printVec3(vert.position))) meshDisplay.pickFaceCallback = pickFace # Start the viewer running if show: meshDisplay.startMainLoop() return mesh
def main(): # Get the path for the mesh to load from the program argument if (len(sys.argv) == 3 and sys.argv[1] == 'simple'): filename = sys.argv[2] simpleTest = True elif (len(sys.argv) == 3 and sys.argv[1] == 'fancy'): filename = sys.argv[2] simpleTest = False else: print( "ERROR: Incorrect call syntax. Proper syntax is 'python Assignment5.py MODE path/to/your/mesh.obj', where MODE is either 'simple' or 'fancy'" ) exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename)) # Create a viewer object winName = 'DDG Assignment5 -- ' + os.path.basename(filename) meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) ###################### BEGIN YOUR CODE # DDGSpring216 Assignment 5 # # In this programming assignment you will implement Helmholtz-Hodge decomposition of covectors. # # The relevant mathematics and algorithm are described in section 8.1 of the course notes. # You will also need to implement the core operators in discrete exterior calculus, described mainly in # section 3.6 of the course notes. # # This code can be run with python Assignment5.py MODE /path/to/you/mesh.obj. MODE should be # either 'simple' or 'fancy', corresponding to the complexity of the input field omega that is given. # It might be easier to debug your algorithm on the simple field first. The assignment code will read in your input # mesh, generate a field 'omega' as input, run your algorithm, then display the results. # The results can be viewed as streamlines on the surface that flow with the covector field (toggle with 'p'), # or, as actual arrows on the faces (toggle with 'l'). The keys '1'-'4' will switch between the input, exact, # coexact, and harmonic fields (respectively). # # A few hints: # - Try performing some basic checks on your operators if things don't seem right. For instance, applying the # exterior derivative twice to anything should always yield zero. # - The streamline visualization is easy to look at, but can be deceiving at times. For instance, streamlines # are not very meaningful where the actual covectors are near 0. Try looking at the actual arrows in that case # ('l'). # - Many inputs will not have any harmonic components, especially genus 0 inputs. Don't stress if the harmonic # component of your output is exactly or nearly zero. # Implement the body of each of these functions... def assignEdgeOrientations(mesh): """ Assign edge orientations to each edge on the mesh. This method will be called from the assignment code, you do not need to explicitly call it in any of your methods. After this method, the following values should be defined: - edge.orientedHalfEdge (a reference to one of the halfedges touching that edge) - halfedge.orientationSign (1.0 if that halfedge agrees with the orientation of its edge, or -1.0 if not). You can use this to make much of your subsequent code cleaner. This is a pretty simple method to implement, any choice of orientation is acceptable. """ pass # remove once you have implemented def diagonalInverse(A): """ Returns the inverse of a sparse diagonal matrix. Makes a copy of the matrix. We will need to invert several diagonal matrices for the algorithm, but scipy does not offer a fast method for inverting diagonal matrices, which is a very easy special case. As such, this is a useful helper method for you. Note that the diagonal inverse is not well-defined if any of the diagonal elements are 0.0. This needs to be acconuted for when you construct the matrices. """ return None # placeholder @property @cacheGeometry def circumcentricDualArea(self): """ Compute the area of the circumcentric dual cell for this vertex. Returns a positive scalar. This gets called on a vertex, so 'self' will be a reference to the vertex. The image on page 78 of the course notes may help you visualize this. """ return 0.0 # placeholder Vertex.circumcentricDualArea = circumcentricDualArea def buildHodgeStar0Form(mesh, vertexIndex): """ Build a sparse matrix encoding the Hodge operator on 0-forms for this mesh. Returns a sparse, diagonal matrix corresponding to vertices. The discrete hodge star is a diagonal matrix where each entry is the (area of the dual element) / (area of the primal element). You will probably want to make use of the Vertex.circumcentricDualArea property you just defined. By convention, the area of a vertex is 1.0. """ return None # placeholder def buildHodgeStar1Form(mesh, edgeIndex): """ Build a sparse matrix encoding the Hodge operator on 1-forms for this mesh. Returns a sparse, diagonal matrix corresponding to edges. The discrete hodge star is a diagonal matrix where each entry is the (area of the dual element) / (area of the primal element). The solution to exercise 26 from the previous homework will be useful here. Note that for some geometries, some entries of hodge1 operator may be exactly 0. This can create a problem when we go to invert the matrix. To numerically sidestep this issue, you probably want to add a small value (like 10^-8) to these diagonal elements to ensure all are nonzero without significantly changing the result. """ return None # placeholder def buildHodgeStar2Form(mesh, faceIndex): """ Build a sparse matrix encoding the Hodge operator on 2-forms for this mesh Returns a sparse, diagonal matrix corresponding to faces. The discrete hodge star is a diagonal matrix where each entry is the (area of the dual element) / (area of the primal element). By convention, the area of a vertex is 1.0. """ return None # placeholder def buildExteriorDerivative0Form(mesh, edgeIndex, vertexIndex): """ Build a sparse matrix encoding the exterior derivative on 0-forms. Returns a sparse matrix. See section 3.6 of the course notes for an explanation of DEC. """ return None # placeholder def buildExteriorDerivative1Form(mesh, faceIndex, edgeIndex): """ Build a sparse matrix encoding the exterior derivative on 1-forms. Returns a sparse matrix. See section 3.6 of the course notes for an explanation of DEC. """ return None # placeholder def decomposeField(mesh): """ Decompose a covector in to exact, coexact, and harmonic components The input mesh will have a scalar named 'omega' on its edges (edge.omega) representing a discretized 1-form. This method should apply Helmoltz-Hodge decomposition algorithm (as described on page 107-108 of the course notes) to compute the exact, coexact, and harmonic components of omega. This method should return its results by storing three new scalars on each edge, as the 3 decomposed components: edge.exactComponent, edge.coexactComponent, and edge.harmonicComponent. Here are the primary steps you will need to perform for this method: - Create indexer objects for the vertices, faces, and edges. Note that the mesh has handy helper functions pre-defined for each of these: mesh.enumerateEdges() etc. - Build all of the operators we will need using the methods you implemented above: hodge0, hodge1, hodge2, d0, and d1. You should also compute their inverses and transposes, as appropriate. - Build a vector which represents the input covector (from the edge.omega values) - Perform a linear solve for the exact component, as described in the algorithm - Perform a linear solve for the coexact component, as described in the algorithm - Compute the harmonic component as the part which is neither exact nor coexact - Store your resulting exact, coexact, and harmonic components on the mesh edges This method will be called by the assignment code, you do not need to call it yourself. """ pass # remove once you have implemented ###################### END YOUR CODE ### More prep functions def generateFieldConstant(mesh): print("\n=== Using constant field as arbitrary direction field") for vert in mesh.verts: vert.vector = vert.projectToTangentSpace(Vector3D(1.4, 0.2, 2.4)) def generateFieldSimple(mesh): for face in mesh.faces: face.vector = face.center + Vector3D( -face.center[2], face.center[1], face.center[0]) face.vector = face.projectToTangentSpace(face.vector) def gradFromPotential(mesh, potAttr, gradAttr): # Simply compute gradient from potential for vert in mesh.verts: sumVal = Vector3D(0.0, 0.0, 0.0) sumWeight = 0.0 vertVal = getattr(vert, potAttr) for he in vert.adjacentHalfEdges(): sumVal += he.edge.cotanWeight * (getattr(he.vertex, potAttr) - vertVal) * he.vector sumWeight += he.edge.cotanWeight setattr(vert, gradAttr, normalize(sumVal)) def generateInterestingField(mesh): print( "\n=== Generating a hopefully-interesting field which has all three types of components\n" ) # Somewhat cheesy hack: # We want this function to generate the exact same result on repeated runs of the program to make # debugging easier. This means ensuring that calls to random.sample() return the exact same result # every time. Normally we could just set a seed for the RNG, and this work work if we were sampling # from a list. However, mesh.verts is a set, and Python does not guarantee consistency of iteration # order between runs of the program (since the default hash uses the memory address, which certainly # changes). Rather than doing something drastic like implementing a custom hash function on the # mesh class, we'll just build a separate data structure where vertices are sorted by position, # which allows reproducible sampling (as long as positions are distinct). sortedVertList = list(mesh.verts) sortedVertList.sort( key=lambda x: (x.position[0], x.position[1], x.position[2])) random.seed(777) # Generate curl-free (ish) component curlFreePotentialVerts = random.sample( sortedVertList, max((2, len(mesh.verts) / 1000))) potential = 1.0 bVals = {} for vert in curlFreePotentialVerts: bVals[vert] = potential potential *= -1 smoothPotential = solvePoisson(mesh, bVals) mesh.applyVertexValue(smoothPotential, "curlFreePotential") gradFromPotential(mesh, "curlFreePotential", "curlFreeVecGen") # Generate divergence-free (ish) component divFreePotentialVerts = random.sample(sortedVertList, max((2, len(mesh.verts) / 1000))) potential = 1.0 bVals = {} for vert in divFreePotentialVerts: bVals[vert] = potential potential *= -1 smoothPotential = solvePoisson(mesh, bVals) mesh.applyVertexValue(smoothPotential, "divFreePotential") gradFromPotential(mesh, "divFreePotential", "divFreeVecGen") for vert in mesh.verts: normEu = eu.Vector3(*vert.normal) vecEu = eu.Vector3(*vert.divFreeVecGen) vert.divFreeVecGen = vecEu.rotate_around( normEu, pi / 2.0) # Rotate the field by 90 degrees # Combine the components for face in mesh.faces: face.vector = Vector3D(0.0, 0.0, 0.0) for vert in face.adjacentVerts(): face.vector = 1.0 * vert.curlFreeVecGen + 1.0 * vert.divFreeVecGen face.vector = face.projectToTangentSpace(face.vector) # clear out leftover attributes to not confuse people for vert in mesh.verts: del vert.curlFreeVecGen del vert.curlFreePotential del vert.divFreeVecGen del vert.divFreePotential # Verify the orientations were defined. Need to do this early, since they are needed for setup def checkOrientationDefined(mesh): """Verify that edges have oriented halfedges and halfedges have orientation signs""" for edge in mesh.edges: if not hasattr(edge, 'orientedHalfEdge'): print( "ERROR: Edges do not have orientedHalfEdge defined. Cannot proceed" ) exit() for he in mesh.halfEdges: if not hasattr(he, 'orientationSign'): print( "ERROR: halfedges do not have orientationSign defined. Cannot proceed" ) exit() # Verify the correct properties are defined after the assignment is run def checkResultTypes(mesh): for edge in mesh.edges: # Check exact if not hasattr(edge, 'exactComponent'): print( "ERROR: Edges do not have edge.exactComponent defined. Cannot proceed" ) exit() if not isinstance(edge.exactComponent, float): print( "ERROR: edge.exactComponent is defined, but has the wrong type. Type is " + str(type(edge.exactComponent)) + " when if should be 'float'") exit() # Check cocoexact if not hasattr(edge, 'coexactComponent'): print( "ERROR: Edges do not have edge.coexactComponent defined. Cannot proceed" ) exit() if not isinstance(edge.coexactComponent, float): print( "ERROR: edge.coexactComponent is defined, but has the wrong type. Type is " + str(type(edge.coexactComponent)) + " when if should be 'float'") exit() # Check harmonic if not hasattr(edge, 'harmonicComponent'): print( "ERROR: Edges do not have edge.harmonicComponent defined. Cannot proceed" ) exit() if not isinstance(edge.harmonicComponent, float): print( "ERROR: edge.harmonicComponent is defined, but has the wrong type. Type is " + str(type(edge.harmonicComponent)) + " when if should be 'float'") exit() # Visualization related def covectorToFaceVectorWhitney(mesh, covectorName, vectorName): for face in mesh.faces: pi = face.anyHalfEdge.vertex.position pj = face.anyHalfEdge.next.vertex.position pk = face.anyHalfEdge.next.next.vertex.position eij = pj - pi ejk = pk - pj eki = pi - pk N = cross(eij, -eki) A = 0.5 * norm(N) N /= 2 * A wi = getattr(face.anyHalfEdge.edge, covectorName) * face.anyHalfEdge.orientationSign wj = getattr(face.anyHalfEdge.next.edge, covectorName) * face.anyHalfEdge.next.orientationSign wk = getattr( face.anyHalfEdge.next.next.edge, covectorName) * face.anyHalfEdge.next.next.orientationSign # s = (1.0 / (6.0 * A)) * cross(N, wi*(eki-ejk) + wj*(eij-eki) + wk*(ejk-eij)) s = (1.0 / (6.0 * A)) * cross( N, wi * (ejk - eij) + wj * (eki - ejk) + wk * (eij - eki)) setattr(face, vectorName, s) def flat(mesh, vectorFieldName, oneFormName): """ Given a vector field defined on faces, compute the corresponding (integrated) 1-form on edges. """ for edge in mesh.edges: oe = edge.orientedHalfEdge if not oe.isReal: val2 = getattr(edge.orientedHalfEdge.twin.face, vectorFieldName) meanVal = val2 elif not oe.twin.isReal: val1 = getattr(edge.orientedHalfEdge.face, vectorFieldName) meanVal = val1 else: val1 = getattr(edge.orientedHalfEdge.face, vectorFieldName) val2 = getattr(edge.orientedHalfEdge.twin.face, vectorFieldName) meanVal = 0.5 * (val1 + val2) setattr(edge, oneFormName, dot(edge.orientedHalfEdge.vector, meanVal)) ### Actual main method: # get ready assignEdgeOrientations(mesh) checkOrientationDefined(mesh) # Generate a vector field on the surface if simpleTest: generateFieldSimple(mesh) else: generateInterestingField(mesh) flat(mesh, 'vector', 'omega') # Apply the decomposition from this assignment print("\n=== Decomposing field in to components") decomposeField(mesh) print("=== Done decomposing field ===\n\n") # Verify everything necessary is defined for the output checkResultTypes(mesh) # Convert the covectors to face vectors for visualization covectorToFaceVectorWhitney(mesh, "exactComponent", "omega_exact_component") covectorToFaceVectorWhitney(mesh, "coexactComponent", "omega_coexact_component") covectorToFaceVectorWhitney(mesh, "harmonicComponent", "omega_harmonic_component") covectorToFaceVectorWhitney(mesh, "omega", "omega_original") # Register a vector toggle to switch between the vectors we just defined vectorList = [{ 'vectorAttr': 'omega_original', 'key': '1', 'colormap': 'Spectral', 'vectorDefinedAt': 'face' }, { 'vectorAttr': 'omega_exact_component', 'key': '2', 'colormap': 'Blues', 'vectorDefinedAt': 'face' }, { 'vectorAttr': 'omega_coexact_component', 'key': '3', 'colormap': 'Reds', 'vectorDefinedAt': 'face' }, { 'vectorAttr': 'omega_harmonic_component', 'key': '4', 'colormap': 'Greens', 'vectorDefinedAt': 'face' }] meshDisplay.registerVectorToggleCallbacks(vectorList) # Start the GUI meshDisplay.startMainLoop()
