def solveLP(self):
# Check simple case when gradient of plane is contained within the convex polygon.
gradient = compute_plane_gradient(
Point3D(self.triangle[0][0], self.triangle[0][1], 'h0'),
Point3D(self.triangle[1][0], self.triangle[1][1], 'h1'),
Point3D(self.triangle[2][0], self.triangle[2][1], 'h2'))
# Get all adjacent pairs of points that make up all sides of the polygon.
self.adj_polygon_points = zip(
*[[self.polygon[-1]] +
self.polygon[:-1], self.polygon, self.polygon[1:] +
[self.polygon[0]]])
# Vectors of coefficients.
(h1_coefs, h2_coefs, h3_coefs, rhs_coefs) = ([], [], [], [])
for (prev_point, curr_point, next_point) in self.adj_polygon_points:
# Choose clockwise order when constructing line of eq. a x + b y + c = 0 from 2 given
# points. Then the semi-plane containing the convex polygon will be given by:
# a x + b y + c <= 0.
line = Line(Point2D(next_point), Point2D(curr_point))
self.polygon_lines.append(line)
line_equation_poly = poly(str(line.equation()),
gens=sp.symbols(['x', 'y']))
# Check if a x + b y + c <= 0 holds; if not, flip the line equation -- critical step!
if line_equation_poly.eval(prev_point) > 0:
line_equation_poly = -line_equation_poly
gradient_poly = poly(str(line_equation_poly.eval(gradient)),
gens=sp.symbols(['h0', 'h1', 'h2']))
h1_coefs.append(float(gradient_poly.diff('h0').eval((0, 0, 0))))
h2_coefs.append(float(gradient_poly.diff('h1').eval((0, 0, 0))))
h3_coefs.append(float(gradient_poly.diff('h2').eval((0, 0, 0))))
rhs_coefs.append(-float(gradient_poly.eval((0, 0, 0))))
# Construct LP problem.
# Objective function: Minimise/Maximise h1 + h2 + h3
objective_function_vector = matrix([1.0, 1.0, 1.0])
# Coefficient matrix.
ones_matrix = spmatrix([1.0, 1.0, 1.0], range(3), range(3))
heights_coef_matrix = matrix([h1_coefs, h2_coefs, h3_coefs])
bounds_coef_matrix = matrix([ones_matrix, -ones_matrix])
coef_matrix = sparse([heights_coef_matrix, bounds_coef_matrix])
# Column vector of right hand sides.
column_vector = matrix(rhs_coefs + [self.bounds[1]] * 3 +
[-self.bounds[0]] * 3)
min_sol = solvers.lp(objective_function_vector, coef_matrix,
column_vector)
is_consistent = min_sol['x'] is not None
if is_consistent:
self.min_heights = np.array(min_sol['x'])
print np.around(self.min_heights, decimals=2)
max_sol = solvers.lp(-objective_function_vector, coef_matrix,
column_vector)
self.max_heights = np.array(max_sol['x'])
print np.around(self.max_heights, decimals=2)
else:
# Try to determine triangulation of the given triangle.
print "Triangulating..."
# Perform triangulation w.r.t to a fixed vertex.
return self.triangulate(0) or self.triangulate(
1) or self.triangulate(2)
def solveLP(self):
# Check simple case when gradient of plane is contained within the convex polygon.
gradient = compute_plane_gradient(Point3D(self.triangle[0][0], self.triangle[0][1], 'h0'),
Point3D(self.triangle[1][0], self.triangle[1][1], 'h1'),
Point3D(self.triangle[2][0], self.triangle[2][1], 'h2'))
# Get all adjacent pairs of points that make up all sides of the polygon.
self.adj_polygon_points = zip(*[[self.polygon[-1]] + self.polygon[:-1],
self.polygon,
self.polygon[1:] + [self.polygon[0]]])
# Vectors of coefficients.
(h1_coefs, h2_coefs, h3_coefs, rhs_coefs) = ([], [], [], [])
for (prev_point, curr_point, next_point) in self.adj_polygon_points:
# Choose clockwise order when constructing line of eq. a x + b y + c = 0 from 2 given
# points. Then the semi-plane containing the convex polygon will be given by:
# a x + b y + c <= 0.
line = Line(Point2D(next_point), Point2D(curr_point))
self.polygon_lines.append(line)
line_equation_poly = poly(str(line.equation()), gens=sp.symbols(['x', 'y']))
# Check if a x + b y + c <= 0 holds; if not, flip the line equation -- critical step!
if line_equation_poly.eval(prev_point) > 0:
line_equation_poly = -line_equation_poly
gradient_poly = poly(str(line_equation_poly.eval(gradient)),
gens=sp.symbols(['h0', 'h1', 'h2']))
h1_coefs.append(float(gradient_poly.diff('h0').eval((0, 0, 0))))
h2_coefs.append(float(gradient_poly.diff('h1').eval((0, 0, 0))))
h3_coefs.append(float(gradient_poly.diff('h2').eval((0, 0, 0))))
rhs_coefs.append(-float(gradient_poly.eval((0, 0, 0))))
