def perpendicular_plane(self, pt): """ Plane perpendicular to the given plane and passing through the point pt. Parameters ========== pt: Point3D Returns ======= Plane Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(1,4,6), normal_vector=[2, 4, 6]) >>> a.perpendicular_plane(Point3D(2, 3, 5)) Plane(Point3D(2, 3, 5), [2, 8, -6]) """ a = Matrix(self.normal_vector) b, c = pt, self.p1 d = Matrix(b.direction_ratio(c)) e = list(d.cross(a)) return Plane(pt, normal_vector=e)
def is_perpendicular(self, l): """is the given geometric entity perpendicualar to the given plane? Parameters ========== LinearEntity3D or Plane Returns ======= Boolean Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6)) >>> b = Plane(Point3D(2, 2, 2), normal_vector=(-1, 2, -1)) >>> a.is_perpendicular(b) True """ from sympy.geometry.line3d import LinearEntity3D if isinstance(l, LinearEntity3D): a = Matrix(l.direction_ratio) b = Matrix(self.normal_vector) if a.cross(b).is_zero: return True else: return False elif isinstance(l, Plane): a = Matrix(l.normal_vector) b = Matrix(self.normal_vector) if a.dot(b) == 0: return True else: return False else: return False
def is_perpendicular(self, l): """is the given geometric entity perpendicualar to the given plane? Parameters ========== LinearEntity3D or Plane Returns ======= Boolean Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6)) >>> b = Plane(Point3D(2, 2, 2), normal_vector=(-1, 2, -1)) >>> a.is_perpendicular(b) True """ from sympy.geometry.line import LinearEntity3D if isinstance(l, LinearEntity3D): a = Matrix(l.direction_ratio) b = Matrix(self.normal_vector) if a.cross(b).is_zero_matrix: return True else: return False elif isinstance(l, Plane): a = Matrix(l.normal_vector) b = Matrix(self.normal_vector) if a.dot(b) == 0: return True else: return False else: return False
def is_parallel(self, l): """Is the given geometric entity parallel to the plane? Parameters ========== LinearEntity3D or Plane Returns ======= Boolean Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6)) >>> b = Plane(Point3D(3,1,3), normal_vector=(4, 8, 12)) >>> a.is_parallel(b) True """ from sympy.geometry.line3d import LinearEntity3D if isinstance(l, LinearEntity3D): a = l.direction_ratio b = self.normal_vector c = sum([i * j for i, j in zip(a, b)]) if c == 0: return True else: return False elif isinstance(l, Plane): a = Matrix(l.normal_vector) b = Matrix(self.normal_vector) if a.cross(b).is_zero: return True else: return False
def is_parallel(self, l): """Is the given geometric entity parallel to the plane? Parameters ========== LinearEntity3D or Plane Returns ======= Boolean Examples ======== >>> from sympy import Plane, Point3D >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6)) >>> b = Plane(Point3D(3,1,3), normal_vector=(4, 8, 12)) >>> a.is_parallel(b) True """ from sympy.geometry.line import LinearEntity3D if isinstance(l, LinearEntity3D): a = l.direction_ratio b = self.normal_vector c = sum([i * j for i, j in zip(a, b)]) if c == 0: return True else: return False elif isinstance(l, Plane): a = Matrix(l.normal_vector) b = Matrix(self.normal_vector) if a.cross(b).is_zero_matrix: return True else: return False
def are_coplanar(*lines): """ Returns True if the given lines are coplanar otherwise False Parameters ========== lines: list Returns ======= Boolean Examples ======== >>> from sympy import Plane, Point3D, Line3D >>> a = Line3D(Point3D(5, 0, 0), Point3D(1, -1, 1)) >>> b = Line3D(Point3D(0, -2, 0), Point3D(3, 1, 1)) >>> c = Line3D(Point3D(0, -1, 0), Point3D(5, -1, 9)) >>> Plane.are_coplanar(a, b, c) False """ if all(isinstance(i, Line3D) for i in lines): if len(lines) < 2: return False a = Matrix(lines[0].direction_ratio) b = Matrix(lines[1].direction_ratio) c = list(a.cross(b)) d = Plane(lines[0].p1, normal_vector=c) for i in lines[2:]: if i not in d: return False return True else: ValueError('Enter Lines only')
def intersection(self, o): """ The intersection with other geometrical entity. Parameters ========== Point, Point3D, LinearEntity, LinearEntity3D, Plane Returns ======= List Examples ======== >>> from sympy import Point, Point3D, Line, Line3D, Plane >>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1)) >>> b = Point3D(1, 2, 3) >>> a.intersection(b) [Point3D(1, 2, 3)] >>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2)) >>> a.intersection(c) [Point3D(2, 2, 2)] >>> d = Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3)) >>> e = Plane(Point3D(2, 0, 0), normal_vector=(3, 4, -3)) >>> d.intersection(e) [Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))] """ from sympy.geometry.line import LinearEntity, LinearEntity3D if not isinstance(o, GeometryEntity): o = Point(o, dim=3) if isinstance(o, Point): if o in self: return [o] else: return [] if isinstance(o, (LinearEntity, LinearEntity3D)): # recast to 3D p1, p2 = o.p1, o.p2 if isinstance(o, Segment): o = Segment3D(p1, p2) elif isinstance(o, Ray): o = Ray3D(p1, p2) elif isinstance(o, Line): o = Line3D(p1, p2) else: raise ValueError("unhandled linear entity: %s" % o.func) if o in self: return [o] else: t = Dummy() # unnamed else it may clash with a symbol in o a = Point3D(o.arbitrary_point(t)) p1, n = self.p1, Point3D(self.normal_vector) # TODO: Replace solve with solveset, when this line is tested c = solve((a - p1).dot(n), t) if not c: return [] else: c = [i for i in c if i.is_real is not False] if len(c) > 1: c = [i for i in c if i.is_real] if len(c) != 1: raise Undecidable("not sure which point is real") p = a.subs(t, c[0]) if p not in o: return [] # e.g. a segment might not intersect a plane return [p] if isinstance(o, Plane): if self.equals(o): return [self] if self.is_parallel(o): return [] else: x, y, z = map(Dummy, "xyz") a, b = Matrix([self.normal_vector]), Matrix([o.normal_vector]) c = list(a.cross(b)) d = self.equation(x, y, z) e = o.equation(x, y, z) result = list(linsolve([d, e], x, y, z))[0] for i in (x, y, z): result = result.subs(i, 0) return [Line3D(Point3D(result), direction_ratio=c)]
def test_sparse_matrix(): def sparse_eye(n): return SparseMatrix.eye(n) def sparse_zeros(n): return SparseMatrix.zeros(n) # creation args raises(TypeError, lambda: SparseMatrix(1, 2)) a = SparseMatrix(((1, 0), (0, 1))) assert SparseMatrix(a) == a from sympy.matrices import MutableSparseMatrix, MutableDenseMatrix a = MutableSparseMatrix([]) b = MutableDenseMatrix([1, 2]) assert a.row_join(b) == b assert a.col_join(b) == b assert type(a.row_join(b)) == type(a) assert type(a.col_join(b)) == type(a) # make sure 0 x n matrices get stacked correctly sparse_matrices = [SparseMatrix.zeros(0, n) for n in range(4)] assert SparseMatrix.hstack(*sparse_matrices) == Matrix(0, 6, []) sparse_matrices = [SparseMatrix.zeros(n, 0) for n in range(4)] assert SparseMatrix.vstack(*sparse_matrices) == Matrix(6, 0, []) # test element assignment a = SparseMatrix(((1, 0), (0, 1))) a[3] = 4 assert a[1, 1] == 4 a[3] = 1 a[0, 0] = 2 assert a == SparseMatrix(((2, 0), (0, 1))) a[1, 0] = 5 assert a == SparseMatrix(((2, 0), (5, 1))) a[1, 1] = 0 assert a == SparseMatrix(((2, 0), (5, 0))) assert a._smat == {(0, 0): 2, (1, 0): 5} # test_multiplication a = SparseMatrix(( (1, 2), (3, 1), (0, 6), )) b = SparseMatrix(( (1, 2), (3, 0), )) c = a * b assert c[0, 0] == 7 assert c[0, 1] == 2 assert c[1, 0] == 6 assert c[1, 1] == 6 assert c[2, 0] == 18 assert c[2, 1] == 0 try: eval('c = a @ b') except SyntaxError: pass else: assert c[0, 0] == 7 assert c[0, 1] == 2 assert c[1, 0] == 6 assert c[1, 1] == 6 assert c[2, 0] == 18 assert c[2, 1] == 0 x = Symbol("x") c = b * Symbol("x") assert isinstance(c, SparseMatrix) assert c[0, 0] == x assert c[0, 1] == 2 * x assert c[1, 0] == 3 * x assert c[1, 1] == 0 c = 5 * b assert isinstance(c, SparseMatrix) assert c[0, 0] == 5 assert c[0, 1] == 2 * 5 assert c[1, 0] == 3 * 5 assert c[1, 1] == 0 #test_power A = SparseMatrix([[2, 3], [4, 5]]) assert (A**5)[:] == [6140, 8097, 10796, 14237] A = SparseMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]]) assert (A**3)[:] == [290, 262, 251, 448, 440, 368, 702, 954, 433] # test_creation x = Symbol("x") a = SparseMatrix([[x, 0], [0, 0]]) m = a assert m.cols == m.rows assert m.cols == 2 assert m[:] == [x, 0, 0, 0] b = SparseMatrix(2, 2, [x, 0, 0, 0]) m = b assert m.cols == m.rows assert m.cols == 2 assert m[:] == [x, 0, 0, 0] assert a == b S = sparse_eye(3) S.row_del(1) assert S == SparseMatrix([[1, 0, 0], [0, 0, 1]]) S = sparse_eye(3) S.col_del(1) assert S == SparseMatrix([[1, 0], [0, 0], [0, 1]]) S = SparseMatrix.eye(3) S[2, 1] = 2 S.col_swap(1, 0) assert S == SparseMatrix([[0, 1, 0], [1, 0, 0], [2, 0, 1]]) a = SparseMatrix(1, 2, [1, 2]) b = a.copy() c = a.copy() assert a[0] == 1 a.row_del(0) assert a == SparseMatrix(0, 2, []) b.col_del(1) assert b == SparseMatrix(1, 1, [1]) assert SparseMatrix([[1, 2, 3], [1, 2], [1]]) == Matrix([[1, 2, 3], [1, 2, 0], [1, 0, 0]]) assert SparseMatrix(4, 4, {(1, 1): sparse_eye(2)}) == Matrix([[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0]]) raises(ValueError, lambda: SparseMatrix(1, 1, {(1, 1): 1})) assert SparseMatrix(1, 2, [1, 2]).tolist() == [[1, 2]] assert SparseMatrix(2, 2, [1, [2, 3]]).tolist() == [[1, 0], [2, 3]] raises(ValueError, lambda: SparseMatrix(2, 2, [1])) raises(ValueError, lambda: SparseMatrix(1, 1, [[1, 2]])) assert SparseMatrix([.1]).has(Float) # autosizing assert SparseMatrix(None, {(0, 1): 0}).shape == (0, 0) assert SparseMatrix(None, {(0, 1): 1}).shape == (1, 2) assert SparseMatrix(None, None, {(0, 1): 1}).shape == (1, 2) raises(ValueError, lambda: SparseMatrix(None, 1, [[1, 2]])) raises(ValueError, lambda: SparseMatrix(1, None, [[1, 2]])) raises(ValueError, lambda: SparseMatrix(3, 3, { (0, 0): ones(2), (1, 1): 2 })) # test_determinant x, y = Symbol('x'), Symbol('y') assert SparseMatrix(1, 1, [0]).det() == 0 assert SparseMatrix([[1]]).det() == 1 assert SparseMatrix(((-3, 2), (8, -5))).det() == -1 assert SparseMatrix(((x, 1), (y, 2 * y))).det() == 2 * x * y - y assert SparseMatrix(((1, 