def main(): # Get the path for the mesh to load from the program argument if(len(sys.argv) == 3 and sys.argv[1] == 'simple'): filename = sys.argv[2] simpleTest = True elif(len(sys.argv) == 3 and sys.argv[1] == 'fancy'): filename = sys.argv[2] simpleTest = False else: print("ERROR: Incorrect call syntax. Proper syntax is 'python Assignment5.py MODE path/to/your/mesh.obj', where MODE is either 'simple' or 'fancy'") exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename)) # Create a viewer object winName = 'DDG Assignment5 -- ' + os.path.basename(filename) meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) ###################### BEGIN YOUR CODE # DDGSpring216 Assignment 5 # # In this programming assignment you will implement Helmholtz-Hodge decomposition of covectors. # # The relevant mathematics and algorithm are described in section 8.1 of the course notes. # You will also need to implement the core operators in discrete exterior calculus, described mainly in # section 3.6 of the course notes. # # This code can be run with python Assignment5.py MODE /path/to/you/mesh.obj. MODE should be # either 'simple' or 'fancy', corresponding to the complexity of the input field omega that is given. # It might be easier to debug your algorithm on the simple field first. The assignment code will read in your input # mesh, generate a field 'omega' as input, run your algorithm, then display the results. # The results can be viewed as streamlines on the surface that flow with the covector field (toggle with 'p'), # or, as actual arrows on the faces (toggle with 'l'). The keys '1'-'4' will switch between the input, exact, # coexact, and harmonic fields (respectively). # # A few hints: # - Try performing some basic checks on your operators if things don't seem right. For instance, applying the # exterior derivative twice to anything should always yield zero. # - The streamline visualization is easy to look at, but can be deceiving at times. For instance, streamlines # are not very meaningful where the actual covectors are near 0. Try looking at the actual arrows in that case # ('l'). # - Many inputs will not have any harmonic components, especially genus 0 inputs. Don't stress if the harmonic # component of your output is exactly or nearly zero. # Implement the body of each of these functions... def assignEdgeOrientations(mesh): """ Assign edge orientations to each edge on the mesh. This method will be called from the assignment code, you do not need to explicitly call it in any of your methods. After this method, the following values should be defined: - edge.orientedHalfEdge (a reference to one of the halfedges touching that edge) - halfedge.orientationSign (1.0 if that halfedge agrees with the orientation of its edge, or -1.0 if not). You can use this to make much of your subsequent code cleaner. This is a pretty simple method to implement, any choice of orientation is acceptable. """ pass # remove once you have implemented def diagonalInverse(A): """ Returns the inverse of a sparse diagonal matrix. Makes a copy of the matrix. We will need to invert several diagonal matrices for the algorithm, but scipy does not offer a fast method for inverting diagonal matrices, which is a very easy special case. As such, this is a useful helper method for you. Note that the diagonal inverse is not well-defined if any of the diagonal elements are 0.0. This needs to be acconuted for when you construct the matrices. """ return None # placeholder @property @cacheGeometry def circumcentricDualArea(self): """ Compute the area of the circumcentric dual cell for this vertex. Returns a positive scalar. This gets called on a vertex, so 'self' will be a reference to the vertex. The image on page 78 of the course notes may help you visualize this. """ return 0.0 # placeholder Vertex.circumcentricDualArea = circumcentricDualArea def buildHodgeStar0Form(mesh, vertexIndex): """ Build a sparse matrix encoding the Hodge operator on 0-forms for this mesh. Returns a sparse, diagonal matrix corresponding to vertices. The discrete hodge star is a diagonal matrix where each entry is the (area of the dual element) / (area of the primal element). You will probably want to make use of the Vertex.circumcentricDualArea property you just defined. By convention, the area of a vertex is 1.0. """ return None # placeholder def buildHodgeStar1Form(mesh, edgeIndex): """ Build a sparse matrix encoding the Hodge operator on 1-forms for this mesh. Returns a sparse, diagonal matrix corresponding to edges. The discrete hodge star is a diagonal matrix where each entry is the (area of the dual element) / (area of the primal element). The solution to exercise 26 from the previous homework will be useful here. Note that for some geometries, some entries of hodge1 operator may be exactly 0. This can create a problem when we go to invert the matrix. To numerically sidestep this issue, you probably want to add a small value (like 10^-8) to these diagonal elements to ensure all are nonzero without significantly changing the result. """ return None # placeholder def buildHodgeStar2Form(mesh, faceIndex): """ Build a sparse matrix encoding the Hodge operator on 2-forms for this mesh Returns a sparse, diagonal matrix corresponding to faces. The discrete hodge star is a diagonal matrix where each entry is the (area of the dual element) / (area of the primal element). By convention, the area of a vertex is 1.0. """ return None # placeholder def buildExteriorDerivative0Form(mesh, edgeIndex, vertexIndex): """ Build a sparse matrix encoding the exterior derivative on 0-forms. Returns a sparse matrix. See section 3.6 of the course notes for an explanation of DEC. """ return None # placeholder def buildExteriorDerivative1Form(mesh, faceIndex, edgeIndex): """ Build a sparse matrix encoding the exterior derivative on 1-forms. Returns a sparse matrix. See section 3.6 of the course notes for an explanation of DEC. """ return None # placeholder def decomposeField(mesh): """ Decompose a covector in to exact, coexact, and harmonic components The input mesh will have a scalar named 'omega' on its edges (edge.omega) representing a discretized 1-form. This method should apply Helmoltz-Hodge decomposition algorithm (as described on page 107-108 of the course notes) to compute the exact, coexact, and harmonic components of omega. This method should return its results by storing three new scalars on each edge, as the 3 decomposed components: edge.exactComponent, edge.coexactComponent, and edge.harmonicComponent. Here are the primary steps you will need to perform for this method: - Create indexer objects for the vertices, faces, and edges. Note that the mesh has handy helper functions pre-defined for each of these: mesh.enumerateEdges() etc. - Build all of the operators we will need using the methods you implemented above: hodge0, hodge1, hodge2, d0, and d1. You should also compute their inverses and transposes, as appropriate. - Build a vector which represents the input covector (from the edge.omega values) - Perform a linear solve for the exact component, as described in the algorithm - Perform a linear solve for the coexact component, as described in the algorithm - Compute the harmonic component as the part which is neither exact nor coexact - Store your resulting exact, coexact, and harmonic components on the mesh edges This method will be called by the assignment code, you do not need to call it yourself. """ pass # remove once you have implemented ###################### END YOUR CODE ### More prep functions def generateFieldConstant(mesh): print("\n=== Using constant field as arbitrary direction field") for vert in mesh.verts: vert.vector = vert.projectToTangentSpace(Vector3D(1.4, 0.2, 2.4)) def generateFieldSimple(mesh): for face in mesh.faces: face.vector = face.center + Vector3D(-face.center[2], face.center[1], face.center[0]) face.vector = face.projectToTangentSpace(face.vector) def gradFromPotential(mesh, potAttr, gradAttr): # Simply compute gradient from potential for vert in mesh.verts: sumVal = Vector3D(0.0,0.0,0.0) sumWeight = 0.0 vertVal = getattr(vert, potAttr) for he in vert.adjacentHalfEdges(): sumVal += he.edge.cotanWeight * (getattr(he.vertex, potAttr) - vertVal) * he.vector sumWeight += he.edge.cotanWeight setattr(vert, gradAttr, normalize(sumVal)) def generateInterestingField(mesh): print("\n=== Generating a hopefully-interesting field which has all three types of components\n") # Somewhat cheesy hack: # We want this function to generate the exact same result on repeated runs of the program to make # debugging easier. This means ensuring that calls to random.sample() return the exact same result # every time. Normally we could just set a seed for the RNG, and this work work if we were sampling # from a list. However, mesh.verts is a set, and Python does not guarantee consistency of iteration # order between runs of the program (since the default hash uses the memory address, which certainly # changes). Rather than doing something drastic like implementing a custom hash function on the # mesh class, we'll just build a separate data structure where vertices are sorted by position, # which allows reproducible sampling (as long as positions are distinct). sortedVertList = list(mesh.verts) sortedVertList.sort(key= lambda x : (x.position[0], x.position[1], x.position[2])) random.seed(777) # Generate curl-free (ish) component curlFreePotentialVerts = random.sample(sortedVertList, max((2,len(mesh.verts)/1000))) potential = 1.0 bVals = {} for vert in curlFreePotentialVerts: bVals[vert] = potential potential *= -1 smoothPotential = solvePoisson(mesh, bVals) mesh.applyVertexValue(smoothPotential, "curlFreePotential") gradFromPotential(mesh, "curlFreePotential", "curlFreeVecGen") # Generate divergence-free (ish) component divFreePotentialVerts = random.sample(sortedVertList, max((2,len(mesh.verts)/1000))) potential = 1.0 bVals = {} for vert in divFreePotentialVerts: bVals[vert] = potential potential *= -1 smoothPotential = solvePoisson(mesh, bVals) mesh.applyVertexValue(smoothPotential, "divFreePotential") gradFromPotential(mesh, "divFreePotential", "divFreeVecGen") for vert in mesh.verts: normEu = eu.Vector3(*vert.normal) vecEu = eu.Vector3(*vert.divFreeVecGen) vert.divFreeVecGen = vecEu.rotate_around(normEu, pi / 2.0) # Rotate the field by 90 degrees # Combine the components for face in mesh.faces: face.vector = Vector3D(0.0, 0.0, 0.0) for vert in face.adjacentVerts(): face.vector = 1.0 * vert.curlFreeVecGen + 1.0 * vert.divFreeVecGen face.vector = face.projectToTangentSpace(face.vector) # clear out leftover attributes to not confuse people for vert in mesh.verts: del vert.curlFreeVecGen del vert.curlFreePotential del vert.divFreeVecGen del vert.divFreePotential # Verify the orientations were defined. Need to do this early, since they are needed for setup def checkOrientationDefined(mesh): """Verify that edges have oriented halfedges and halfedges have orientation signs""" for edge in mesh.edges: if not hasattr(edge, 'orientedHalfEdge'): print("ERROR: Edges do not have orientedHalfEdge defined. Cannot proceed") exit() for he in mesh.halfEdges: if not hasattr(he, 'orientationSign'): print("ERROR: halfedges do not have orientationSign defined. Cannot proceed") exit() # Verify the correct properties are defined after the assignment is run def checkResultTypes(mesh): for edge in mesh.edges: # Check exact if not hasattr(edge, 'exactComponent'): print("ERROR: Edges do not have edge.exactComponent defined. Cannot proceed") exit() if not isinstance(edge.exactComponent, float): print("ERROR: edge.exactComponent is defined, but has the wrong type. Type is " + str(type(edge.exactComponent)) + " when if should be 'float'") exit() # Check cocoexact if not hasattr(edge, 'coexactComponent'): print("ERROR: Edges do not have edge.coexactComponent defined. Cannot proceed") exit() if not isinstance(edge.coexactComponent, float): print("ERROR: edge.coexactComponent is defined, but has the wrong type. Type is " + str(type(edge.coexactComponent)) + " when if should be 'float'") exit() # Check harmonic if not hasattr(edge, 'harmonicComponent'): print("ERROR: Edges do not have edge.harmonicComponent defined. Cannot proceed") exit() if not isinstance(edge.harmonicComponent, float): print("ERROR: edge.harmonicComponent is defined, but has the wrong type. Type is " + str(type(edge.harmonicComponent)) + " when if should be 'float'") exit() # Visualization related def covectorToFaceVectorWhitney(mesh, covectorName, vectorName): for face in mesh.faces: pi = face.anyHalfEdge.vertex.position pj = face.anyHalfEdge.next.vertex.position pk = face.anyHalfEdge.next.next.vertex.position eij = pj - pi ejk = pk - pj eki = pi - pk N = cross(eij, -eki) A = 0.5 * norm(N) N /= 2*A wi = getattr(face.anyHalfEdge.edge, covectorName) * face.anyHalfEdge.orientationSign wj = getattr(face.anyHalfEdge.next.edge, covectorName) * face.anyHalfEdge.next.orientationSign wk = getattr(face.anyHalfEdge.next.next.edge, covectorName) * face.anyHalfEdge.next.next.orientationSign # s = (1.0 / (6.0 * A)) * cross(N, wi*(eki-ejk) + wj*(eij-eki) + wk*(ejk-eij)) s = (1.0 / (6.0 * A)) * cross(N, wi*(ejk-eij) + wj*(eki-ejk) + wk*(eij-eki)) setattr(face, vectorName, s) def flat(mesh, vectorFieldName, oneFormName): """ Given a vector field defined on faces, compute the corresponding (integrated) 1-form on edges. """ for edge in mesh.edges: oe = edge.orientedHalfEdge if not oe.isReal: val2 = getattr(edge.orientedHalfEdge.twin.face, vectorFieldName) meanVal = val2 elif not oe.twin.isReal: val1 = getattr(edge.orientedHalfEdge.face, vectorFieldName) meanVal = val1 else: val1 = getattr(edge.orientedHalfEdge.face, vectorFieldName) val2 = getattr(edge.orientedHalfEdge.twin.face, vectorFieldName) meanVal = 0.5 * (val1 + val2) setattr(edge, oneFormName, dot(edge.orientedHalfEdge.vector, meanVal)) ### Actual main method: # get ready assignEdgeOrientations(mesh) checkOrientationDefined(mesh) # Generate a vector field on the surface if simpleTest: generateFieldSimple(mesh) else: generateInterestingField(mesh) flat(mesh, 'vector', 'omega') # Apply the decomposition from this assignment print("\n=== Decomposing field in to components") decomposeField(mesh) print("=== Done decomposing field ===\n\n") # Verify everything necessary is defined for the output checkResultTypes(mesh) # Convert the covectors to face vectors for visualization covectorToFaceVectorWhitney(mesh, "exactComponent", "omega_exact_component") covectorToFaceVectorWhitney(mesh, "coexactComponent", "omega_coexact_component") covectorToFaceVectorWhitney(mesh, "harmonicComponent", "omega_harmonic_component") covectorToFaceVectorWhitney(mesh, "omega", "omega_original") # Register a vector toggle to switch between the vectors we just defined vectorList = [ {'vectorAttr':'omega_original', 'key':'1', 'colormap':'Spectral', 'vectorDefinedAt':'face'}, {'vectorAttr':'omega_exact_component', 'key':'2', 'colormap':'Blues', 'vectorDefinedAt':'face'}, {'vectorAttr':'omega_coexact_component', 'key':'3', 'colormap':'Reds', 'vectorDefinedAt':'face'}, {'vectorAttr':'omega_harmonic_component', 'key':'4', 'colormap':'Greens', 'vectorDefinedAt':'face'} ] meshDisplay.registerVectorToggleCallbacks(vectorList) # Start the GUI meshDisplay.startMainLoop()