# Construct LP problem.
# Objective function: Minimise/Maximise h1 + h2 + h3
objective_function_vector = matrix([1.0, 1.0, 1.0])
# Coefficient matrix.
ones_matrix = spmatrix([1.0, 1.0, 1.0], range(3), range(3))
heights_coef_matrix = matrix([h1_coefs, h2_coefs, h3_coefs])
bounds_coef_matrix = matrix([ones_matrix, -ones_matrix])
coef_matrix = sparse([heights_coef_matrix, bounds_coef_matrix])
# Column vector of right hand sides.
column_vector = matrix(rhs_coefs + [self.bounds[1]] * 3 + [-self.bounds[0]] * 3)
min_sol = solvers.lp(objective_function_vector, coef_matrix, column_vector)
is_consistent = min_sol['x'] is not None
if is_consistent:
self.min_heights = np.array(min_sol['x'])
print np.around(self.min_heights, decimals=2)
max_sol = solvers.lp(-objective_function_vector, coef_matrix, column_vector)
self.max_heights = np.array(max_sol['x'])
print np.around(self.max_heights, decimals=2)
else:
# Try to determine triangulation of the given triangle.
print "Triangulating..."
# Perform triangulation w.r.t to a fixed vertex.
return self.triangulate(0) or self.triangulate(1) or self.triangulate(2)
def triangulate(self, vertex):
others = [0, 1, 2, 0, 1]
others = others[(vertex + 1):(vertex + 3)]
segment = Segment(self.triangle[others[0]], self.triangle[others[1]])
vertex_str = 'h' + str(vertex)
segment_heights_str = ['h' + str(other) for other in others]
new_points = []
counter = 2
for line in self.polygon_lines:
perp = line.perpendicular_line(
Point2D(self.triangle[vertex][0], self.triangle[vertex][1]))
intersections = perp.intersection(segment)
if len(intersections) > 0:
counter = counter + 1
new_points.append(
Point3D(intersections[0].x, intersections[0].y,
'h' + str(counter)))
no_vars = counter + 1
all_side_points = [Point3D(segment.points[0][0],
segment.points[0][1],
segment_heights_str[0])] + new_points + \
[Point3D(segment.points[1][0],
segment.points[1][1],
segment_heights_str[1])]
all_pairs = zip(all_side_points, all_side_points[1:])
h_coefs = matrix(0.0, (len(all_pairs) * len(self.polygon), no_vars))
rhs_coefs = []
for (idx1, pair) in enumerate(all_pairs):
# Compute gradient for each triangulation.
gradient = compute_plane_gradient(
Point3D(self.triangle[vertex][0], self.triangle[vertex][1],
vertex_str), pair[0], pair[1])
heights_str = [str(pair[0][2]), str(pair[1][2])]
hs = [int(heights_str[0][1:]), int(heights_str[1][1:])]
# Ensure that each gradient is with the convex polygon.
for (idx2, (prev_point, curr_point,
next_point)) in enumerate(self.adj_polygon_points):
line = Line(Point2D(next_point), Point2D(curr_point))
line_equation_poly = poly(str(line.equation()),
gens=sp.symbols(['x', 'y']))
# Flip equation if necessary.
if line_equation_poly.eval(prev_point) > 0:
line_equation_poly = -line_equation_poly
gradient_poly = poly(str(line_equation_poly.eval(gradient)),
gens=sp.symbols([
vertex_str, heights_str[0],
heights_str[1]
]))
row = idx1 * len(self.polygon) + idx2
h_coefs[row, 0] = float(
gradient_poly.diff(vertex_str).eval((0, 0, 0)))
h_coefs[row, hs[0]] = float(
gradient_poly.diff(heights_str[0]).eval((0, 0, 0)))
h_coefs[row, hs[1]] = float(
gradient_poly.diff(heights_str[1]).eval((0, 0, 0)))
rhs_coefs.append(-float(gradient_poly.eval((0, 0, 0))))