1, 1), (1, 2, 3), (1, 3, 6))).det() == 1 assert SparseMatrix(((3, -2, 0, 5), (-2, 1, -2, 2), (0, -2, 5, 0), (5, 0, 3, 4))).det() == -289 assert SparseMatrix(((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16))).det() == 0 assert SparseMatrix(((3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0), (0, 0, 0, 3, 2), (2, 0, 0, 0, 3))).det() == 275 assert SparseMatrix(((1, 0, 1, 2, 12), (2, 0, 1, 1, 4), (2, 1, 1, -1, 3), (3, 2, -1, 1, 8), (1, 1, 1, 0, 6))).det() == -55 assert SparseMatrix(((-5, 2, 3, 4, 5), (1, -4, 3, 4, 5), (1, 2, -3, 4, 5), (1, 2, 3, -2, 5), (1, 2, 3, 4, -1))).det() == 11664 assert SparseMatrix(((2, 7, -1, 3, 2), (0, 0, 1, 0, 1), (-2, 0, 7, 0, 2), (-3, -2, 4, 5, 3), (1, 0, 0, 0, 1))).det() == 123 # test_slicing m0 = sparse_eye(4) assert m0[:3, :3] == sparse_eye(3) assert m0[2:4, 0:2] == sparse_zeros(2) m1 = SparseMatrix(3, 3, lambda i, j: i + j) assert m1[0, :] == SparseMatrix(1, 3, (0, 1, 2)) assert m1[1:3, 1] == SparseMatrix(2, 1, (2, 3)) m2 = SparseMatrix([[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11], [12, 13, 14, 15]]) assert m2[:, -1] == SparseMatrix(4, 1, [3, 7, 11, 15]) assert m2[-2:, :] == SparseMatrix([[8, 9, 10, 11], [12, 13, 14, 15]]) assert SparseMatrix([[1, 2], [3, 4]])[[1], [1]] == Matrix([[4]]) # test_submatrix_assignment m = sparse_zeros(4) m[2:4, 2:4] = sparse_eye(2) assert m == SparseMatrix([(0, 0, 0, 0), (0, 0, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]) assert len(m._smat) == 2 m[:2, :2] = sparse_eye(2) assert m == sparse_eye(4) m[:, 0] = SparseMatrix(4, 1, (1, 2, 3, 4)) assert m == SparseMatrix([(1, 0, 0, 0), (2, 1, 0, 0), (3, 0, 1, 0), (4, 0, 0, 1)]) m[:, :] = sparse_zeros(4) assert m == sparse_zeros(4) m[:, :] = ((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)) assert m == SparseMatrix( ((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16))) m[:2, 0] = [0, 0] assert m == SparseMatrix( ((0, 2, 3, 4), (0, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16))) # test_reshape m0 = sparse_eye(3) assert m0.reshape(1, 9) == SparseMatrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1)) m1 = SparseMatrix(3, 4, lambda i, j: i + j) assert m1.reshape(4, 3) == \ SparseMatrix([(0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)]) assert m1.reshape(2, 6) == \ SparseMatrix([(0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)]) # test_applyfunc m0 = sparse_eye(3) assert m0.applyfunc(lambda x: 2 * x) == sparse_eye(3) * 2 assert m0.applyfunc(lambda x: 0) == sparse_zeros(3) # test__eval_Abs assert abs(SparseMatrix(((x, 1), (y, 2 * y)))) == SparseMatrix( ((Abs(x), 1), (Abs(y), 2 * Abs(y)))) # test_LUdecomp testmat = SparseMatrix([[0, 2, 5, 3], [3, 3, 7, 4], [8, 4, 0, 2], [-2, 6, 3, 4]]) L, U, p = testmat.LUdecomposition() assert L.is_lower assert U.is_upper assert (L * U).permute_rows(p, 'backward') - testmat == sparse_zeros(4) testmat = SparseMatrix([[6, -2, 7, 4], [0, 3, 6, 7], [1, -2, 7, 4], [-9, 2, 6, 3]]) L, U, p = testmat.LUdecomposition() assert L.is_lower assert U.is_upper assert (L * U).permute_rows(p, 'backward') - testmat == sparse_zeros(4) x, y, z = Symbol('x'), Symbol('y'), Symbol('z') M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z))) L, U, p = M.LUdecomposition() assert L.is_lower assert U.is_upper assert (L * U).permute_rows(p, 'backward') - M == sparse_zeros(3) # test_LUsolve A = SparseMatrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]]) x = SparseMatrix(3, 1, [3, 7, 5]) b = A * x soln = A.LUsolve(b) assert soln == x A = SparseMatrix([[0, -1, 2], [5, 10, 7], [8, 3, 4]]) x = SparseMatrix(3, 1, [-1, 2, 5]) b = A * x soln = A.LUsolve(b) assert soln == x # test_inverse A = sparse_eye(4) assert A.inv() == sparse_eye(4) assert A.inv(method="CH") == sparse_eye(4) assert A.inv(method="LDL") == sparse_eye(4) A = SparseMatrix([[2, 3, 5], [3, 6, 2], [7, 2, 6]]) Ainv = SparseMatrix(Matrix(A).inv()) assert A * Ainv == sparse_eye(3) assert A.inv(method="CH") == Ainv assert A.inv(method="LDL") == Ainv A = SparseMatrix([[2, 3, 5], [3, 6, 2], [5, 2, 6]]) Ainv = SparseMatrix(Matrix(A).inv()) assert A * Ainv == sparse_eye(3) assert A.inv(method="CH") == Ainv assert A.inv(method="LDL") == Ainv # test_cross v1 = Matrix(1, 3, [1, 2, 3]) v2 = Matrix(1, 3, [3, 4, 5]) assert v1.cross(v2) == Matrix(1, 3, [-2, 4, -2]) assert v1.norm(2)**2 == 14 # conjugate a = SparseMatrix(((1, 2 + I), (3, 4))) assert a.C == SparseMatrix([[1, 2 - I], [3, 4]]) # mul assert a * Matrix(2, 2, [1, 0, 0, 1]) == a assert a + Matrix(2, 2, [1, 1, 1, 1]) == SparseMatrix([[2, 3 + I], [4, 5]]) # col join assert a.col_join(sparse_eye(2)) == SparseMatrix([[1, 2 + I], [3, 4], [1, 0], [0, 1]]) # symmetric assert not a.is_symmetric(simplify=False) # test_cofactor assert sparse_eye(3) == sparse_eye(3).cofactor_matrix() test = SparseMatrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]]) assert test.cofactor_matrix() == \ SparseMatrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]]) test = SparseMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) assert test.cofactor_matrix() == \ SparseMatrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]]) # test_jacobian x = Symbol('x') y = Symbol('y') L = SparseMatrix(1, 2, [x**2 * y, 2 * y**2 + x * y]) syms = [x, y] assert L.jacobian(syms) == Matrix([[2 * x * y, x**2], [y, 4 * y + x]]) L = SparseMatrix(1, 2, [x, x**2 * y**3]) assert L.jacobian(syms) == SparseMatrix([[1, 0], [2 * x * y**3, x**2 * 3 * y**2]]) # test_QR A = Matrix([[1, 2], [2, 3]]) Q, S = A.QRdecomposition() R = Rational assert Q == Matrix([[5**R(-1, 2), (R(2) / 5) * (R(1) / 5)**R(-1, 2)], [2 * 5**R(-1, 2), (-R(1) / 5) * (R(1) / 5)**R(-1, 2)]]) assert S == Matrix([[5**R(1, 2), 8 * 5**R(-1, 2)], [0, (R(1) / 5)**R(1, 2)]]) assert Q * S == A assert Q.T * Q == sparse_eye(2) R = Rational # test nullspace # first test reduced row-ech form M = SparseMatrix([[5, 7, 2, 1], [1, 6, 2, -1]]) out, tmp = M.rref() assert out == Matrix([[1, 0, -R(2) / 23, R(13) / 23], [0, 1, R(8) / 23, R(-6) / 23]]) M = SparseMatrix([[1, 3, 0, 2, 6, 3, 1], [-2, -6, 0, -2, -8, 3, 1], [3, 9, 0, 0, 6, 6, 2], [-1, -3, 0, 1, 0, 9, 3]]) out, tmp = M.rref() assert out == Matrix([[1, 3, 0, 0, 2, 0, 0], [0, 0, 0, 1, 2, 0, 0], [0, 0, 0, 0, 0, 1, R(1) / 3], [0, 0, 0, 0, 0, 0, 0]]) # now check the vectors basis = M.nullspace() assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0]) assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0]) assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0]) assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1) / 3, 1]) # test eigen x = Symbol('x') y = Symbol('y') sparse_eye3 = sparse_eye(3) assert sparse_eye3.charpoly(x) == PurePoly(((x - 1)**3)) assert sparse_eye3.charpoly(y) == PurePoly(((y - 1)**3)) # test values M = Matrix([(0, 1, -1), (1, 1, 0), (-1, 0, 1)]) vals = M.eigenvals() assert sorted(vals.keys()) == [-1, 1, 2] R = Rational M = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) assert M.eigenvects() == [ (1, 3, [Matrix([1, 0, 0]), Matrix([0, 1, 0]), Matrix([0, 0, 1])]) ] M = Matrix([[5, 0, 2], [3, 2, 0], [0, 0, 1]]) assert M.eigenvects() == [(1, 1, [Matrix([R(-1) / 2, R(3) / 2, 1])]), (2, 1, [Matrix([0, 1, 0])]), (5, 1, [Matrix([1, 1, 0])])] assert M.zeros(3, 5) == SparseMatrix(3, 5, {}) A = SparseMatrix( 10, 10, { (0, 0): 18, (0, 9): 12, (1, 4): 18, (2, 7): 16, (3, 9): 12, (4, 2): 19, (5, 7): 16, (6, 2): 12, (9, 7): 18 }) assert A.row_list() == [(0, 0, 18), (0, 9, 12), (1, 4, 18), (2, 7, 16), (3, 9, 12), (4, 2, 19), (5, 7, 16), (6, 2, 12), (9, 7, 18)] assert A.col_list() == [(0, 0, 18), (4, 2, 19), (6, 2, 12), (1, 4, 18), (2, 7, 16), (5, 7, 16), (9, 7, 18), (0, 9, 12), (3, 9, 12)] assert SparseMatrix.eye(2).nnz() == 2
def test_sparse_matrix(): def sparse_eye(n): return SparseMatrix.eye(n) def sparse_zeros(n): return SparseMatrix.zeros(n) # creation args raises(TypeError, lambda: SparseMatrix(1, 2)) a = SparseMatrix(( (1, 0), (0, 1) )) assert SparseMatrix(a) == a from sympy.matrices import MutableSparseMatrix, MutableDenseMatrix a = MutableSparseMatrix([]) b = MutableDenseMatrix([1, 2]) assert a.row_join(b) == b assert a.col_join(b) == b assert type(a.row_join(b)) == type(a) assert type(a.col_join(b)) == type(a) # make sure 0 x n matrices get stacked correctly sparse_matrices = [SparseMatrix.zeros(0, n) for n in range(4)] assert SparseMatrix.hstack(*sparse_matrices) == Matrix(0, 6, []) sparse_matrices = [SparseMatrix.zeros(n, 0) for n in range(4)] assert SparseMatrix.vstack(*sparse_matrices) == Matrix(6, 0, []) # test element assignment a = SparseMatrix(( (1, 0), (0, 1) )) a[3] = 4 assert a[1, 1] == 4 a[3] = 1 a[0, 0] = 2 assert a == SparseMatrix(( (2, 0), (0, 1) )) a[1, 0] = 5 assert a == SparseMatrix(( (2, 0), (5, 1) )) a[1, 1] = 0 assert a == SparseMatrix(( (2, 0), (5, 0) )) assert a._smat == {(0, 0): 2, (1, 0): 5} # test_multiplication a = SparseMatrix(( (1, 2), (3, 1), (0, 6), )) b = SparseMatrix(( (1, 2), (3, 0), )) c = a*b assert c[0, 0] == 7 assert c[0, 1] == 2 assert c[1, 0] == 6 assert c[1, 1] == 6 assert c[2, 0] == 18 assert c[2, 1] == 0 try: eval('c = a @ b') except SyntaxError: pass else: assert c[0, 0] == 7 assert c[0, 1] == 2 assert c[1, 0] == 6 assert c[1, 1] == 6 assert c[2, 0] == 18 assert c[2, 1] == 0 x = Symbol("x") c = b * Symbol("x") assert isinstance(c, SparseMatrix) assert c[0, 0] == x assert c[0, 1] == 2*x assert c[1, 0] == 3*x assert c[1, 1] == 0 c = 5 * b assert isinstance(c, SparseMatrix) assert c[0, 0] == 5 assert c[0, 1] == 2*5 assert c[1, 0] == 3*5 assert c[1, 1] == 0 #test_power A = SparseMatrix([[2, 3], [4, 5]]) assert (A**5)[:] == [6140, 8097, 10796, 14237] A = SparseMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]]) assert (A**3)[:] == [290, 262, 251, 448, 440, 368, 702, 954, 433] # test_creation x = Symbol("x") a = SparseMatrix([[x, 0], [0, 0]]) m = a assert m.cols == m.rows assert m.cols == 2 assert m[:] == [x, 0, 0, 0] b = SparseMatrix(2, 2, [x, 0, 0, 0]) m = b assert m.cols == m.rows assert m.cols == 2 assert