def main(): # Get the path for the mesh to load, either from the program argument if # one was given, or a dialog otherwise if(len(sys.argv) > 1): filename = sys.argv[1] else: print("ERROR: No file name specified. Proper syntax is 'python Assignment2.py path/to/your/mesh.obj'.") exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename)) # Create a viewer object winName = 'DDG Assignment2 -- ' + os.path.basename(filename) meshDisplay = MeshDisplay(windowTitle=winName, width=400, height=300) meshDisplay.setMesh(mesh) ###################### BEGIN YOUR CODE # implement the body of each of these functions @property @cacheGeometry def faceArea(self): """ Compute the area of a face. Though not directly requested, this will be useful when computing face-area weighted normals below. This method gets called on a face, so 'self' is a reference to the face at which we will compute the area. """ v = list(self.adjacentVerts()) a = 0.5 * norm(cross(v[1].position - v[0].position, v[2].position - v[0].position)) return a def faceArea2(self): """ use area vector to compute the polygon area """ sum_areavector = [0.0, 0.0, 0.0] verts = list(self.adjacentVerts()) LEN = len(verts) for (i, v) in enumerate(verts): sum_areavector += 0.5 * cross(verts[i].position, verts[(i+1)%LEN].position) return norm(sum_areavector) @property @cacheGeometry def faceNormal(self): """ Compute normal at a face of the mesh. Unlike at vertices, there is one very obvious way to do this, since a face uniquely defines a plane. This method gets called on a face, so 'self' is a reference to the face at which we will compute the normal. """ v = list(self.adjacentVerts()) n = normalize(cross(v[1].position - v[0].position, v[2].position - v[0].position)) return n @property @cacheGeometry def vertexNormal_EquallyWeighted(self): """ Compute a vertex normal using the 'equally weighted' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ normalSum = np.array([0.0,0.0,0.0]) for face in self.adjacentFaces(): normalSum += face.normal * 1.0 n = normalize(normalSum) return n @property @cacheGeometry def vertexNormal_AreaWeighted(self): """ Compute a vertex normal using the 'face area weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ normalSum = np.array([0.0,0.0,0.0]) for face in self.adjacentFaces(): normalSum += face.normal * face.area n = normalize(normalSum) return n @property @cacheGeometry def vertexNormal_AngleWeighted(self): """ Compute a vertex normal using the 'tip angle weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ normalSum = np.array([0.0,0.0,0.0]) for face in self.adjacentFaces(): v = list(face.adjacentVerts()) v0 = v1 = v2 = Vertex() if v[0].id == self.id: v0 = v[0]; v1 = v[1]; v2 = v[2]; if v[1].id == self.id: v0 = v[1]; v1 = v[2]; v2 = v[0]; if v[2].id == self.id: v0 = v[2]; v1 = v[0]; v2 = v[1]; a = v1.position - v0.position b = v2.position - v0.position theta = acos(np.dot((a/norm(a)),(b/norm(b)))) normalSum += face.normal * theta n = normalize(normalSum) return n #@property #@cacheGeometry def cotan(self): """ Compute the cotangent of the angle opposite a halfedge. This is not directly required, but will be useful when computing the mean curvature normals below. This method gets called on a halfedge, so 'self' is a reference to the halfedge at which we will compute the cotangent. """ if self.next.next.next is not self: raise ValueError("ERROR: halfedge.cotan() is only well-defined on a triangle") if self.isReal: # Relevant vectors v0 = self.next.next.vector v1 = -self.next.vector # Nifty vector equivalent of cot(theta) val = np.dot(v0, v1) / norm(cross(v0, v1)) return val else: return 0.0 # placeholder value @property @cacheGeometry def vertexNormal_MeanCurvature(self): """ Compute a vertex normal using the 'mean curvature' method. Be sure to understand the relationship between this method and the area gradient method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ # the vertex normals we get from the mean curvature vector are precisely # the same as the ones we get from the area gradient # areaGrad(p_i) = 0.5 * SUM((cot a_j + cot b_j)(p_i - p_j)) sum_normal = [0.0, 0.0, 0.0] halfedges = list(self.adjacentHalfEdges_CounterClockwise()) for he in halfedges: sum_normal += (cotan(he.twin) + cotan(he)) * (-he.vector) # (p_i - p_j) = -he.vector n = normalize(0.5 * sum_normal) return n @property @cacheGeometry def vertexNormal_SphereInscribed(self): """ Compute a vertex normal using the 'inscribed sphere' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ # Ns = 1/c * SUM(e(j) x e(j+1) / (|e(j)|^2 * |e(j+1)|^2)) sum_normal = [0.0, 0.0, 0.0] halfedges = list(self.adjacentHalfEdges_CounterClockwise()) LEN = len(halfedges) for j in range(0, LEN-1): # [0, LEN-1) normj = norm(halfedges[j].vector) normj1 = norm(halfedges[j+1].vector) sum_normal += cross(halfedges[j].vector, halfedges[j+1].vector) / (normj*normj * normj1*normj1) n = normalize(sum_normal) return n # But it seems that I should use -n to return the correct normal value @property @cacheGeometry def angleDefect(self): """ Compute the angle defect of a vertex, d(v) (see Assignment 1 Exercise 8). This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the angle defect. """ sum_theta = 0.0 # acos(np.dot((a/norm(a)),(b/norm(b)))) for face in self.adjacentFaces(): v = list(face.adjacentVerts()) v0 = v1 = v2 = Vertex() if v[0].id == self.id: v0 = v[0]; v1 = v[1]; v2 = v[2]; if v[1].id == self.id: v0 = v[1]; v1 = v[2]; v2 = v[0]; if v[2].id == self.id: v0 = v[2]; v1 = v[0]; v2 = v[1]; a = v1.position - v0.position b = v2.position - v0.position theta = acos(np.dot((a/norm(a)),(b/norm(b)))) sum_theta += theta return 2.0 * pi - sum_theta def totalGaussianCurvature(): """ Compute the total Gaussian curvature in the mesh, meaning the sum of Gaussian curvature at each vertex. Note that you can access the mesh with the 'mesh' variable. """ sum_ = 0.0 for v in mesh.verts: sum_ += v.angleDefect return sum_ def gaussianCurvatureFromGaussBonnet(): """ Compute the total Gaussian curvature that the mesh should have, given that the Gauss-Bonnet theorem holds (see Assignment 1 Exercise 9). Note that you can access the mesh with the 'mesh' variable. The mesh includes members like 'mesh.verts' and 'mesh.faces', which are sets of the vertices (resp. faces) in the mesh. """ X = len(mesh.verts) - len(mesh.edges) + len(mesh.faces) return 2.0 * pi * X ###################### END YOUR CODE # Set these newly-defined methods as the methods to use in the classes Face.normal = faceNormal Face.area = faceArea Vertex.normal = vertexNormal_AreaWeighted Vertex.vertexNormal_EquallyWeighted = vertexNormal_EquallyWeighted Vertex.vertexNormal_AreaWeighted = vertexNormal_AreaWeighted Vertex.vertexNormal_AngleWeighted = vertexNormal_AngleWeighted Vertex.vertexNormal_MeanCurvature = vertexNormal_MeanCurvature Vertex.vertexNormal_SphereInscribed = vertexNormal_SphereInscribed Vertex.angleDefect = angleDefect HalfEdge.cotan = cotan ## Functions which will be called by keypresses to visualize these definitions def toggleFaceVectors(): print("\nToggling vertex vector display") if toggleFaceVectors.val: toggleFaceVectors.val = False meshDisplay.setVectors(None) else: toggleFaceVectors.val = True meshDisplay.setVectors('normal', vectorDefinedAt='face') meshDisplay.generateVectorData() toggleFaceVectors.val = False # ridiculous Python scoping hack meshDisplay.registerKeyCallback('1', toggleFaceVectors, docstring="Toggle drawing face normal vectors") def toggleVertexVectors(): print("\nToggling vertex vector display") if toggleVertexVectors.val: toggleVertexVectors.val = False meshDisplay.setVectors(None) else: toggleVertexVectors.val = True meshDisplay.setVectors('normal', vectorDefinedAt='vertex') meshDisplay.generateVectorData() toggleVertexVectors.val = False # ridiculous Python scoping hack meshDisplay.registerKeyCallback('2', toggleVertexVectors, docstring="Toggle drawing vertex normal vectors") def toggleDefect(): print("\nToggling angle defect display") if toggleDefect.val: toggleDefect.val = False meshDisplay.setShapeColorToDefault() else: toggleDefect.val = True meshDisplay.setShapeColorFromScalar("angleDefect", cmapName="seismic",vMinMax=[-pi/8,pi/8]) meshDisplay.generateFaceData() toggleDefect.val = False meshDisplay.registerKeyCallback('3', toggleDefect, docstring="Toggle drawing angle defect coloring") def useEquallyWeightedNormals(): mesh.staticGeometry = False print("\nUsing equally-weighted normals") Vertex.normal = vertexNormal_EquallyWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback('4', useEquallyWeightedNormals, docstring="Use equally-weighted normal computation") def useAreaWeightedNormals(): mesh.staticGeometry = False print("\nUsing area-weighted normals") Vertex.normal = vertexNormal_AreaWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback('5', useAreaWeightedNormals, docstring="Use area-weighted normal computation") def useAngleWeightedNormals(): mesh.staticGeometry = False print("\nUsing angle-weighted normals") Vertex.normal = vertexNormal_AngleWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback('6', useAngleWeightedNormals, docstring="Use angle-weighted normal computation") def useMeanCurvatureNormals(): mesh.staticGeometry = False print("\nUsing mean curvature normals") Vertex.normal = vertexNormal_MeanCurvature mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback('7', useMeanCurvatureNormals, docstring="Use mean curvature normal computation") def useSphereInscribedNormals(): mesh.staticGeometry = False print("\nUsing sphere-inscribed normals") Vertex.normal = vertexNormal_SphereInscribed mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback('8', useSphereInscribedNormals, docstring="Use sphere-inscribed normal computation") def computeDiscreteGaussBonnet(): print("\nComputing total curvature:") computed = totalGaussianCurvature() predicted = gaussianCurvatureFromGaussBonnet() print(" Total computed curvature: " + str(computed)) print(" Predicted value from Gauss-Bonnet is: " + str(predicted)) print(" Error is: " + str(abs(computed - predicted))) meshDisplay.registerKeyCallback('z', computeDiscreteGaussBonnet, docstring="Compute total curvature") def deformShape(): print("\nDeforming shape") mesh.staticGeometry = False # Get the center and scale of the shape center = meshDisplay.dataCenter scale = meshDisplay.scaleFactor # Rotate according to swirly function ax = eu.Vector3(-1.0,.75,0.5) for v in mesh.verts: vec = v.position - center theta = 0.8 * norm(vec) / scale newVec = np.array(eu.Vector3(*vec).rotate_around(ax, theta)) v.position = center + newVec mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback('x', deformShape, docstring="Apply a swirly deformation to the shape") ## Register pick functions that output useful information on click def pickVert(vert): print(" Position:" + printVec3(vert.position)) print(" Angle defect: {:.5f}".format(vert.angleDefect)) print(" Normal (equally weighted): " + printVec3(vert.vertexNormal_EquallyWeighted)) print(" Normal (area weighted): " + printVec3(vert.vertexNormal_AreaWeighted)) print(" Normal (angle weighted): " + printVec3(vert.vertexNormal_AngleWeighted)) print(" Normal (sphere-inscribed): " + printVec3(vert.vertexNormal_SphereInscribed)) print(" Normal (mean curvature): " + printVec3(vert.vertexNormal_MeanCurvature)) meshDisplay.pickVertexCallback = pickVert def pickFace(face): print(" Face area : {:.5f}".format(face.area)) print(" Face area2: {:.5f}".format(faceArea2(face))) print(" Normal: " + printVec3(face.normal)) print(" Vertex positions: ") for (i, vert) in enumerate(face.adjacentVerts()): print(" v{}: {}".format((i+1),printVec3(vert.position))) meshDisplay.pickFaceCallback = pickFace # Start the viewer running meshDisplay.startMainLoop()
def main(inputfile, show=False, StaticGeometry=False, partString='part1'): # Get the path for the mesh to load from the program argument if (len(sys.argv) == 3): partString = sys.argv[1] if partString not in ['part1', 'part2', 'part3']: print( "ERROR part specifier not recognized. Should be one of 'part1', 'part2', or 'part3'" ) exit() filename = sys.argv[2] elif inputfile is not None: filename = inputfile else: print( "ERROR: Incorrect call syntax. Proper syntax is 'python Assignment3.py partN path/to/your/mesh.obj'." ) exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename), staticGeometry=StaticGeometry) # Create a viewer object winName = 'DDG Assignment3 ' + partString + '-- ' + os.path.basename( filename) meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) ###################### BEGIN YOUR CODE # implement the body of each of these functions ############################ # assignment 2 code: ############################ @property @cacheGeometry def faceArea(self): """ Compute the area of a face. Though not directly requested, this will be useful when computing face-area weighted normals below. This method gets called on a face, so 'self' is a reference to the face at which we will compute the area. """ v = list(self.adjacentVerts()) a = 0.5 * norm( cross(v[1].position - v[0].position, v[2].position - v[0].position)) return a @property @cacheGeometry def vertexNormal_EquallyWeighted(self): """ Compute a vertex normal using the 'equally weighted' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. http://brickisland.net/cs177/?p=217 Perhaps the simplest way to get vertex normals is to just add up the neighboring face normals: """ normalSum = np.array([0.0, 0.0, 0.0]) for face in self.adjacentFaces(): normalSum += face.normal n = normalize(normalSum) #issue: # two different tessellations of the same geometry # can produce very different vertex normals return n @property @cacheGeometry def vertexNormal_AreaWeighted(self): """ Compute a vertex normal using the 'face area weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. The area-weighted normal vector for this vertex""" normalSum = np.array([0.0, 0.0, 0.0]) for face in self.adjacentFaces(): normalSum += face.normal * face.area n = normalize(normalSum) #print 'computed vertexNormal_AreaWeighted n = ',n return n @property @cacheGeometry def vertexNormal_AngleWeighted(self): """ element type : vertex Compute a vertex normal using the 'Tip-Angle Weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. A simple way to reduce dependence on the tessellation is to weigh face normals by their corresponding tip angles theta, i.e., the interior angles incident on the vertex of interest: """ normalSum = np.array([0.0, 0.0, 0.0]) for face in self.adjacentFaces(): vl = list(face.adjacentVerts()) vl.remove(self) v1 = vl[0].position - self.position v2 = vl[1].position - self.position # norm ->no need for check: # it doesn not matter what the sign is? #area = norm(cross(v1, v2)) ##if area < 0.0000000001*max((norm(v1),norm(v2))): #if area < 0.: # area *= -1. alpha = np.arctan2(norm(cross(v1, v2)), dot(v1, v2)) #print v1 #print v2 #print alpha #print '' normalSum += face.normal * alpha n = normalize(normalSum) return n @property @cacheGeometry def faceNormal(self): """ Compute normal at a face of the mesh. Unlike at vertices, there is one very obvious way to do this, since a face uniquely defines a plane. This method gets called on a face, so 'self' is a reference to the face at which we will compute the normal. """ v = list(self.adjacentVerts()) n = normalize( cross(v[1].position - v[0].position, v[2].position - v[0].position)) return n @property @cacheGeometry def cotan(self): """ element type : halfedge Compute the cotangent of the angle OPPOSITE this halfedge. This is not directly required, but will be useful when computing the mean curvature normals below. This method gets called on a halfedge, so 'self' is a reference to the halfedge at which we will compute the cotangent. https://math.stackexchange.com/questions/2041099/ angle-between-vectors-given-cross-and-dot-product see half edge here: Users/lukemcculloch/Documents/Coding/Python/ DifferentialGeometry/course-master/libddg_userguide.pdf """ # Validate that this is on a triangle if self.next.next.next is not self: raise ValueError( "ERROR: halfedge.cotan() is only well-defined on a triangle") if self.isReal: # Relevant vectors A = -self.next.vector B = self.next.next.vector # Nifty vector equivalent of cot(theta) val = np.dot(A, B) / norm(cross(A, B)) return val else: return 0.0 @property @cacheGeometry def angleDefect(self): """ angleDefect <=> local Gaussian Curvature element type : vertex Compute the angle defect of a vertex, d(v) (see Assignment 1 Exercise 8). This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the angle defect. """ """ el = list(self.adjacentEdges()) evpl = list(self.adjacentEdgeVertexPairs()) fl = list(self.adjacentFaces()) vl = list(self.adjacentVerts()) https://scicomp.stackexchange.com/questions/27689/ numerically-stable-way-of-computing-angles-between-vectors #""" hl = list(self.adjacentHalfEdges()) lenhl = len(hl) hl.append(hl[0]) alpha = 0. for i in range(lenhl): v1 = hl[i].vector v2 = hl[i + 1].vector alpha += np.arctan2(norm(cross(v1, v2)), dot(v1, v2)) #dv = 2.