ones = [1.0] * no_vars
# Final LP Problem.
# Objective function.
objective_function_vector = matrix(ones)
# Coefficient matrix.
ones_matrix = spmatrix(ones, range(no_vars), range(no_vars))
heights_coef_matrix = h_coefs
bounds_coef_matrix = matrix([ones_matrix, -ones_matrix])
coef_matrix = sparse([heights_coef_matrix, bounds_coef_matrix])
# Column vector of right hand sides.
column_vector = matrix(rhs_coefs + \
[self.bounds[1]] * no_vars + [-self.bounds[0]] * no_vars)
min_sol = solvers.lp(objective_function_vector, coef_matrix,
column_vector)
is_consistent = min_sol['x'] is not None
if is_consistent:
self.min_heights = np.array(min_sol['x'])
print np.around(self.min_heights, decimals=2)
max_sol = solvers.lp(-objective_function_vector, coef_matrix,
column_vector)
self.max_heights = np.array(max_sol['x'])
print np.around(self.max_heights, decimals=2)
return is_consistent
def triangulate(self, vertex):
others = [0, 1, 2, 0, 1]
others = others[(vertex + 1):(vertex + 3)]
segment = Segment(self.triangle[others[0]], self.triangle[others[1]])
vertex_str = 'h' + str(vertex)
segment_heights_str = ['h' + str(other) for other in others]
new_points = []
counter = 2
for line in self.polygon_lines:
perp = line.perpendicular_line(Point2D(self.triangle[vertex][0],
self.triangle[vertex][1]))
intersections = perp.intersection(segment)
if len(intersections) > 0:
counter = counter + 1
new_points.append(Point3D(intersections[0].x,
intersections[0].y,
'h' + str(counter)))
no_vars = counter + 1
all_side_points = [Point3D(segment.points[0][0],
segment.points[0][1],
segment_heights_str[0])] + new_points + \
[Point3D(segment.points[1][0],
segment.points[1][1],
segment_heights_str[1])]
all_pairs = zip(all_side_points, all_side_points[1:])
h_coefs = matrix(0.0, (len(all_pairs) * len(self.polygon), no_vars))
rhs_coefs = []
for (idx1, pair) in enumerate(all_pairs):
# Compute gradient for each triangulation.
gradient = compute_plane_gradient(
Point3D(self.triangle[vertex][0], self.triangle[vertex][1], vertex_str),
pair[0],
pair[1])
heights_str = [str(pair[0][2]), str(pair[1][2])]
hs = [int(heights_str[0][1:]), int(heights_str[1][1:])]
# Ensure that each gradient is with the convex polygon.
for (idx2, (prev_point, curr_point, next_point)) in enumerate(self.adj_polygon_points):
line = Line(Point2D(next_point), Point2D(curr_point))
line_equation_poly = poly(str(line.equation()), gens=sp.symbols(['x', 'y']))
# Flip equation if necessary.
if line_equation_poly.eval(prev_point) > 0:
line_equation_poly = -line_equation_poly
gradient_poly = poly(str(line_equation_poly.eval(gradient)),
gens=sp.symbols([vertex_str, heights_str[0], heights_str[1]]))
row = idx1 * len(self.polygon) + idx2
h_coefs[row, 0] = float(gradient_poly.diff(vertex_str).eval((0, 0, 0)))
h_coefs[row, hs[0]] = float(gradient_poly.diff(heights_str[0]).eval((0, 0, 0)))
h_coefs[row, hs[1]] = float(gradient_poly.diff(heights_str[1]).eval((0, 0, 0)))
rhs_coefs.append(-float(gradient_poly.eval((0, 0, 0))))
ones = [1.0] * no_vars
# Final LP Problem.
# Objective function.
objective_function_vector = matrix(ones)
# Coefficient matrix.
ones_matrix = spmatrix(ones, range(no_vars), range(no_vars))
heights_coef_matrix = h_coefs
bounds_coef_matrix = matrix([ones_matrix, -ones_matrix])
coef_matrix = sparse([heights_coef_matrix, bounds_coef_matrix])
# Column vector of right hand sides.
column_vector = matrix(rhs_coefs + \
[self.bounds[1]] * no_vars + [-self.bounds[0]] * no_vars)
min_sol = solvers.lp(objective_function_vector, coef_matrix, column_vector)
is_consistent = min_sol['x'] is not None
if is_consistent:
self.min_heights = np.array(min_sol['x'])
print np.around(self.min_heights, decimals=2)
max_sol = solvers.lp(-objective_function_vector, coef_matrix, column_vector)
self.max_heights = np.array(max_sol['x'])
print np.around(self.max_heights, decimals=2)
return is_consistent