m[:] == [x, 0, 0, 0] assert a == b S = sparse_eye(3) S.row_del(1) assert S == SparseMatrix([ [1, 0, 0], [0, 0, 1]]) S = sparse_eye(3) S.col_del(1) assert S == SparseMatrix([ [1, 0], [0, 0], [0, 1]]) S = SparseMatrix.eye(3) S[2, 1] = 2 S.col_swap(1, 0) assert S == SparseMatrix([ [0, 1, 0], [1, 0, 0], [2, 0, 1]]) a = SparseMatrix(1, 2, [1, 2]) b = a.copy() c = a.copy() assert a[0] == 1 a.row_del(0) assert a == SparseMatrix(0, 2, []) b.col_del(1) assert b == SparseMatrix(1, 1, [1]) # test_determinant x, y = Symbol('x'), Symbol('y') assert SparseMatrix(1, 1, [0]).det() == 0 assert SparseMatrix([[1]]).det() == 1 assert SparseMatrix(((-3, 2), (8, -5))).det() == -1 assert SparseMatrix(((x, 1), (y, 2*y))).det() == 2*x*y - y assert SparseMatrix(( (1, 1, 1), (1, 2, 3), (1, 3, 6) )).det() == 1 assert SparseMatrix(( ( 3, -2, 0, 5), (-2, 1, -2, 2), ( 0, -2, 5, 0), ( 5, 0, 3, 4) )).det() == -289 assert SparseMatrix(( ( 1, 2, 3, 4), ( 5, 6, 7, 8), ( 9, 10, 11, 12), (13, 14, 15, 16) )).det() == 0 assert SparseMatrix(( (3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0), (0, 0, 0, 3, 2), (2, 0, 0, 0, 3) )).det() == 275 assert SparseMatrix(( (1, 0, 1, 2, 12), (2, 0, 1, 1, 4), (2, 1, 1, -1, 3), (3, 2, -1, 1, 8), (1, 1, 1, 0, 6) )).det() == -55 assert SparseMatrix(( (-5, 2, 3, 4, 5), ( 1, -4, 3, 4, 5), ( 1, 2, -3, 4, 5), ( 1, 2, 3, -2, 5), ( 1, 2, 3, 4, -1) )).det() == 11664 assert SparseMatrix(( ( 2, 7, -1, 3, 2), ( 0, 0, 1, 0, 1), (-2, 0, 7, 0, 2), (-3, -2, 4, 5, 3), ( 1, 0, 0, 0, 1) )).det() == 123 # test_slicing m0 = sparse_eye(4) assert m0[:3, :3] == sparse_eye(3) assert m0[2:4, 0:2] == sparse_zeros(2) m1 = SparseMatrix(3, 3, lambda i, j: i + j) assert m1[0, :] == SparseMatrix(1, 3, (0, 1, 2)) assert m1[1:3, 1] == SparseMatrix(2, 1, (2, 3)) m2 = SparseMatrix( [[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11], [12, 13, 14, 15]]) assert m2[:, -1] == SparseMatrix(4, 1, [3, 7, 11, 15]) assert m2[-2:, :] == SparseMatrix([[8, 9, 10, 11], [12, 13, 14, 15]]) assert SparseMatrix([[1, 2], [3, 4]])[[1], [1]] == Matrix([[4]]) # test_submatrix_assignment m = sparse_zeros(4) m[2:4, 2:4] = sparse_eye(2) assert m == SparseMatrix([(0, 0, 0, 0), (0, 0, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]) assert len(m._smat) == 2 m[:2, :2] = sparse_eye(2) assert m == sparse_eye(4) m[:, 0] = SparseMatrix(4, 1, (1, 2, 3, 4)) assert m == SparseMatrix([(1, 0, 0, 0), (2, 1, 0, 0), (3, 0, 1, 0), (4, 0, 0, 1)]) m[:, :] = sparse_zeros(4) assert m == sparse_zeros(4) m[:, :] = ((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)) assert m == SparseMatrix((( 1, 2, 3, 4), ( 5, 6, 7, 8), ( 9, 10, 11, 12), (13, 14, 15, 16))) m[:2, 0] = [0, 0] assert m == SparseMatrix((( 0, 2, 3, 4), ( 0, 6, 7, 8), ( 9, 10, 11, 12), (13, 14, 15, 16))) # test_reshape m0 = sparse_eye(3) assert m0.reshape(1, 9) == SparseMatrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1)) m1 = SparseMatrix(3, 4, lambda i, j: i + j) assert m1.reshape(4, 3) == \ SparseMatrix([(0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)]) assert m1.reshape(2, 6) == \ SparseMatrix([(0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)]) # test_applyfunc m0 = sparse_eye(3) assert m0.applyfunc(lambda x: 2*x) == sparse_eye(3)*2 assert m0.applyfunc(lambda x: 0 ) == sparse_zeros(3) # test__eval_Abs assert abs(SparseMatrix(((x, 1), (y, 2*y)))) == SparseMatrix(((Abs(x), 1), (Abs(y), 2*Abs(y)))) # test_LUdecomp testmat = SparseMatrix([[ 0, 2, 5, 3], [ 3, 3, 7, 4], [ 8, 4, 0, 2], [-2, 6, 3, 4]]) L, U, p = testmat.LUdecomposition() assert L.is_lower assert U.is_upper assert (L*U).permute_rows(p, 'backward') - testmat == sparse_zeros(4) testmat = SparseMatrix([[ 6, -2, 7, 4], [ 0, 3, 6, 7], [ 1, -2, 7, 4], [-9, 2, 6, 3]]) L, U, p = testmat.LUdecomposition() assert L.is_lower assert U.is_upper assert (L*U).permute_rows(p, 'backward') - testmat == sparse_zeros(4) x, y, z = Symbol('x'), Symbol('y'), Symbol('z') M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z))) L, U, p = M.LUdecomposition() assert L.is_lower assert U.is_upper assert (L*U).permute_rows(p, 'backward') - M == sparse_zeros(3) # test_LUsolve A = SparseMatrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]]) x = SparseMatrix(3, 1, [3, 7, 5]) b = A*x soln = A.LUsolve(b) assert soln == x A = SparseMatrix([[0, -1, 2], [5, 10, 7], [8, 3, 4]]) x = SparseMatrix(3, 1, [-1, 2, 5]) b = A*x soln = A.LUsolve(b) assert soln == x # test_inverse A = sparse_eye(4) assert A.inv() == sparse_eye(4) assert A.inv(method="CH") == sparse_eye(4) assert A.inv(method="LDL") == sparse_eye(4) A = SparseMatrix([[2, 3, 5], [3, 6, 2], [7, 2, 6]]) Ainv = SparseMatrix(Matrix(A).inv()) assert A*Ainv == sparse_eye(3) assert A.inv(method="CH") == Ainv assert A.inv(method="LDL") == Ainv A = SparseMatrix([[2, 3, 5], [3, 6, 2], [5, 2, 6]]) Ainv = SparseMatrix(Matrix(A).inv()) assert A*Ainv == sparse_eye(3) assert A.inv(method="CH") == Ainv assert A.inv(method="LDL") == Ainv # test_cross v1 = Matrix(1, 3, [1, 2, 3]) v2 = Matrix(1, 3, [3, 4, 5]) assert v1.cross(v2) == Matrix(1, 3, [-2, 4, -2]) assert v1.norm(2)**2 == 14 # conjugate a = SparseMatrix(((1, 2 + I), (3, 4))) assert a.C == SparseMatrix([ [1, 2 - I], [3, 4] ]) # mul assert a*Matrix(2, 2, [1, 0, 0, 1]) == a assert a + Matrix(2, 2, [1, 1, 1, 1]) == SparseMatrix([ [2, 3 + I], [4, 5] ]) # col join assert a.col_join(sparse_eye(2)) == SparseMatrix([ [1, 2 + I], [3, 4], [1, 0], [0, 1] ]) # symmetric assert not a.is_symmetric(simplify=False) # test_cofactor assert sparse_eye(3) == sparse_eye(3).cofactor_matrix() test = SparseMatrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]]) assert test.cofactor_matrix() == \ SparseMatrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]]) test = SparseMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) assert test.cofactor_matrix() == \ SparseMatrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]]) # test_jacobian x = Symbol('x') y = Symbol('y') L = SparseMatrix(1, 2, [x**2*y, 2*y**2 + x*y]) syms = [x, y] assert L.jacobian(syms) == Matrix([[2*x*y, x**2], [y, 4*y + x]]) L = SparseMatrix(1, 2, [x, x**2*y**3]) assert L.jacobian(syms) == SparseMatrix([[1, 0], [2*x*y**3, x**2*3*y**2]]) # test_QR A = Matrix([[1, 2], [2, 3]]) Q, S = A.QRdecomposition() R = Rational assert Q == Matrix([ [ 5**R(-1, 2), (R(2)/5)*(R(1)/5)**R(-1, 2)], [2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]]) assert S == Matrix([ [5**R(1, 2), 8*5**R(-1, 2)], [ 0, (R(1)/5)**R(1, 2)]]) assert Q*S == A assert Q.T * Q == sparse_eye(2) R = Rational # test nullspace # first test reduced row-ech form M = SparseMatrix([[5, 7, 2, 1], [1, 6, 2, -1]]) out, tmp = M.rref() assert out == Matrix([[1, 0, -R(2)/23, R(13)/23], [0, 1, R(8)/23, R(-6)/23]]) M = SparseMatrix([[ 1, 3, 0, 2, 6, 3, 1], [-2, -6, 0, -2, -8, 3, 1], [ 3, 9, 0, 0, 6, 6, 2], [-1, -3, 0, 1, 0, 9, 3]]) out, tmp = M.rref() assert out == Matrix([[1, 3, 0, 0, 2, 0, 0], [0, 0, 0, 1, 2, 0, 0], [0, 0, 0, 0, 0, 1, R(1)/3], [0, 0, 0, 0, 0, 0, 0]]) # now check the vectors basis = M.nullspace() assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0]) assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0]) assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0]) assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1)/3, 1]) # test eigen x = Symbol('x') y = Symbol('y') sparse_eye3 = sparse_eye(3) assert sparse_eye3.charpoly(x) == PurePoly(((x - 1)**3)) assert sparse_eye3.charpoly(y) == PurePoly(((y - 1)**3)) # test values M = Matrix([( 0, 1, -1), ( 1, 1, 0), (-1, 0, 1)]) vals = M.eigenvals() assert sorted(vals.keys()) == [-1, 1, 2] R = Rational M = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) assert M.eigenvects() == [(1, 3, [ Matrix([1, 0, 0]), Matrix([0, 1, 0]), Matrix([0, 0, 1])])] M = Matrix([[5, 0, 2], [3, 2, 0], [0, 0, 1]]) assert M.eigenvects() == [(1, 1, [Matrix([R(-1)/2, R(3)/2, 1])]), (2, 1, [Matrix([0, 1, 0])]), (5, 1, [Matrix([1, 1, 0])])] assert M.zeros(3, 5) == SparseMatrix(3, 5, {}) A = SparseMatrix(10, 10, {(0, 0): 18, (0, 9): 12, (1, 4): 18, (2, 7): 16, (3, 9): 12, (4, 2): 19, (5, 7): 16, (6, 2): 12, (9, 7): 18}) assert A.row_list() == [(0, 0, 18), (0, 9, 12), (1, 4, 18), (2, 7, 16), (3, 9, 12), (4, 2, 19), (5, 7, 16), (6, 2, 12), (9, 7, 18)] assert A.col_list() == [(0, 0, 18), (4, 2, 19), (6, 2, 12), (1, 4, 18), (2, 7, 16), (5, 7, 16), (9, 7, 18), (0, 9, 12), (3, 9, 12)] assert SparseMatrix.eye(2).nnz() == 2
def intersection(self, o): """ The intersection with other geometrical entity. Parameters ========== Point, Point3D, LinearEntity, LinearEntity3D, Plane Returns ======= List Examples ======== >>> from sympy import Point, Point3D, Line, Line3D, Plane >>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1)) >>> b = Point3D(1, 2, 3) >>> a.intersection(b) [Point3D(1, 2, 3)] >>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2)) >>> a.intersection(c) [Point3D(2, 2, 2)] >>> d = Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3)) >>> e = Plane(Point3D(2, 0, 0), normal_vector=(3, 4, -3)) >>> d.intersection(e) [Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))] """ from sympy.geometry.line import LinearEntity, LinearEntity3D if not isinstance(o, GeometryEntity): o = Point(o, dim=3) if isinstance(o, Point): if o in self: return [o] else: return [] if isinstance(o, (LinearEntity, LinearEntity3D)): if o in self: p1, p2 = o.p1, o.p2 if isinstance(o, Segment): o = Segment3D(p1, p2) elif isinstance(o, Ray): o = Ray3D(p1, p2) elif isinstance(o, Line): o = Line3D(p1, p2) else: raise ValueError('unhandled linear entity: %s' % o.func) return [o] else: x, y, z = map(Dummy, 'xyz') t = Dummy() # unnamed else it may clash with a symbol in o a = Point3D(o.arbitrary_point(t)) b = self.equation(x, y, z) # TODO: Replace solve with solveset, when this line is tested c = solve(b.subs(list(zip((x, y, z), a.args))), t) if not c: return [] else: p = a.subs(t, c[0]) if p not in self: return [] # e.g. a segment might not intersect a plane return [p] if isinstance(o, Plane): if self.equals(o): return [self] if self.is_parallel(o): return [] else: x, y, z = map(Dummy, 'xyz') a, b = Matrix([self.normal_vector]), Matrix([o.normal_vector]) c = list(a.cross(b)) d = self.equation(x, y, z) e = o.equation(x, y, z) result = list(linsolve([d, e], x, y, z))[0] for i in (x, y, z): result = result.subs(i, 0) return [Line3D(Point3D(result), direction_ratio=c)]