*np.pi - alpha return 2. * np.pi - alpha def totalGaussianCurvature(): """ Compute the total Gaussian curvature in the mesh, meaning the sum of Gaussian curvature at each vertex. Note that you can access the mesh with the 'mesh' variable. """ tot = 0. for vel in mesh.verts: tot += vel.angleDefect return tot def gaussianCurvatureFromGaussBonnet(): """ Compute the total Gaussian curvature that the mesh should have, given that the Gauss-Bonnet theorem holds (see Assignment 1 Exercise 9). Note that you can access the mesh with the 'mesh' variable. The mesh includes members like 'mesh.verts' and 'mesh.faces', which are sets of the vertices (resp. faces) in the mesh. """ V = len(mesh.verts) E = len(mesh.edges) F = len(mesh.faces) EulerChar = V - E + F return 2. * np.pi * EulerChar ############################ # Part 0: Helper functions # ############################ # Implement a few useful functions that you will want in the remainder of # the assignment. @property @cacheGeometry def cotanWeight(self): """ Return the cotangent weight for an edge. Since this gets called on an edge, 'self' will be a reference to an edge. This will be useful in the problems below. Don't forget, everything you implemented for the last homework is now available as part of the library (normals, areas, etc). (Moving forward, Vertex.normal will mean area-weighted normals, unless otherwise specified) """ val = 0.0 if self.anyHalfEdge.isReal: val += self.anyHalfEdge.cotan if self.anyHalfEdge.twin.isReal: val += self.anyHalfEdge.twin.cotan val *= 0.5 return val @property @cacheGeometry def vertex_Laplace(self): """ element type : vertex Compute a vertex normal using the 'mean curvature' method. del del phi = 2NH -picked up negative sign due to cross products pointing into the page? -no they are normalized. -picked up a negative sign due to the cotan(s) being defined for pj, instead of pi. But how did it change anything? """ hl = list(self.adjacentHalfEdges()) pi = self.position sumj = 0. ot = 1. / 3. for hlfedge in hl: pj = hlfedge.vertex.position ct1 = hlfedge.cotan ct2 = hlfedge.twin.cotan sumj += (ct1 + ct2) * (pj - pi) #laplace = .5*sumj return normalize(-.5 * sumj) ## ##******************************************************* ## @property @cacheGeometry def dualArea(self): """ Return the dual area associated with a vertex. Since this gets called on a vertex, 'self' will be a reference to a vertex. Recall that the dual area can be defined as 1/3 the area of the surrounding faces. http://brickisland.net/DDGFall2017/ 'the barycentric dual area associated with a vertex i is equal to one-third the area of all triangles ijk touching i.' """ fl = list(self.adjacentFaces()) area_star = 0. for ff in fl: area_star += ff.area / 3. return area_star def enumerateVertices(mesh): """ Assign a unique index from 0 to (N-1) to each vertex in the mesh. Should return a dictionary containing mappings {vertex ==> index}. You will want to use this function in your solutions below. """ # index_map = {} # index = 0 # for vv in mesh.verts: # index_map[vv] = index # index += 1 return mesh.enumerateVertices @property @cacheGeometry def adjacency(self): index_map = enumerateVertices(self) nrows = ncols = len(mesh.verts) adjacency = np.zeros((nrows, ncols), int) for vv in mesh.verts: ith = index_map[vv] avlist = list(vv.adjacentVerts()) for av in avlist: jth = index_map[av] adjacency[ith, jth] = 1 return adjacency ################################# # Part 1: Dense Poisson Problem # ################################# # Solve a Poisson problem on the mesh. The primary function here # is solvePoissonProblem_dense(), it will get called when you run # python Assignment3.py part1 /path/to/your/mesh.obj # and specify density values with the mouse (the press space to solve). # # Note that this code will be VERY slow on large meshes, because it uses # dense matrices. def buildLaplaceMatrix_dense(mesh, index_map=None): """ Build a Laplace operator for the mesh, with a dense representation 'index' is a dictionary mapping {vertex ==> index} TLM renamed to index_map Returns the resulting matrix. """ if index_map is None: # index_map = mesh.enumerateVertices() index_map = enumerateVertices(mesh) nrows = ncols = len(mesh.verts) adjacency = np.zeros((nrows, ncols), int) for vv in mesh.verts: ith = index_map[vv] avlist = list(vv.adjacentVerts()) for av in avlist: jth = index_map[av] adjacency[ith, jth] = 1 Laplacian = np.zeros((nrows, ncols), float) for vi in mesh.verts: ith = index_map[vi] ll = list(vi.adjacentEdgeVertexPairs()) for edge, vj in ll: jth = index_map[vj] # Laplacian[ith,jth] = np.dot(vj.normal, # edge.cotanWeight*(vj.position - # vi.position) # ) if ith == jth: pass #Laplacian[ith,jth] = edge.cotanWeight else: Laplacian[ith, jth] = edge.cotanWeight Laplacian[ith, ith] = -sum(Laplacian[ith]) return Laplacian def buildMassMatrix_dense(mesh, index): """ Build a mass matrix for the mesh. Returns the resulting matrix. """ nrows = ncols = len(mesh.verts) #MassMatrix = np.zeros((nrows),float) MassMatrix = np.zeros((nrows, ncols), float) for i, vert in enumerate(mesh.verts): #MassMatrix[i,i] = 1./vert.dualArea MassMatrix[i, i] = vert.dualArea return MassMatrix def solvePoissonProblem_dense(mesh, densityValues): """ Solve a Poisson problem on the mesh. The results should be stored on the vertices in a variable named 'solutionVal'. You will want to make use of your buildLaplaceMatrix_dense() function from above. densityValues is a dictionary mapping {vertex ==> value} that specifies densities. The density is implicitly zero at every vertex not in this dictionary. When you run this program with 'python Assignment3.py part1 path/to/your/mesh.obj', you will get to click on vertices to specify density conditions. See the assignment document for more details. """ index_map = enumerateVertices(mesh) L = buildLaplaceMatrix_dense(mesh, index_map) M = buildMassMatrix_dense(mesh, index_map) #M <= 2D rho = np.zeros((len(mesh.verts)), float) for key in densityValues: #index_val = index_map[key] print 'key dual area = ', key.dualArea rho[index_map[key]] = densityValues[key] #*key.dualArea # # SwissArmyLaplacian, # page 179 Cu = Mf is better conditioned sol_vec = np.linalg.solve(L, np.dot(M, rho)) #sparse attempts: #sol_vec = linsolve.spsolve(L, rho) #sol_vec = dsolve.spsolve(L, rho, use_umfpack=False) #sol_vec = dsolve.spsolve(L, rho, use_umfpack=True) for vert in mesh.verts: key = index_map[vert] #print 'TLM sol_vec = ',sol_vec[key] vert.solutionVal = sol_vec[key] if rho[key]: vert.densityVal = rho[key] else: vert.densityVal = 0. return ################################## # Part 2: Sparse Poisson Problem # ################################## # Solve a Poisson problem on the mesh. The primary function here # is solvePoissonProblem_sparse(), it will get called when you run # python Assignment3.py part2 /path/to/your/mesh.obj # and specify density values with the mouse (the press space to solve). # # This will be very similar to the previous part. Be sure to see the wiki # for notes about the nuances of sparse matrix computation. Now, your code # should scale well to larger meshes! def buildLaplaceMatrix_sparse(mesh, index_map=None): """ Build a laplace operator for the mesh, with a sparse representation. This will be nearly identical to the dense method. 'index' is a dictionary mapping {vertex ==> index} Returns the resulting sparse matrix. """ if index_map is None: # index_map = mesh.enumerateVertices() index_map = enumerateVertices(mesh) nrows = ncols = len(mesh.verts) adjacency = np.zeros((nrows, ncols), int) for vv in mesh.verts: ith = index_map[vv] avlist = list(vv.adjacentVerts()) for av in avlist: jth = index_map[av] adjacency[ith, jth] = 1 Laplacian = np.zeros((nrows, ncols), float) for vi in mesh.verts: ith = index_map[vi] ll = list(vi.adjacentEdgeVertexPairs()) for edge, vj in ll: jth = index_map[vj] # Laplacian[ith,jth] = np.dot(vj.normal, # edge.cotanWeight*(vj.position - # vi.position) # ) if ith == jth: pass #Laplacian[ith,jth] = edge.cotanWeight else: Laplacian[ith, jth] = edge.cotanWeight Laplacian[ith, ith] = -sum(Laplacian[ith]) return csr_matrix(Laplacian) def buildMassMatrix_sparse(mesh, index): """ Build a sparse mass matrix for the system. Returns the resulting sparse matrix. """ nrows = ncols = len(mesh.verts) MassMatrix = np.zeros((nrows), float) #for i,vert in enumerate(mesh.verts): # MassMatrix[i] = vert.dualArea return MassMatrix def solvePoissonProblem_sparse(mesh, densityValues): """ Solve a Poisson problem on the mesh, using sparse matrix operations. This will be nearly identical to the dense method. The results should be stored on the vertices in a variable named 'solutionVal'. densityValues is a dictionary mapping {vertex ==> value} that specifies any densities. The density is implicitly zero at every vertex not in this dictionary. Note: Be sure to look at the notes on the github wiki about sparse matrix computation in Python. When you run this program with 'python Assignment3.py part2 path/to/your/mesh.obj', you will get to click on vertices to specify density conditions. See the assignment document for more details. """ index_map = enumerateVertices(mesh) L = buildLaplaceMatrix_sparse(mesh, index_map) M = buildMassMatrix_dense(mesh, index_map) #M <= 2D rho = np.zeros((len(mesh.verts)), float) for key in densityValues: #index_val = index_map[key] print 'key dual area = ', key.dualArea rho[index_map[key]] = densityValues[key] #*key.dualArea # convert to sparse matrix (CSR method) #Lsparse = csr_matrix(L) #iL = np.linalg.inv(L) #sol_vec = np.dot(iL,rho) #sol_vec = np.linalg.solve(L, rho) #sol_vec = linsolve.spsolve(L, rho) #sol_vec = linsolve.spsolve(L, np.dot(M,rho) ) #sol_vec = dsolve.spsolve(L, rho, use_umfpack=False) sol_vec = dsolve.spsolve(L, np.dot(M, rho), use_umfpack=True) for vert in mesh.verts: key = index_map[vert] #print 'TLM sol_vec = ',sol_vec[key] vert.solutionVal = sol_vec[key] if rho[key]: vert.densityVal = rho[key] else: vert.densityVal = 0. return ############################### # Part 3: Mean Curvature Flow # ############################### # Perform mean curvature flow on the mesh. The primary function here # is meanCurvatureFlow(), which will get called when you run # python Assignment3.py part3 /path/to/your/mesh.obj # You can adjust the step size with the 'z' and 'x' keys, and press space # to perform one step of flow. # # Of course, you will want to use sparse matrices here, so your code # scales well to larger meshes. def buildMeanCurvatureFlowOperator(mesh, index=None, h=None): """ Construct the (sparse) mean curvature operator matrix for the mesh. It might be helpful to use your buildLaplaceMatrix_sparse() and buildMassMatrix_sparse() methods from before. Returns the resulting matrix. """ nrows = ncols = len(mesh.verts) ##MassMatrix = np.zeros((nrows),float) #MassMatrix = np.zeros((nrows,ncols),float) #for i,vert in enumerate(mesh.verts): # MassMatrix[i] = 1./vert.dualArea # #MassMatrix[i,i] = 1./vert.dualArea Laplacian = np.zeros((nrows, ncols), float) for vi in mesh.verts: ith = index[vi] ll = list(vi.adjacentEdgeVertexPairs()) for edge, vj in ll: jth = index[vj] # Laplacian[ith,jth] = np.dot(vj.normal, # edge.cotanWeight*(vj.position - # vi.position) # ) if ith == jth: pass #Laplacian[ith,jth] = edge.cotanWeight else: Laplacian[ith, jth] = edge.cotanWeight Laplacian[ith, ith] = -sum(Laplacian[ith]) return csr_matrix(Laplacian) def meanCurvatureFlow_use_numpy_solve(mesh, h): """ Perform mean curvature flow on the mesh. The result of this operation is updated positions for the vertices; you should conclude by modifying the position variables for the mesh vertices. h is the step size for the backwards euler integration. When you run this program with 'python Assignment3.py part3 path/to/your/mesh.obj', you can press the space bar to perform this operation and z/x to change the step size. Recall that before you modify the positions of the mesh, you will need to set mesh.staticGeometry = False, which disables caching optimizations but allows you to modfiy the geometry. After you are done modfiying positions, you should set mesh.staticGeometry = True to re-enable these optimizations. You should probably have mesh.staticGeometry = True while you assemble your operator, or it will be very slow. """ # index_map = mesh.enumerateVertices() index_map = enumerateVertices(mesh) nrows = ncols = len(mesh.verts) Id = np.identity(nrows, float) M = buildMassMatrix_dense(mesh, index_map) #M <= 2D MCF = buildMeanCurvatureFlowOperator(mesh, index=index_map, h=h) # # SwissArmyLaplacian, # page 181 (I-hC)u = u is not symmetric # (M-hC)u = Mu is better conditioned #---------------------------------------------- Mi = np.linalg.inv(M) L = np.matmul(Mi, MCF) #UpdateOperator = np.linalg.inv(Id-h*L) #---------------------------------------------- #UpdateOperator = np.linalg.inv(M-h*MCF) LHS = M - h * MCF UpdateOperator = np.linalg.inv(LHS) #UpdateOperator = np.matmul(UpdateOperator,M) vertices = np.zeros((nrows, 3), float) for i, vert in enumerate(mesh.verts): vertices[i] = vert.position LHS = Id - h * L UpdateOperator = np.linalg.solve(LHS, vertices) vertices = UpdateOperator for i, vert in enumerate(mesh.verts): #key = index_map[vert] vert.position = vertices[i] # # vertices = np.dot(UpdateOperator,vertices) # for i,vert in enumerate(mesh.verts): # key = index_map[vert] # vert.position = vertices[i] return def meanCurvatureFlow(mesh, h): """ Perform mean curvature flow on the mesh. The result of this operation is updated positions for the vertices; you should conclude by modifying the position variables for the mesh vertices. h is the step size for the backwards euler integration. When you run this program with 'python Assignment3.py part3 path/to/your/mesh.obj', you can press the space bar to perform this operation and z/x to change the step size. Recall that before you modify the positions of the mesh, you will need to set mesh.staticGeometry = False, which disables caching optimizations but allows you to modfiy the geometry. After you are done modfiying positions, you should set mesh.staticGeometry = True to re-enable these optimizations. You should probably have mesh.staticGeometry = True while you assemble your operator, or it will be very slow. """ # index_map = mesh.enumerateVertices() index_map = enumerateVertices(mesh) nrows = ncols = len(mesh.verts) #Id = np.identity(nrows,float) M = buildMassMatrix_dense(mesh, index_map) #M <= 2D Msp = csr_matrix(M) #pure cotan operator: MCF = buildMeanCurvatureFlowOperator(mesh, index=index_map, h=h) # # SwissArmyLaplacian, # page 181 (I-hC)u = u is not symmetric # (M-hC)u = Mu is better conditioned #---------------------------------------------- #Mi = np.linalg.inv(M) #L = np.matmul(Mi,MCF) #UpdateOperator = np.linalg.inv(Id-h*L) #---------------------------------------------- #LHS = M-h*MCF LHS = Msp - MCF.multiply(h) #UpdateOperator = np.linalg.inv(LHS) #UpdateOperator = np.matmul(UpdateOperator,M) UpdateOperator = dsolve.spsolve(LHS, M, use_umfpack=True) vertices = np.zeros((nrows, 3), float) for i, vert in enumerate(mesh.verts): vertices[i] = vert.position #https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.linalg.cho_solve.html #UpdateOperator = scipy.linalg.cho_solve( # scipy.linalg.cho_factor(LHS), # np.dot(M,vertices)) #P, L, U = scipy.linalg.lu(LHS) # for non symmetric, numpy solve, style: # LHS = Id-h*L # UpdateOperator = np.linalg.solve(LHS, vertices) # vertices = UpdateOperator # for i,vert in enumerate(mesh.verts): # #key = index_map[vert] # vert.position = vertices[i] # vertices = np.dot(UpdateOperator, vertices) for i, vert in enumerate(mesh.verts): #key = index_map[vert] vert.position = vertices[i] return ###################### END YOUR CODE # from assignment 2: Face.normal = faceNormal Face.area = faceArea Vertex.normal = vertexNormal_AreaWeighted Vertex.vertexNormal_EquallyWeighted = vertexNormal_EquallyWeighted Vertex.vertexNormal_AreaWeighted = vertexNormal_AreaWeighted Vertex.vertexNormal_AngleWeighted = vertexNormal_AngleWeighted ## Vertex.vertex_Laplace = vertex_Laplace # #Vertex.vertexNormal_SphereInscribed = vertexNormal_SphereInscribed Vertex.angleDefect = angleDefect HalfEdge.cotan = cotan def toggleDefect(): print("\nToggling angle defect display") if toggleDefect.val: toggleDefect.val = False meshDisplay.setShapeColorToDefault() else: toggleDefect.val = True meshDisplay.setShapeColorFromScalar("angleDefect", cmapName="seismic") #,vMinMax=[-pi/8,pi/8]) meshDisplay.generateFaceData() toggleDefect.val = False meshDisplay.registerKeyCallback( '3', toggleDefect, docstring="Toggle drawing angle defect coloring") def computeDiscreteGaussBonnet(): print("\nComputing total curvature:") computed = totalGaussianCurvature() predicted = gaussianCurvatureFromGaussBonnet() print(" Total computed curvature: " + str(computed)) print(" Predicted value from Gauss-Bonnet is: " + str(predicted)) print(" Error is: " + str(abs(computed - predicted))) meshDisplay.registerKeyCallback('z', computeDiscreteGaussBonnet, docstring="Compute total curvature") ###################### Assignment 3 stuff Edge.cotanWeight = cotanWeight Vertex.dualArea = dualArea # A pick function for choosing density conditions densityValues = dict() def pickVertBoundary(vert): """ See MeshDisplay callbacks, pickVertexCallback for how this works! self.pickVertexCallback <== pickVertBoundary(vert) self.pickVertexCallback(pickObject = your_vertex) """ value = 1.0 if pickVertBoundary.isHigh else -1.0 print(" Selected vertex at position:" + printVec3(vert.position)) print(" as a density with value = " + str(value)) densityValues[vert] = value print 'densityValues = ', densityValues pickVertBoundary.isHigh = not pickVertBoundary.isHigh pickVertBoundary.isHigh = True # Run in part1 mode if partString == 'part1': print("\n\n === Executing assignment 2 part 1") print(""" Please click on vertices of the mesh to specify density conditions. Alternating clicks will specify high-value (= 1.0) and low-value (= -1.0) density conditions. You may select as many density vertices as you want, but >= 2 are necessary to yield an interesting solution. When you are done, press the space bar to execute your solver and view the results. """) meshDisplay.pickVertexCallback = pickVertBoundary meshDisplay.drawVertices = True def executePart1Callback(): print("\n=== Solving Poisson problem with your dense solver\n") # Print and check the density values print("Density values:") for key in densityValues: print(" " + str(key) + " = " + str(densityValues[key])) #if len(densityValues) < 2: # print("Aborting solve, not enough density vertices specified") # return # Call the solver print("\nSolving problem...") t0 = time.time() solvePoissonProblem_dense(mesh, densityValues) tSolve = time.time() - t0 print("...solution completed.") print("Solution took {:.5f} seconds.".format(tSolve)) print("Visualizing results...") # Error out intelligently if nothing is stored on vert.solutionVal for vert in mesh.verts: if not hasattr(vert, 'solutionVal'): print( "ERROR: At least one vertex does not have the attribute solutionVal defined." ) exit() if not isinstance(vert.solutionVal, float): print( "ERROR: The data stored at vertex.solutionVal is not of type float." ) print(" The data has type=" + str(type(vert.solutionVal))) print(" The data looks like vert.solutionVal=" + str(vert.solutionVal)) exit() # Visualize the result # meshDisplay.setShapeColorFromScalar("solutionVal", # definedOn='vertex', # cmapName="seismic", # vMinMax=[-10.0,10.0]) meshDisplay.setShapeColorFromScalar("solutionVal", definedOn='vertex', cmapName="seismic") meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( ' ', executePart1Callback, docstring="Solve the Poisson problem and view the results") def showdensity(): # Visualize the result # meshDisplay.setShapeColorFromScalar("densityVal", # definedOn='vertex', # cmapName="seismic", # vMinMax=[-1.0,1.0]) meshDisplay.setShapeColorFromScalar("densityVal", definedOn='vertex', cmapName="seismic") meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( 'b', showdensity, docstring="Show the density map for the Poisson Problem") # Start the GUI if show: meshDisplay.startMainLoop() # Run in part2 mode elif partString == 'part2': print("\n\n === Executing assignment 2 part 2") print(""" Please click on vertices of the mesh to specify density conditions. Alternating clicks will specify high-value (= 1.0) and low-value (= -1.0) density conditions. You may select as many density vertices as you want, but >= 2 are necessary to yield an interesting solution. When you are done, press the space bar to execute your solver and view the results. """) meshDisplay.pickVertexCallback = pickVertBoundary meshDisplay.drawVertices = True def executePart2Callback(): print("\n=== Solving Poisson problem with your sparse solver\n") # Print and check the density values print("Density values:") for key in densityValues: print(" " + str(key) + " = " + str(densityValues[key])) #if len(densityValues) < 2: # print("Aborting solve, not enough density vertices specified") # return # Call the solver print("\nSolving problem...") t0 = time.time() solvePoissonProblem_sparse(mesh, densityValues) tSolve = time.time() - t0 print("...solution completed.") print("Solution took {:.5f} seconds.".format(tSolve)) print("Visualizing results...") # Error out intelligently if nothing is stored on vert.solutionVal for vert in mesh.verts: if not hasattr(vert, 'solutionVal'): print( "ERROR: At least one vertex does not have the attribute solutionVal defined." ) exit() if not isinstance(vert.solutionVal, float): print( "ERROR: The data stored at vertex.solutionVal is not of type float." ) print(" The data has type=" + str(type(vert.solutionVal))) print(" The data looks like vert.solutionVal=" + str(vert.solutionVal)) exit() # Visualize the result # meshDisplay.setShapeColorFromScalar("solutionVal", definedOn='vertex', cmapName="seismic", vMinMax=[-1.0,1.0]) meshDisplay.setShapeColorFromScalar("solutionVal", definedOn='vertex', cmapName="seismic") meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( ' ', executePart2Callback, docstring="Solve the Poisson problem and view the results") # Start the GUI if show: meshDisplay.startMainLoop() # Run in part3 mode elif partString == 'part3': print("\n\n === Executing assignment 2 part 3") print(""" Press the space bar to perform one step of mean curvature flow smoothing, using your solver. Pressing the 'z' and 'x' keys will decrease and increase the step size (h), respectively. """) stepSize = [0.01] def increaseStepsize(): stepSize[0] += 0.001 print("Increasing step size. New size h=" + str(stepSize[0])) def decreaseStepsize(): stepSize[0] -= 0.001 print("Decreasing step size. New size h=" + str(stepSize[0])) meshDisplay.registerKeyCallback( 'z', decreaseStepsize, docstring="Increase the value of the step size (h) by 0.1") meshDisplay.registerKeyCallback( 'x', increaseStepsize, docstring="Decrease the value of the step size (h) by 0.1") def smoothingStep(): print("\n=== Performing mean curvature smoothing step\n") print(" Step size h=" + str(stepSize[0])) # Call the solver print(" Solving problem...") t0 = time.time() meanCurvatureFlow(mesh, stepSize[0]) tSolve = time.time() - t0 print(" ...solution completed.") print(" Solution took {:.5f} seconds.".format(tSolve)) print("Updating display...") meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( ' ', smoothingStep, docstring="Perform one step of your mean curvature flow on the mesh" ) # Start the GUI if show: meshDisplay.startMainLoop() return mesh, meshDisplay
def main(): # Get the path for the mesh to load, either from the program argument if # one was given, or a dialog otherwise if (len(sys.argv) > 1): filename = sys.argv[1] else: print( "ERROR: No file name specified. Proper syntax is 'python Assignment2.py path/to/your/mesh.obj'." ) exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename)) # Create a viewer object winName = 'DDG Assignment2 -- ' + os.path.basename(filename) meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) ###################### BEGIN YOUR CODE # implement the body of each of these functions @property @cacheGeometry def faceArea(self): """ Compute the area of a face. Though not directly requested, this will be useful when computing face-area weighted normals below. This method gets called on a face, so 'self' is a reference to the face at which we will compute the area. """ return 0.0 # placeholder value @property @cacheGeometry def faceNormal(self): """ Compute normal at a face of the mesh. Unlike at vertices, there is one very obvious way to do this, since a face uniquely defines a plane. This method gets called on a face, so 'self' is a reference to the face at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def vertexNormal_EquallyWeighted(self): """ Compute a vertex normal using the 'equally weighted' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def vertexNormal_AreaWeighted(self): """ Compute a vertex normal using the 'face area weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def vertexNormal_AngleWeighted(self): """ Compute a vertex normal using the 'tip angle weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def cotan(self): """ Compute the cotangent of the angle opposite a halfedge. This is not directly required, but will be useful when computing the mean curvature normals below. This method gets called on a halfedge, so 'self' is a reference to the halfedge at which we will compute the cotangent. """ return 0.0 # placeholder value @property @cacheGeometry def vertexNormal_MeanCurvature(self): """ Compute a vertex normal using the 'mean curvature' method. Be sure to understand the relationship between this method and the area gradient method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def vertexNormal_SphereInscribed(self): """ Compute a vertex normal using the 'inscribed sphere' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def angleDefect(self): """ Compute the angle defect of a vertex, d(v) (see Assignment 1 Exercise 8). This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the angle defect. """ return 0.0 # placeholder value def totalGaussianCurvature(): """ Compute the total Gaussian curvature in the mesh, meaning the sum of Gaussian curvature at each vertex. Note that you can access the mesh with the 'mesh' variable. """ return 0.0 # placeholder value def gaussianCurvatureFromGaussBonnet(): """ Compute the total Gaussian curvature that the mesh should have, given that the Gauss-Bonnet theorem holds (see Assignment 1 Exercise 9). Note that you can access the mesh with the 'mesh' variable. The mesh includes members like 'mesh.verts' and 'mesh.faces', which are sets of the vertices (resp. faces) in the mesh. """ return 0.0 # placeholder value ###################### END YOUR CODE # Set these newly-defined methods as the methods to use in the classes Face.normal = faceNormal Face.area = faceArea Vertex.normal = vertexNormal_AreaWeighted Vertex.vertexNormal_EquallyWeighted = vertexNormal_EquallyWeighted Vertex.vertexNormal_AreaWeighted = vertexNormal_AreaWeighted Vertex.vertexNormal_AngleWeighted = vertexNormal_AngleWeighted Vertex.vertexNormal_MeanCurvature = vertexNormal_MeanCurvature Vertex.vertexNormal_SphereInscribed = vertexNormal_SphereInscribed Vertex.angleDefect = angleDefect HalfEdge.cotan = cotan ## Functions which will be called by keypresses to visualize these definitions def toggleFaceVectors(): print("\nToggling vertex vector display") if toggleFaceVectors.val: toggleFaceVectors.val = False meshDisplay.setVectors(None) else: toggleFaceVectors.val = True meshDisplay.setVectors('normal', vectorDefinedAt='face') meshDisplay.generateVectorData() toggleFaceVectors.val = False # ridiculous Python scoping hack meshDisplay.registerKeyCallback( '1', toggleFaceVectors, docstring="Toggle drawing face normal vectors") def toggleVertexVectors(): print("\nToggling vertex vector display") if toggleVertexVectors.val: toggleVertexVectors.val = False meshDisplay.setVectors(None) else: toggleVertexVectors.val = True meshDisplay.setVectors('normal', vectorDefinedAt='vertex') meshDisplay.generateVectorData() toggleVertexVectors.val = False # ridiculous Python scoping hack meshDisplay.registerKeyCallback( '2', toggleVertexVectors, docstring="Toggle drawing vertex normal vectors") def toggleDefect(): print("\nToggling angle defect display") if toggleDefect.val: toggleDefect.val = False meshDisplay.setShapeColorToDefault() else: toggleDefect.val = True meshDisplay.setShapeColorFromScalar("angleDefect", cmapName="seismic", vMinMax=[-pi / 8, pi / 8]) meshDisplay.generateFaceData() toggleDefect.val = False meshDisplay.registerKeyCallback( '3', toggleDefect, docstring="Toggle drawing angle defect coloring") def useEquallyWeightedNormals(): mesh.staticGeometry = False print("\nUsing equally-weighted normals") Vertex.normal = vertexNormal_EquallyWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '4', useEquallyWeightedNormals, docstring="Use equally-weighted normal computation") def useAreaWeightedNormals(): mesh.staticGeometry = False print("\nUsing area-weighted normals") Vertex.normal = vertexNormal_AreaWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '5', useAreaWeightedNormals, docstring="Use area-weighted normal computation") def useAngleWeightedNormals(): mesh.staticGeometry = False print("\nUsing angle-weighted normals") Vertex.normal = vertexNormal_AngleWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '6', useAngleWeightedNormals, docstring="Use angle-weighted normal computation") def useMeanCurvatureNormals(): mesh.staticGeometry = False print("\nUsing mean curvature normals") Vertex.normal = vertexNormal_MeanCurvature mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '7', useMeanCurvatureNormals, docstring="Use mean curvature normal computation") def useSphereInscribedNormals(): mesh.staticGeometry = False print("\nUsing sphere-inscribed normals") Vertex.normal = vertexNormal_SphereInscribed mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( '8', useSphereInscribedNormals, docstring="Use sphere-inscribed normal computation") def computeDiscreteGaussBonnet(): print("\nComputing total curvature:") computed = totalGaussianCurvature() predicted = gaussianCurvatureFromGaussBonnet() print(" Total computed curvature: " + str(computed)) print(" Predicted value from Gauss-Bonnet is: " + str(predicted)) print(" Error is: " + str(abs(computed - predicted))) meshDisplay.registerKeyCallback('z', computeDiscreteGaussBonnet, docstring="Compute total curvature") def deformShape(): print("\nDeforming shape") mesh.staticGeometry = False # Get the center and scale of the shape center = meshDisplay.dataCenter scale = meshDisplay.scaleFactor # Rotate according to swirly function ax = eu.Vector3(-1.0, .75, 0.5) for v in mesh.verts: vec = v.position - center theta = 0.8 * norm(vec) / scale newVec = np.array(eu.Vector3(*vec).rotate_around(ax, theta)) v.position = center + newVec mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback( 'x', deformShape, docstring="Apply a swirly deformation to the shape") ## Register pick functions that output useful information on click def pickVert(vert): print(" Position:" + printVec3(vert.position)) print(" Angle defect: {:.5f}".format(vert.angleDefect)) print(" Normal (equally weighted): " + printVec3(vert.vertexNormal_EquallyWeighted)) print(" Normal (area weighted): " + printVec3(vert.vertexNormal_AreaWeighted)) print(" Normal (angle weighted): " + printVec3(vert.vertexNormal_AngleWeighted)) print(" Normal (sphere-inscribed): " + printVec3(vert.vertexNormal_SphereInscribed)) print(" Normal (mean curvature): " + printVec3(vert.vertexNormal_MeanCurvature)) meshDisplay.pickVertexCallback = pickVert def pickFace(face): print(" Face area: {:.5f}".format(face.area)) print(" Normal: " + printVec3(face.normal)) print(" Vertex positions: ") for (i, vert) in enumerate(face.adjacentVerts()): print(" v{}: {}".format((i + 1), printVec3(vert.position))) meshDisplay.pickFaceCallback = pickFace # Start the viewer running meshDisplay.startMainLoop()
def main(): # Get the path for the mesh to load, either from the program argument if # one was given, or a dialog otherwise if len(sys.argv) > 1: filename = sys.argv[1] else: print("ERROR: No file name specified. Proper syntax is 'python Assignment2.py path/to/your/mesh.obj'.") exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename)) # Create a viewer object winName = "DDG Assignment2 -- " + os.path.basename(filename) meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) ###################### BEGIN YOUR CODE # implement the body of each of these functions @property @cacheGeometry def faceArea(self): """ Compute the area of a face. Though not directly requested, this will be useful when computing face-area weighted normals below. This method gets called on a face, so 'self' is a reference to the face at which we will compute the area. """ return 0.0 # placeholder value @property @cacheGeometry def faceNormal(self): """ Compute normal at a face of the mesh. Unlike at vertices, there is one very obvious way to do this, since a face uniquely defines a plane. This method gets called on a face, so 'self' is a reference to the face at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def vertexNormal_EquallyWeighted(self): """ Compute a vertex normal using the 'equally weighted' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def vertexNormal_AreaWeighted(self): """ Compute a vertex normal using the 'face area weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def vertexNormal_AngleWeighted(self): """ Compute a vertex normal using the 'tip angle weights' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def cotan(self): """ Compute the cotangent of the angle opposite a halfedge. This is not directly required, but will be useful when computing the mean curvature normals below. This method gets called on a halfedge, so 'self' is a reference to the halfedge at which we will compute the cotangent. """ return 0.0 # placeholder value @property @cacheGeometry def vertexNormal_MeanCurvature(self): """ Compute a vertex normal using the 'mean curvature' method. Be sure to understand the relationship between this method and the area gradient method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def vertexNormal_SphereInscribed(self): """ Compute a vertex normal using the 'inscribed sphere' method. This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the normal. """ return Vector3D(0.0, 0.0, 0.0) # placeholder value @property @cacheGeometry def angleDefect(self): """ Compute the angle defect of a vertex, d(v) (see Assignment 1 Exercise 8). This method gets called on a vertex, so 'self' is a reference to the vertex at which we will compute the angle defect. """ return 0.0 # placeholder value def totalGaussianCurvature(): """ Compute the total Gaussian curvature in the mesh, meaning the sum of Gaussian curvature at each vertex. Note that you can access the mesh with the 'mesh' variable. """ return 0.0 # placeholder value def gaussianCurvatureFromGaussBonnet(): """ Compute the total Gaussian curvature that the mesh should have, given that the Gauss-Bonnet theorem holds (see Assignment 1 Exercise 9). Note that you can access the mesh with the 'mesh' variable. The mesh includes members like 'mesh.verts' and 'mesh.faces', which are sets of the vertices (resp. faces) in the mesh. """ return 0.0 # placeholder value ###################### END YOUR CODE # Set these newly-defined methods as the methods to use in the classes Face.normal = faceNormal Face.area = faceArea Vertex.normal = vertexNormal_AreaWeighted Vertex.vertexNormal_EquallyWeighted = vertexNormal_EquallyWeighted Vertex.vertexNormal_AreaWeighted = vertexNormal_AreaWeighted Vertex.vertexNormal_AngleWeighted = vertexNormal_AngleWeighted Vertex.vertexNormal_MeanCurvature = vertexNormal_MeanCurvature Vertex.vertexNormal_SphereInscribed = vertexNormal_SphereInscribed Vertex.angleDefect = angleDefect HalfEdge.cotan = cotan ## Functions which will be called by keypresses to visualize these definitions def toggleFaceVectors(): print("\nToggling vertex vector display") if toggleFaceVectors.val: toggleFaceVectors.val = False meshDisplay.setVectors(None) else: toggleFaceVectors.val = True meshDisplay.setVectors("normal", vectorDefinedAt="face") meshDisplay.generateVectorData() toggleFaceVectors.val = False # ridiculous Python scoping hack meshDisplay.registerKeyCallback("1", toggleFaceVectors, docstring="Toggle drawing face normal vectors") def toggleVertexVectors(): print("\nToggling vertex vector display") if toggleVertexVectors.val: toggleVertexVectors.val = False meshDisplay.setVectors(None) else: toggleVertexVectors.val = True meshDisplay.setVectors("normal", vectorDefinedAt="vertex") meshDisplay.generateVectorData() toggleVertexVectors.val = False # ridiculous Python scoping hack meshDisplay.registerKeyCallback("2", toggleVertexVectors, docstring="Toggle drawing vertex normal vectors") def toggleDefect(): print("\nToggling angle defect display") if toggleDefect.val: toggleDefect.val = False meshDisplay.setShapeColorToDefault() else: toggleDefect.val = True meshDisplay.setShapeColorFromScalar("angleDefect", cmapName="seismic", vMinMax=[-pi / 8, pi / 8]) meshDisplay.generateFaceData() toggleDefect.val = False meshDisplay.registerKeyCallback("3", toggleDefect, docstring="Toggle drawing angle defect coloring") def useEquallyWeightedNormals(): mesh.staticGeometry = False print("\nUsing equally-weighted normals") Vertex.normal = vertexNormal_EquallyWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback("4", useEquallyWeightedNormals, docstring="Use equally-weighted normal computation") def useAreaWeightedNormals(): mesh.staticGeometry = False print("\nUsing area-weighted normals") Vertex.normal = vertexNormal_AreaWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback("5", useAreaWeightedNormals, docstring="Use area-weighted normal computation") def useAngleWeightedNormals(): mesh.staticGeometry = False print("\nUsing angle-weighted normals") Vertex.normal = vertexNormal_AngleWeighted mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback("6", useAngleWeightedNormals, docstring="Use angle-weighted normal computation") def useMeanCurvatureNormals(): mesh.staticGeometry = False print("\nUsing mean curvature normals") Vertex.normal = vertexNormal_MeanCurvature mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback("7", useMeanCurvatureNormals, docstring="Use mean curvature normal computation") def useSphereInscribedNormals(): mesh.staticGeometry = False print("\nUsing sphere-inscribed normals") Vertex.normal = vertexNormal_SphereInscribed mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback("8", useSphereInscribedNormals, docstring="Use sphere-inscribed normal computation") def computeDiscreteGaussBonnet(): print("\nComputing total curvature:") computed = totalGaussianCurvature() predicted = gaussianCurvatureFromGaussBonnet() print(" Total computed curvature: " + str(computed)) print(" Predicted value from Gauss-Bonnet is: " + str(predicted)) print(" Error is: " + str(abs(computed - predicted))) meshDisplay.registerKeyCallback("z", computeDiscreteGaussBonnet, docstring="Compute total curvature") def deformShape(): print("\nDeforming shape") mesh.staticGeometry = False # Get the center and scale of the shape center = meshDisplay.dataCenter scale = meshDisplay.scaleFactor # Rotate according to swirly function ax = eu.Vector3(-1.0, 0.75, 0.5) for v in mesh.verts: vec = v.position - center theta = 0.8 * norm(vec) / scale newVec = np.array(eu.Vector3(*vec).rotate_around(ax, theta)) v.position = center + newVec mesh.staticGeometry = True meshDisplay.generateAllMeshValues() meshDisplay.registerKeyCallback("x", deformShape, docstring="Apply a swirly deformation to the shape") ## Register pick functions that output useful information on click def pickVert(vert): print(" Position:" + printVec3(vert.position)) print(" Angle defect: {:.5f}".format(vert.angleDefect)) print(" Normal (equally weighted): " + printVec3(vert.vertexNormal_EquallyWeighted)) print(" Normal (area weighted): " + printVec3(vert.vertexNormal_AreaWeighted)) print(" Normal (angle weighted): " + printVec3(vert.vertexNormal_AngleWeighted)) print(" Normal (sphere-inscribed): " + printVec3(vert.vertexNormal_SphereInscribed)) print(" Normal (mean curvature): " + printVec3(vert.vertexNormal_MeanCurvature)) meshDisplay.pickVertexCallback = pickVert def pickFace(face): print(" Face area: {:.5f}".format(face.area)) print(" Normal: " + printVec3(face.normal)) print(" Vertex positions: ") for (i, vert) in enumerate(face.adjacentVerts()): print(" v{}: {}".format((i + 1), printVec3(vert.position))) meshDisplay.pickFaceCallback = pickFace # Start the viewer running meshDisplay.startMainLoop()
def main(inputfile, show=False, StaticGeometry=False, partString='part1', is_simple=True): # Get the path for the mesh to load from the program argument if (len(sys.argv) == 3 and sys.argv[1] == 'simple'): filename = sys.argv[2] simpleTest = True elif (len(sys.argv) == 3 and sys.argv[1] == 'fancy'): filename = sys.argv[2] simpleTest = False elif inputfile is not None: filename = inputfile simpleTest = is_simple else: print( "ERROR: Incorrect call syntax. Proper syntax is 'python Assignment5.py MODE path/to/your/mesh.obj', where MODE is either 'simple' or 'fancy'" ) exit() # Read in the mesh mesh = HalfEdgeMesh(readMesh(filename)) # Create a viewer object winName = 'DDG Assignment5 -- ' + os.path.basename(filename) meshDisplay = MeshDisplay(windowTitle=winName) meshDisplay.setMesh(mesh) ###################### BEGIN YOUR CODE # DDGSpring216 Assignment 5 # # In this programming assignment you will implement Helmholtz-Hodge decomposition of covectors. # # The relevant mathematics and algorithm are described in section 8.1 of the course notes. # You will also need to implement the core operators in discrete exterior calculus, described mainly in # section 3.6 of the course notes. # # This code can be run with python Assignment5.py MODE /path/to/you/mesh.obj. MODE should be # either 'simple' or 'fancy', corresponding to the complexity of the input field omega that is given. # It might be easier to debug your algorithm on the simple field first. The assignment code will read in your input # mesh, generate a field 'omega' as input, run your algorithm, then display the results. # The results can be viewed as streamlines on the surface that flow with the covector field (toggle with 'p'), # or, as actual arrows on the faces (toggle with 'l'). The keys '1'-'4' will switch between the input, exact, # coexact, and harmonic fields (respectively). # # A few hints: # - Try performing some basic checks on your operators if things don't seem right. For instance, applying the # exterior derivative twice to anything should always yield zero. # - The streamline visualization is easy to look at, but can be deceiving at times. For instance, streamlines # are not very meaningful where the actual covectors are near 0. Try looking at the actual arrows in that case # ('l'). # - Many inputs will not have any harmonic components, especially genus 0 inputs. Don't stress if the harmonic # component of your output is exactly or nearly zero. # Implement the body of each of these