def intersection(self, o): """ The intersection with other geometrical entity. Parameters ========== Point, Point3D, LinearEntity, LinearEntity3D, Plane Returns ======= List Examples ======== >>> from sympy import Point, Point3D, Line, Line3D, Plane >>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1)) >>> b = Point3D(1, 2, 3) >>> a.intersection(b) [Point3D(1, 2, 3)] >>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2)) >>> a.intersection(c) [Point3D(2, 2, 2)] >>> d = Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3)) >>> e = Plane(Point3D(2, 0, 0), normal_vector=(3, 4, -3)) >>> d.intersection(e) [Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))] """ from sympy.geometry.line3d import LinearEntity3D from sympy.geometry.line import LinearEntity if isinstance(o, (Point, Point3D)): if o in self: return [Point3D(o)] else: return [] if isinstance(o, (LinearEntity, LinearEntity3D)): if o in self: p1, p2 = o.p1, o.p2 if isinstance(o, Segment): o = Segment3D(p1, p2) elif isinstance(o, Ray): o = Ray3D(p1, p2) elif isinstance(o, Line): o = Line3D(p1, p2) else: raise ValueError('unhandled linear entity: %s' % o.func) return [o] else: x, y, z = map(Dummy, 'xyz') t = Dummy() # unnamed else it may clash with a symbol in o a = Point3D(o.arbitrary_point(t)) b = self.equation(x, y, z) c = solve(b.subs(list(zip((x, y, z), a.args))), t) if not c: return [] else: p = a.subs(t, c[0]) if p not in self: return [] # e.g. a segment might not intersect a plane return [p] if isinstance(o, Plane): if o == self: return [self] if self.is_parallel(o): return [] else: x, y, z = map(Dummy, 'xyz') a, b = Matrix([self.normal_vector]), Matrix([o.normal_vector]) c = list(a.cross(b)) d = self.equation(x, y, z) e = o.equation(x, y, z) f = solve((d.subs(z, 0), e.subs(z, 0)), [x, y]) if len(f) == 2: return [Line3D(Point3D(f[x], f[y], 0), direction_ratio=c)] g = solve((d.subs(y, 0), e.subs(y, 0)),[x, z]) if len(g) == 2: return [Line3D(Point3D(g[x], 0, g[z]), direction_ratio=c)] h = solve((d.subs(x, 0), e.subs(x, 0)),[y, z]) if len(h) == 2: return [Line3D(Point3D(0, h[y], h[z]), direction_ratio=c)]
def generate_forward_kinematics(self): # Get link parameters. n_links = self.n n_dofs = 2 * (self.n - 1) link_radius = self.r link_length = self.l # Create transforms gₛₗ(0) from base frame to each link l. g_sl0 = [None] * n_links g_sl0[0] = eye_rat(4) for i in range(1, n_links): g_sl = eye_rat(4) g_sl[0, 3] = Rational(i * (link_length + 2 * link_radius)) if i % 2 == 1: g_sl[0:3, 0:3] = exp3(Matrix(3, 1, [1, 0, 0]), pi / 2) g_sl0[i] = g_sl if DEBUG: print('g_sl0[i]') pprint(g_sl0[i]) # Create revolute joint twists for each DOF, 2 per link. joint_twists = [] w1 = Matrix(3, 1, [0, 0, 1]) w2 = Matrix(3, 1, [0, -1, 0]) for i in range(1, n_links): q1 = Matrix(3, 1, [g_sl0[i - 1][0, 3] + link_length / 2, 0, 0]) x1 = zeros(6, 1) x1[0:3, 0] = -w1.cross(q1) x1[3:6, 0] = w1 for j in range(6): x1[j, 0] = Rational(x1[j, 0]) x2 = zeros(6, 1) q2 = Matrix(3, 1, [g_sl0[i][0, 3] - link_length / 2, 0, 0]) x2[0:3, 0] = -w2.cross(q2) x2[3:6, 0] = w2 for j in range(6): x2[j, 0] = Rational(x2[j, 0]) joint_twists.append([x1, x2]) if DEBUG: print('x1') pprint(x1) print('x2') pprint(x2) # Create symbolic DOFs. q = [] k = 0 for i in range(1, n_links): q1 = symbols('q%d' % k) k += 1 q2 = symbols('q%d' % k) k += 1 q.append([q1, q2]) # Create transforms gₛₗ(θ) from base frame to each link l. g_sl = [None] * n_links g_sl[0] = eye_rat(4) exps = [None] * n_links exps[0] = eye_rat(4) for i in range(1, n_links): q1 = q[i - 1][0] q2 = q[i - 1][1] x1 = joint_twists[i - 1][0] x2 = joint_twists[i - 1][1] exp_x1 = exp6(x1, q1) exp_x2 = exp6(x2, q2) if DEBUG: print('exp x1') pprint(exp_x1) print('exp x2') pprint(exp_x2) # exps[i] = exps[i-1] * exp_x1 * exp_x2 exps[i] = exps[i - 1] * trigsimp(exp_x1 * exp_x2) # exps[i] = trigsimp(exps[i]) # g_sl[i] = trigsimp(exps[i] * g_sl0[i]) # g_sl[i] = exps[i] * g_sl0[i] g_sl[i] = exps[i - 1] * trigsimp(exp_x1 * exp_x2 * g_sl0[i]) if DEBUG: print('g_sl[%d]' % i) pprint(g_sl[i]) # Assign transforms to each link. dofs = [] for q1, q2 in q: dofs.append(q1) dofs.append(q2) for i in range(n_links): self.links[i].set_tf_expr(lambdify(dofs, g_sl[i]))