functions... # def assignEdgeOrientations(mesh): # """ # Assign edge orientations to each edge on the mesh. # # This method will be called from the assignment code, you do not need to explicitly call it in any of your methods. # # After this method, the following values should be defined: # - edge.orientedHalfEdge (a reference to one of the halfedges touching that edge) # - halfedge.orientationSign (1.0 if that halfedge agrees with the orientation of its # edge, or -1.0 if not). You can use this to make much of your subsequent code cleaner. # # This is a pretty simple method to implement, any choice of orientation is acceptable. # """ # for edge in mesh.edges: # edge.orientedHalfEdge = edge.anyHalfEdge # edge.anyHalfEdge.orientationSign = -1.0 # edge.anyHalfEdge.twin.orientationSign = 1.0 # return def diagonalInverse(A): """ Returns the inverse of a sparse diagonal matrix. Makes a copy of the matrix. We will need to invert several diagonal matrices for the algorithm, but scipy does not offer a fast method for inverting diagonal matrices, which is a very easy special case. As such, this is a useful helper method for you. Note that the diagonal inverse is not well-defined if any of the diagonal elements are 0.0. This needs to be acconuted for when you construct the matrices. """ ncol, nrow = np.shape(A) assert ( ncol == nrow ), 'ERROR: Diagonal inverse only make sense for a symmetric matrix' #B = 1./np.diag(A) for i in range(ncol): A[i, i] = 1. / A[i, i] #B[i] return A # @property # @cacheGeometry # def circumcentricArea(self): # """ # Compute the area of the circumcentric dual cell for this vertex. # Returns a positive scalar. # # This gets called on a vertex, so 'self' will be a reference to the vertex. # # The image on page 78 of the course notes may help you visualize this. # (TLM: not sure what this references any more) # # # TLM note for those like me who miss the obvious: # You are not computing the circumcenter! # Go straight to the area! # # real source, slide 62: # http://brickisland.net/DDGFall2017/wp-content/uploads/2017/09/ # CMU_DDG_Fall2017_06_DiscreteExteriorCalculus.pdf # """ # # vl = list(self.adjacentVerts()) # # fl = list(self.adjacentFaces()) # DualArea = 0. # for face in self.adjacentFaces(): # #v1 = face.anyHalfEdge.vertex.position # #v2 = face.anyHalfEdge.next.vertex.position # #v3 = face.anyHalfEdge.next.next.vertex.position # l1 = norm(face.anyHalfEdge.vector) #||v1-v3|| # l2 = norm(face.anyHalfEdge.next.vector) #||v2-v1|| # l3 = norm(face.anyHalfEdge.next.next.vector) #||v3-v2|| # # s = .5*(l1+l2+l3) # DualArea += np.sqrt(s*(s-l1)*(s-l2)*(s-l3)) # # # return DualArea # Vertex.circumcentricArea = circumcentricArea # @property # @cacheGeometry # def circumcentricDualArea(self): # """ # Compute the area of the circumcentric dual cell for this vertex. # Returns a positive scalar. # # This gets called on a vertex, so 'self' will be a reference to the vertex. # # The image on page 78 of the course notes may help you visualize this. # (TLM: not sure what this references any more) # # # TLM note for those like me who miss the obvious: # You are not computing the circumcenter! # Go straight to the area! # # real source, slide 62: # http://brickisland.net/DDGFall2017/wp-content/uploads/2017/09/ # CMU_DDG_Fall2017_06_DiscreteExteriorCalculus.pdf # """ # DualArea = 0. # for hedge in self.adjacentHalfEdges(): # cak = hedge.cotan # lik = norm(hedge.vector) # caj = hedge.next.next.cotan # lij = norm(hedge.next.next.vector) # # DualArea += (lij**2 *cak) + (lik**2 * caj) # # # return DualArea/8. # Vertex.circumcentricDualArea = circumcentricDualArea # def buildHodgeStar0Form(mesh, vertexIndex): # """ # Build a sparse matrix encoding the Hodge operator on 0-forms for this mesh. # Returns a sparse, diagonal matrix corresponding to vertices. # # The discrete hodge star is a diagonal matrix where each entry is # the (area of the dual element) / (area of the primal element). You will probably # want to make use of the Vertex.circumcentricDualArea property you just defined. # # TLM as seen in notes: # By convention, the area of a vertex is 1.0. # """ # nrows = ncols = len(mesh.verts) # vertex_area = 1.0 # # Hodge0Form = np.zeros((nrows,ncols),float) # for i,vert in enumerate(mesh.verts): # vi = vertexIndex[vert] # Hodge0Form[vi,vi] = vert.circumcentricDualArea #/primal vertex_area # #Hodge0Form[vi,vi] = vert.barycentricDualArea #/primal vertex_area # return Hodge0Form # # # def buildHodgeStar1Form(mesh, edgeIndex): # """ # Build a sparse matrix encoding the Hodge operator on 1-forms for this mesh. # Returns a sparse, diagonal matrix corresponding to edges. # # The discrete hodge star is a diagonal matrix where each entry is # the (area of the dual element) / (area of the primal element). The solution # to exercise 26 from the previous homework will be useful here. # # TLM: cotan formula again. see ddg notes page 89 # see also source slide 56 (did you mean slide 62?): # http://brickisland.net/DDGFall2017/wp-content/uploads/2017/09/ # CMU_DDG_Fall2017_06_DiscreteExteriorCalculus.pdf # # Note that for some geometries, some entries of hodge1 operator may be exactly 0. # This can create a problem when we go to invert the matrix. To numerically sidestep # this issue, you probably want to add a small value (like 10^-8) to these diagonal # elements to ensure all are nonzero without significantly changing the result. # """ # nrows = ncols = len(mesh.edges) # Hodge1Form = np.zeros((nrows,ncols),float) # # for i,edge in enumerate(mesh.edges): # ei = edgeIndex[edge] # w = (( edge.anyHalfEdge.cotan + edge.anyHalfEdge.twin.cotan ) *.5) + 1.e-8 # #Hodge1Form[ei,ei] = edge.cotanWeight + 1.e-8 # Hodge1Form[ei,ei] = w # return Hodge1Form # # # def buildHodgeStar2Form(mesh, faceIndex): # """ # Build a sparse matrix encoding the Hodge operator on 2-forms for this mesh # Returns a sparse, diagonal matrix corresponding to faces. # # The discrete hodge star is a diagonal matrix where each entry is # the (area of the dual element) / (area of the primal element). # # # TLM hint hint!, vertex is => (dual) vertex: # By convention, the area of a vertex is 1.0. # # # TLM: see also source slide 57: # http://brickisland.net/DDGFall2017/wp-content/uploads/2017/09/ # CMU_DDG_Fall2017_06_DiscreteExteriorCalculus.pdf # """ # nrows = ncols = len(mesh.faces) # Hodge2Form = np.zeros((nrows,ncols),float) # # for i,face in enumerate(mesh.faces): # fi = faceIndex[face] # Hodge2Form[fi,fi] = 1./face.area # #Hodge2Form[fi,fi] = 1./face.AreaToDualVertexCicumcentric #circumcentric # return Hodge2Form # # # def buildExteriorDerivative0Form(mesh, edgeIndex, vertexIndex): # """ # Build a sparse matrix encoding the exterior derivative on 0-forms. # Returns a sparse matrix. # # See section 3.6 of the course notes for an explanation of DEC. # # 0form -> 1form # # In [2]: ed # Out[2]: <Edge #0> # # In [3]: ed.anyHalfEdge # Out[3]: <HalfEdge #11661> # # In [4]: ed.anyHalfEdge.vertex # Out[4]: <Vertex #0> # # In [5]: ed.anyHalfEdge.vertex.position # Out[5]: array([1.25, 0. , 0. ]) # # In [6]: ed.anyHalfEdge.twin.vertex.position # Out[6]: array([ 1.246147, 0. , -0.098074]) # # In [7]: ed.anyHalfEdge.vertex.position - ed.anyHalfEdge.twin.vertex.position # Out[7]: array([0.003853, 0. , 0.098074]) # # In [8]: ed.anyHalfEdge.vector # Out[8]: array([0.003853, 0. , 0.098074]) # # ## so ed.anyHalfEdge.vector runs # from anyHalfEdge.twin.vertex # to anyHalfEdge.vertex # # In [9]: ed.anyHalfEdge.twin.vector # Out[9]: array([-0.003853, 0. , -0.098074]) # """ # vert_edge_incidence = np.zeros((mesh.nedges,mesh.nverts),float) # # for vertex in mesh.verts: # # vj = vertexIndex[vertex] # # for edge in vertex.adjacentEdges(): # # ei = edgeIndex[edge] # # # # value = edge.orientedHalfEdge.orientationSign # # if vertex is edge.anyHalfEdge.vertex: # # # then we are at edge.anyHalfEdge.vertex, # # # i.e., the end of this half edge's vector. (not the start of the vector) # # value = -value # # # # vert_edge_incidence[ei,vj] = value # # # for edge in mesh.edges: # ei = edgeIndex[edge] # # vh1 = edge.orientedHalfEdge.vertex # vh2 = edge.orientedHalfEdge.twin.vertex # # ci = vertexIndex[vh1] # cj = vertexIndex[vh2] # # #value = edge.orientedHalfEdge.orientationSign # # vert_edge_incidence[ei,ci] = 1. #-value # vert_edge_incidence[ei,cj] = -1. #value # return csr_matrix( vert_edge_incidence ) # #return vert_edge_incidence # # # def buildExteriorDerivative1FormOLD(mesh, faceIndex, edgeIndex): # """ # Build a sparse matrix encoding the exterior derivative on 1-forms. # Returns a sparse matrix. # # See section 3.6 of the course notes for an explanation of DEC. # """ # edge_face_incidence = np.zeros((mesh.nfaces,mesh.nedges),float) # for face in mesh.faces: # fi = faceIndex[face] # v = list(face.adjacentVerts()) #0,1,2 # #tv = [] # for edge in face.adjacentEdges(): # ej = edgeIndex[edge] # value = edge.orientedHalfEdge.orientationSign # # #anyHalfEdge vector goes from # # anyHalfEdge.twin.vertex to anyHalfEdge.vertex # edge_start = edge.anyHalfEdge.twin.vertex # edge_end = edge.anyHalfEdge.vertex # if edge_start is v[0]: # if edge_end is v[1]: # value = value # else: # value = -value # elif edge_start is v[1]: # if edge_end is v[2]: # value = value # else: # value = -value # else: # assert(edge_start is v[2]) # if edge_end is v[0]: # value = value # else: # value = -value # ## tv.append([edge,value]) # edge_face_incidence[fi,ej] = value # return edge_face_incidence # def buildExteriorDerivative1Form(mesh, faceIndex, edgeIndex): # """ # Build a sparse matrix encoding the exterior derivative on 1-forms. # Returns a sparse matrix. # # See section 3.6 of the course notes for an explanation of DEC. # """ # edge_face_incidence = np.zeros((mesh.nfaces,mesh.nedges),float) # for face in mesh.faces: # fi = faceIndex[face] # #v = list(face.adjacentVerts()) #0,1,2 # #tv = [] # for he in face.adjacentHalfEdges(): # ej = edgeIndex[he.edge] # #value = he.orientationSign # #tv.append([edge,value]) # if he is he.edge.orientedHalfEdge: # edge_face_incidence[fi,ej] = 1. #value # else: # edge_face_incidence[fi,ej] = -1. #-value # # #return edge_face_incidence # return csr_matrix( edge_face_incidence ) # def decomposeField(mesh): # """ # Decompose a covector in to exact, coexact, and harmonic components # # The input mesh will have a scalar named 'omega' on its edges (edge.omega) # representing a discretized 1-form. This method should apply Helmoltz-Hodge # decomposition algorithm (as described on page 107-108 of the course notes) # to compute the exact, coexact, and harmonic components of omega. # # This method should return its results by storing three new scalars on each edge, # as the 3 decomposed components: edge.exactComponent, edge.coexactComponent, # and edge.harmonicComponent. # # Here are the primary steps you will need to perform for this method: # # - Create indexer objects for the vertices, faces, and edges. Note that the mesh # has handy helper functions pre-defined # for each of these: mesh.enumerateEdges() etc. # # - Build all of the operators we will need using # the methods you implemented above: # hodge0, hodge1, hodge2, d0, and d1. # You should also compute their inverses and # transposes, as appropriate. # # - Build a vector which represents the input covector (from the edge.omega values) # # - Perform a linear solve for the exact component, as described in the algorithm # # - Perform a linear solve for the coexact component, as described in the algorithm # # - Compute the harmonic component as the part which is neither exact nor coexact # # - Store your resulting exact, coexact, and harmonic components on the mesh edges # # This method will be called by the assignment code, you do not need to call it yourself. # """ # # """1)Create indexer objects for the vertices, faces, and edges. Note that the mesh # has handy helper functions pre-defined for each of these: mesh.enumerateEdges() etc. """ # # t0master = time.time() # edgeIndex = mesh.enumerateEdges # vertexIndex = mesh.enumerateVertices # faceIndex = mesh.enumerateFaces # # """2)Build all of the operators we will need using the methods you implemented above: # hodge0, hodge1, hodge2, d0, and d1. You should also compute their inverses and # transposes, as appropriate.""" # hodge0 = mesh.buildHodgeStar0Form(vertexIndex) # ihodge0 = diagonalInverse(hodge0) # hodge1 = mesh.buildHodgeStar1Form( edgeIndex) # #hodge2 = buildHodgeStar2Form(mesh, faceIndex) # ihodge1 = diagonalInverse(hodge1) # #ihodge2 = diagonalInverse(hodge2) # d0 = mesh.buildExteriorDerivative0Form( # edgeIndex=edgeIndex, # vertexIndex=vertexIndex) # d0T = d0.T # d1 = mesh.buildExteriorDerivative1Form( # faceIndex=faceIndex, # edgeIndex=edgeIndex) # d1T = d1.T # # # print 'shape d0 = ',np.shape(d0) # print 'shape d1 = ',np.shape(d1) # #print 'shape hodge0 = ',np.shape(hodge0) # print 'shape hodge1 = ',np.shape(hodge1) # #print 'shape hodge2 = ',np.shape(hodge2) # # # # omega = np.zeros((mesh.nedges),float) # for edge in mesh.edges: # i = edgeIndex[edge] # omega[i] = edge.omega # # #solve system 1 for d alpha # # page 117-118-119 # # scipy.linalg.cholesky # print 'system 1, alpha' # print 'build LHS...' # #LHS = np.matmul(d0T, # # np.matmul(hodge1,d0)) # t0 = time.time() # LHS = np.dot(ihodge0, # d0T.dot(hodge1)) # ss = np.shape(LHS)[0] # LHS = csr_matrix(LHS) # LHS = LHS.dot(d0) # LHS = LHS + (1.e-8 * csr_matrix(np.identity(ss,float))) # #llt = scipy.linalg.cholesky(LHS,lower=True) # tSolve = time.time() - t0 # print("...sparse alpha LHS completed.") # print("alpha LHS build took {:.5f} seconds.".format(tSolve)) # print 'build RHS...' # #RHS = np.matmul(d0T, # # np.matmul(hodge1,omega)) # RHS = np.dot(ihodge0, # d0T.dot(hodge1.dot(omega)) # ) # print 'type RHS = ',type(RHS) # print 'solve' # #alpha = np.linalg.solve(LHS,RHS) # alpha = dsolve.spsolve(LHS, RHS , # use_umfpack=True) # #alpha = scipy.sparse.linalg.cg(llt,RHS) # #alpha = dsolve.spsolve(csr_matrix(llt), RHS , # # use_umfpack=True) # # print 'solve complete, alpha complete' # # # # # #solve system 2 for delta Beta # # page 117-118-119 # # scipy.linalg.lu # print 'system 2, Beta' # print 'build LHS...' # #LHS = np.matmul(d1, # # np.matmul(ihodge1,d1T)) # t0 = time.time() # LHS = d1.dot(ihodge1) # # #ss = np.shape(LHS)[0] # # LHS = csr_matrix(LHS) # LHS = LHS.dot(d1T) # # #LHS = csr_matrix(LHS) # LHS = LHS #+ 1.e-8 * csr_matrix(np.identity(ss,float)) # # tSolve = time.time() - t0 # print("...sparse Beta LHS build completed.") # print("Beta LHS build took {:.5f} seconds.".format(tSolve)) # print 'build RHS...' # #RHS = np.matmul(d1,omega) # RHS = d1.dot(omega) # print 'solve' # #Beta = np.linalg.solve(LHS,RHS) # Beta = dsolve.spsolve(LHS, RHS , # use_umfpack=True) # # print 'solve complete, transform' # # Beta = np.dot(ihodge2,Beta) # # print 'transform complete, Beta complete' # # # # store exact, coexact, harmonic components on the mesh edges. # print 'decomposition field to mesh' # # # print 'Now push alpha and Beta into 1 forms' # # now push alpha to a 1 form using d: # alpha = d0.dot(alpha) # """ Say we start with a primal 2-form on a primal face. # Applying the star operator takes us to a dual 0-form on a dual vertex. # Taking the differential getsus to a dual 1-form on a dual edge. # And finally, another star operator # brings us to a primal 1-formon a primal edge.""" # #now pull back to a 1 form using the codifferential *d* # # *d* Beta => *d0* # #Beta = np.dot(hodge0, # # np.dot(d0, # # np.dot(hodge2,Beta))) # # the easy way: # Beta = d1T.dot(Beta) # print 'solve complete, transform' # Beta = np.dot(ihodge1,Beta) # print 'transform complete, Beta complete' # # #Beta = np.zeros_like(alpha) # for edge in mesh.edges: # i = edgeIndex[edge] # edge.exactComponent = alpha[i] # edge.coexactComponent = Beta [i] # edge.harmonicComponent = omega[i] - (alpha[i] + Beta[i]) # #edge.harmonicComponent = omega[i] - (Beta[i]) # print 'decomposition complete' # # tSolve = time.time() - t0master # print("...Decomposition completed.") # print("Total Time {:.5f} seconds.".format(tSolve)) def enumerateVertices(mesh): """ Assign a unique index from 0 to (N-1) to each vertex in the mesh. Should return a dictionary containing mappings {vertex ==> index}. You will want to use this function in your solutions below. """ # index_map = {} # index = 0 # for vv in mesh.verts: # index_map[vv] = index # index += 1 return mesh.enumerateVertices @property @cacheGeometry def adjacency(self): index_map = enumerateVertices(self) nrows = ncols = len(mesh.verts) adjacency = np.zeros((nrows, ncols), int) for vv in mesh.verts: ith = index_map[vv] avlist = list(vv.adjacentVerts()) for av in avlist: jth = index_map[av] adjacency[ith, jth] = 1 return adjacency ################################# # Part 1: Dense Poisson Problem # ################################# # Solve a Poisson problem on the mesh. The primary function here # is solvePoissonProblem_dense(), it will get called when you run # python Assignment3.py part1 /path/to/your/mesh.obj # and specify density values with the mouse (the press space to solve). # # Note that this code will be VERY slow on large meshes, because it uses # dense matrices. def buildLaplaceMatrix_sparse(mesh, index_map=None): """ Build a Laplace operator for the mesh, with a dense representation 'index' is a dictionary mapping {vertex ==> index} TLM renamed to index_map Returns the resulting matrix. """ if index_map is None: index_map = mesh.enumerateVertices() nrows = ncols = len(mesh.verts) # adjacency = np.zeros((nrows,ncols),int) # for vv in mesh.verts: # ith = index_map[vv] # avlist = list(vv.adjacentVerts()) # for av in avlist: # jth = index_map[av] # adjacency[ith,jth] = 1 Laplacian = np.zeros((nrows, ncols), float) for vi in mesh.verts: ith = index_map[vi] ll = list(vi.adjacentEdgeVertexPairs()) for edge, vj in ll: jth = index_map[vj] # Laplacian[ith,jth] = np.dot(vj.normal, # edge.cotanWeight*(vj.position - # vi.position) # ) w1 = edge.anyHalfEdge.cotan w2 = edge.anyHalfEdge.twin.cotan W = .5 * (w1 + w2) #W = edge.cotanWeight if ith == jth: pass else: Laplacian[ith, jth] = W Laplacian[ith, ith] = -(sum(Laplacian[ith])) #+ 1.e-8) return csr_matrix(Laplacian) def buildMassMatrix_dense(mesh, index): """ Build a mass matrix for the mesh. Returns the resulting matrix. """ nrows = ncols = len(mesh.verts) #MassMatrix = np.zeros((nrows),float) MassMatrix = np.zeros((nrows, ncols), float) for vert in mesh.verts: i = index[vert] #MassMatrix[i,i] = 1./vert.dualArea MassMatrix[i, i] = vert.barycentricDualArea #MassMatrix[i,i] = vert.circumcentricDualArea return MassMatrix def solvePoisson(mesh, densityValues): """ Solve a Poisson problem on the mesh. The results should be stored on the vertices in a variable named 'solutionVal'. You will want to make use of your buildLaplaceMatrix_dense() function from above. densityValues is a dictionary mapping {vertex ==> value} that specifies densities. The density is implicitly zero at every vertex not in this dictionary. When you run this program with 'python Assignment3.py part1 path/to/your/mesh.obj', you will get to click on vertices to specify density conditions. See the assignment document for more details. """ index_map = mesh.enumerateVertices L = buildLaplaceMatrix_sparse(mesh, index_map) M = buildMassMatrix_dense(mesh, index_map) #M <= 2D totalArea = mesh.totalArea rho = np.zeros((len(mesh.verts), 1), float) for key in densityValues: #index_val = index_map[key] print 'key dual area = ', key.barycentricDualArea rho[index_map[key]] = densityValues[key] #*key.dualArea nRows, nCols = np.shape(M) totalRho = sum(M.dot(rho)) #rhoBar = np.ones((nRows,1),float)*(totalRho/totalArea) rhoBar = totalRho / totalArea rhs = M.dot(rhoBar - rho) #rhs = np.matmul(M,(rho-rhoBar) ) #rhs = np.dot(M,rho) # # SwissArmyLaplacian, # page 179 Cu = Mf is better conditioned # assert(Cu == L) ?? #sol_vec = np.linalg.solve(L, np.dot(M,rho) ) #sparse: #sol_vec = dsolve.spsolve(L, np.dot(M,rho) , use_umfpack=True) # standard: #sol_vec = dsolve.spsolve(L, rhs , use_umfpack=True) #sparse Cholesky solve: llt = skchol.cholesky_AAt(L) #factor sol_vec = llt(rhs) #eigen: #sol_vec = np.zeros((nRows),float) #scipy.sparse.linalg.lobpcg(L,sol_vec,rhs) #@eigensolver #sol_vec = dsolve.spsolve(L, rho , use_umfpack=True) vert_sol = {} for vert in mesh.verts: key = index_map[vert] #print 'TLM sol_vec = ',sol_vec[key] vert.solutionVal = sol_vec[key] vert_sol[vert] = sol_vec[key] if rho[key]: vert.densityVal = rho[key] else: vert.densityVal = 0. return vert_sol ###################### END YOUR CODE ### More prep functions def generateFieldConstant(mesh): print("\n=== Using constant field as arbitrary direction field") for vert in mesh.verts: vert.vector = vert.projectToTangentSpace(Vector3D(1.4, 0.2, 2.4)) def generateFieldSimple(mesh): for face in mesh.faces: face.vector = face.center + Vector3D( -face.center[2], face.center[1], face.center[0]) face.vector = face.projectToTangentSpace(face.vector) def gradFromPotential(mesh, potAttr, gradAttr): # Simply compute gradient from potential for vert in mesh.verts: sumVal = Vector3D(0.0, 0.0, 0.0) sumWeight = 0.0 vertVal = getattr(vert, potAttr) for he in vert.adjacentHalfEdges(): sumVal += he.edge.cotanWeight * (getattr(he.vertex, potAttr) - vertVal) * he.vector sumWeight += he.edge.cotanWeight setattr(vert, gradAttr, normalize(sumVal)) def generateInterestingField(mesh, divscale=1., curlscale=1.): print( "\n=== Generating a hopefully-interesting field which has all three types of components\n" ) # Somewhat cheesy hack: # We want this function to generate the exact same result on repeated runs of the program to make # debugging easier. This means ensuring that calls to random.sample() return the exact same result # every time. Normally we could just set a seed for the RNG, and this work work if we were sampling # from a list. However, mesh.verts is a set, and Python does not guarantee consistency of iteration # order between runs of the program (since the default hash uses the memory address, which certainly # changes). Rather than doing something drastic like implementing a custom hash function on the # mesh class, we'll just build a separate data structure where vertices are sorted by position, # which allows reproducible sampling (as long as positions are distinct). sortedVertList = list(mesh.verts) sortedVertList.sort( key=lambda x: (x.position[0], x.position[1], x.position[2])) random.seed(100) # Generate curl-free (ish) component curlFreePotentialVerts = random.sample( sortedVertList, max((2, len(mesh.verts) / 1000))) potential = divscale bVals = {} for vert in curlFreePotentialVerts: bVals[vert] = potential potential *= -1. smoothPotential = solvePoisson(mesh, bVals) mesh.applyVertexValue(smoothPotential, "curlFreePotential") gradFromPotential(mesh, "curlFreePotential", "curlFreeVecGen") # Generate divergence-free (ish) component divFreePotentialVerts = random.sample(sortedVertList, max((2, len(mesh.verts) / 1000))) potential = curlscale bVals = {} for vert in divFreePotentialVerts: bVals[vert] = potential potential *= -1. smoothPotential = solvePoisson(mesh, bVals) mesh.applyVertexValue(smoothPotential, "divFreePotential") gradFromPotential(mesh, "divFreePotential", "divFreeVecGen") for vert in mesh.verts: normEu = eu.Vector3(*vert.normal) vecEu = eu.Vector3(*vert.divFreeVecGen) vert.divFreeVecGen = vecEu.rotate_around( normEu, pi / 2.0) # Rotate the field by 90 degrees # Combine the components for face in mesh.faces: face.vector = Vector3D(0.0, 0.0, 0.0) for vert in face.adjacentVerts(): face.vector = 1.0 * vert.curlFreeVecGen + 1.0 * vert.divFreeVecGen face.vector = face.projectToTangentSpace(face.vector) # clear out leftover attributes to not confuse people for vert in mesh.verts: del vert.curlFreeVecGen del vert.curlFreePotential del vert.divFreeVecGen del vert.divFreePotential # Verify the orientations were defined. Need to do this early, since they are needed for setup def checkOrientationDefined(mesh): """Verify that edges have oriented halfedges and halfedges have orientation signs""" for edge in mesh.edges: if not hasattr(edge, 'orientedHalfEdge'): print( "ERROR: Edges do not have orientedHalfEdge defined. Cannot proceed" ) exit() for he in mesh.halfEdges: if not hasattr(he, 'orientationSign'): print( "ERROR: halfedges do not have orientationSign defined. Cannot proceed" ) exit() # Verify the correct properties are defined after the assignment is run def checkResultTypes(mesh): for edge in mesh.edges: # Check exact if not hasattr(edge, 'exactComponent'): print( "ERROR: Edges do not have edge.exactComponent defined. Cannot proceed" ) exit() if not isinstance(edge.exactComponent, float): print( "ERROR: edge.exactComponent is defined, but has the wrong type. Type is " + str(type(edge.exactComponent)) + " when if should be 'float'") exit() # Check cocoexact if not hasattr(edge, 'coexactComponent'): print( "ERROR: Edges do not have edge.coexactComponent defined. Cannot proceed" ) exit() if not isinstance(edge.coexactComponent, float): print( "ERROR: edge.coexactComponent is defined, but has the wrong type. Type is " + str(type(edge.coexactComponent)) + " when if should be 'float'") exit() # Check harmonic if not hasattr(edge, 'harmonicComponent'): print( "ERROR: Edges do not have edge.harmonicComponent defined. Cannot proceed" ) exit() if not isinstance(edge.harmonicComponent, float): print( "ERROR: edge.harmonicComponent is defined, but has the wrong type. Type is " + str(type(edge.harmonicComponent)) + " when if should be 'float'") exit() # Visualization related def covectorToFaceVectorWhitney(mesh, covectorName, vectorName): """lookout wedge below! (tlm) this code is okay because it is able to show the initial vector field correctly. """ for face in mesh.faces: pi = face.anyHalfEdge.vertex.position pj = face.anyHalfEdge.next.vertex.position pk = face.anyHalfEdge.next.next.vertex.position eij = pj - pi ejk = pk - pj eki = pi - pk N = cross(eij, -eki) A = 0.5 * norm(N) N /= 2 * A wi = getattr(face.anyHalfEdge.edge, covectorName) * face.anyHalfEdge.orientationSign wj = getattr(face.anyHalfEdge.next.edge, covectorName) * face.anyHalfEdge.next.orientationSign wk = getattr( face.anyHalfEdge.next.next.edge, covectorName) * face.anyHalfEdge.next.next.orientationSign #s = (1.0 / (6.0 * A)) * cross(N, wi*(eki-ejk) + wj*(eij-eki) + wk*(ejk-eij)) s = (1.0 / (6.0 * A)) * cross( N, wi * (ejk - eij) + wj * (eki - ejk) + wk * (eij - eki)) setattr(face, vectorName, s) return # Visualization related def covectorToFaceVectorWhitneyJS(mesh, covectorName, vectorName): """lookout wedge below! (tlm) """ edgeIndex = mesh.enumerateEdges for face in mesh.faces: h = face.anyHalfEdge pi = h.vertex.position pj = h.next.vertex.position pk = h.next.next.vertex.position eij = pj - pi ejk = pk - pj eki = pi - pk #cij = #if h.edge.anyHalfEdge is not h: # cij *= -1. wij = getattr(face.anyHalfEdge.edge, covectorName) #* face.anyHalfEdge.orientationSign wjk = getattr( face.anyHalfEdge.next.edge, covectorName) #* face.anyHalfEdge.next.orientationSign wki = getattr( face.anyHalfEdge.next.next.edge, covectorName) #* face.anyHalfEdge.next.next.orientationSign if h.edge.anyHalfEdge is not h: wij *= -1 if h.next.edge.anyHalfEdge is not h: wjk *= -1 if h.next.next.edge.anyHalfEdge is not h: wki *= -1 #N = cross(eij, -eki) #A = 0.5 * norm(N) #N /= 2*A A = face.area N = face.normal # a = (eki - ejk) * wij b = (eij - eki) * wjk c = (ejk - eij) * wki #pystyle #a=wij*(ejk-eij) #b=wjk*(eki-ejk) #c=wki*(eij-eki) #s = (1.0 / (6.0 * A)) * cross(N, wij*(eki-ejk) + wjk*(eij-eki) + wki*(ejk-eij)) #s = (1.0 / (6.0 * A)) * cross(N, wij*(ejk-eij) + wjk*(eki-ejk) + wki*(eij-eki)) s = cross(N, (a + b + c)) * (1. / (6. * A)) setattr(face, vectorName, s) def flat(mesh, vectorFieldName, oneFormName): """ Given a vector field defined on faces, compute the corresponding (integrated) 1-form on edges. """ for edge in mesh.edges: oe = edge.orientedHalfEdge if not oe.isReal: val2 = getattr(edge.orientedHalfEdge.twin.face, vectorFieldName) meanVal = val2 elif not oe.twin.isReal: val1 = getattr(edge.orientedHalfEdge.face, vectorFieldName) meanVal = val1 else: val1 = getattr(edge.orientedHalfEdge.face, vectorFieldName) val2 = getattr(edge.orientedHalfEdge.twin.face, vectorFieldName) meanVal = 0.5 * (val1 + val2) setattr(edge, oneFormName, dot(edge.orientedHalfEdge.vector, meanVal)) ### Actual main method: # get ready mesh.assignEdgeOrientations() checkOrientationDefined(mesh) # Generate a vector field on the surface if simpleTest: generateFieldSimple(mesh) #generateFieldConstant(mesh) else: generateInterestingField(mesh, divscale=1., curlscale=1.) flat(mesh, 'vector', 'omega') # Apply the decomposition from this assignment print("\n=== Decomposing field in to components") #decomposeField(mesh) #hd = HodgeDecomposition(mesh) mesh.HodgeDecomposition() mesh.hodgeDecomposition.decomposeField() print("=== Done decomposing field ===\n\n") # Verify everything necessary is dfined for the output checkResultTypes(mesh) # # Convert the covectors to face vectors for visualization covectorToFaceVectorWhitney(mesh, "exactComponent", "omega_exact_component") covectorToFaceVectorWhitney(mesh, "coexactComponent", "omega_coexact_component") covectorToFaceVectorWhitney(mesh, "harmonicComponent", "omega_harmonic_component") covectorToFaceVectorWhitney(mesh, "omega", "omega_original") # # # Register a vector toggle to switch between the vectors we just defined vectorList = [{ 'vectorAttr': 'omega_original', 'key': '1', 'colormap': 'Spectral', 'vectorDefinedAt': 'face' }, { 'vectorAttr': 'omega_exact_component', 'key': '2', 'colormap': 'Blues', 'vectorDefinedAt': 'face' }, { 'vectorAttr': 'omega_coexact_component', 'key': '3', 'colormap': 'Reds', 'vectorDefinedAt': 'face' }, { 'vectorAttr': 'omega_harmonic_component', 'key': '4', 'colormap': 'Greens', 'vectorDefinedAt': 'face' }] meshDisplay.registerVectorToggleCallbacks(vectorList) print 'Computing Tree Cotree decomposition' mesh.TreeCotree() print 'Computing Tree Cotree generators' mesh.TreeCotree_compute_generators() print 'Plot Tree Cotree generators' mesh.setup_TreeCotree_plot() # meshDisplay.registerVectorToggleCallbacks( # [{'vectorAttr':'g1', # 'key':'5', # 'colormap':'Oranges', # 'vectorDefinedAt':'face'}, # {'vectorAttr':'g2', # 'key':'6', # 'colormap':'Oranges', # 'vectorDefinedAt':'face'}]) # Start the GUI if show: meshDisplay.startMainLoop() return mesh, meshDisplay