def intersection(self, o): """ The intersection with other geometrical entity. Parameters ========== Point, Point3D, LinearEntity, LinearEntity3D, Plane Returns ======= List Examples ======== >>> from sympy import Point, Point3D, Line, Line3D, Plane >>> a = Plane(Point3D(1, 2, 3), normal_vector=[1, 1, 1]) >>> b = Point3D(1, 2, 3) >>> a.intersection(b) Point3D(1, 2, 3) >>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2)) >>> a.intersection(c) Point3D(2, 2, 2) >>> d = Plane(Point3D(6, 0, 0), normal_vector=[2, -5, 3]) >>> e = Plane(Point3D(2, 0, 0), normal_vector=[3, 4, -3]) >>> d.intersection(e) [Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))] """ from sympy.geometry.line3d import LinearEntity3D from sympy.geometry.line import LinearEntity if isinstance(o, Point) or isinstance(o, Point3D): if o in self: return o else: return [] if isinstance(o, LinearEntity3D): t = symbols('t') x, y, z = symbols("x y z") if o in self: return [o] else: a = o.arbitrary_point(t) b = self.equation(x, y, z) c = solve(b.subs([(x, a.x), (y, a.y), (z, a.z)])) if c == []: return [] else: a = a.subs(t, c[0]) if a in o: return a else: return [] if isinstance(o, LinearEntity): t = symbols('t') x, y, z = symbols("x y z") if o in self: return [o] else: a = self.equation(x, y, z).subs(z, 0) b = o.equation(x, y) c = solve((a, b), [x, y]) if c is {}: return [] else: return [Point(c[x], c[y])] if isinstance(o, Plane): if o == self: return self if self.is_parallel(o): return [] else: x, y, z = symbols("x y z") a, b = Matrix([self.normal_vector]), Matrix([o.normal_vector]) c = list(a.cross(b)) d = self.equation(x, y, z) e = o.equation(x, y, z) f = solve((d.subs(z, 0), e.subs(z, 0)), [x, y]) g = solve((d.subs(y, 0), e.subs(y, 0)), [x, z]) h = solve((d.subs(x, 0), e.subs(x, 0)), [y, z]) if len(f) == 2: return [Line3D(Point3D(f[x], f[y], 0), direction_ratio=c)] if len(g) == 2: return [Line3D(Point3D(g[x], 0, g[z]), direction_ratio=c)] if len(h) == 2: return [Line3D(Point3D(0, h[y], h[z]), direction_ratio=c)]
L_ * (F_1 - F_2), L_ * (F_1 - F_3), # b_ * (M_1 - M_2 + M_3 - M_4) (M_1 - M_2 + M_3 - M_4) ]) omega = Matrix([phi_dot, theta_dot, psi_dot]) radius = L_ / 2.0 Ix = 2 * mass_ * (radius * radius) / 5.0 + 2 * radius * radius * mass_ Iy = 2 * mass_ * (radius * radius) / 5.0 + 2 * radius * radius * mass_ Iz = 2 * mass_ * (radius * radius) / 5.0 + 4 * radius * radius * mass_ # Eigen::Matrix3d I, Iinv; I = Matrix([[Ix, 0, 0], [0, Iy, 0], [0, 0, Iz]]) Iinv = Matrix([[1 / Ix, 0, 0], [0, 1 / Iy, 0], [0, 0, 1 / Iz]]) omega_dot = Iinv * (tau - omega.cross(I * omega)) # Eigen::VectorXd state_dot(12); state_dot = Matrix([ x_dot, y_dot, z_dot, phi_dot, theta_dot, psi_dot, pos_ddot[0], pos_ddot[1], pos_ddot[2], omega_dot[0], omega_dot[1], omega_dot[2] ]) statevars = [ x_, y_, z_, phi, theta, psi, x_dot, y_dot, z_dot, phi_dot, theta_dot, psi_dot ] fx = [] for var in statevars:
def intersection(self, o): """ The intersection with other geometrical entity. Parameters ========== Point, Point3D, LinearEntity, LinearEntity3D, Plane Returns ======= List Examples ======== >>> from sympy import Point, Point3D, Line, Line3D, Plane >>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1)) >>> b = Point3D(1, 2, 3) >>> a.intersection(b) [Point3D(1, 2, 3)] >>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2)) >>> a.intersection(c) [Point3D(2, 2, 2)] >>> d = Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3)) >>> e = Plane(Point3D(2, 0, 0), normal_vector=(3, 4, -3)) >>> d.intersection(e) [Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))] """ from sympy.geometry.line3d import LinearEntity3D from sympy.geometry.line import LinearEntity if isinstance(o, (Point, Point3D)): if o in self: return [Point3D(o)] else: return [] if isinstance(o, (LinearEntity, LinearEntity3D)): if o in self: p1, p2 = o.p1, o.p2 if isinstance(o, Segment): o = Segment3D(p1, p2) elif isinstance(o, Ray): o = Ray3D(p1, p2) elif isinstance(o, Line): o = Line3D(p1, p2) else: raise ValueError('unhandled linear entity: %s' % o.func) return [o] else: x, y, z = map(Dummy, 'xyz') t = Dummy() # unnamed else it may clash with a symbol in o a = Point3D(o.arbitrary_point(t)) b = self.equation(x, y, z) c = solve(b.subs(list(zip((x, y, z), a.args))), t) if not c: return [] else: p = a.subs(t, c[0]) if p not in self: return [] # e.g. a segment might not intersect a plane return [p] if isinstance(o, Plane): if o == self: return [self] if self.is_parallel(o): return [] else: x, y, z = map(Dummy, 'xyz') a, b = Matrix([self.normal_vector]), Matrix([o.normal_vector]) c = list(a.cross(b)) d = self.equation(x, y, z) e = o.equation(x, y, z) f = solve((d.subs(z, 0), e.subs(z, 0)), [x, y]) if len(f) == 2: return [Line3D(Point3D(f[x], f[y], 0), direction_ratio=c)] g = solve((d.subs(y, 0), e.subs(y, 0)), [x, z]) if len(g) == 2: return [Line3D(Point3D(g[x], 0, g[z]), direction_ratio=c)] h = solve((d.subs(x, 0), e.subs(x, 0)), [y, z]) if len(h) == 2: return [Line3D(Point3D(0, h[y], h[z]), direction_ratio=c)]
w = Matrix([0, 0, -65]) # Vector de velocidad angular th = rad(30).evalf() # Ángulo en radianes b = 300 # Base de la placa h = 180 # Alto de la placa # Para la velocidad del punto $Q$ sabemos que: # # $$ \vec{v}_Q = \vec{\omega} \times \vec{r}_{Q/O}$$ # # Entonces: # In[11]: rQO = Matrix([b * cos(th), b * sin(th), 0]) # Vector r_Q/O vQ = w.cross(rQO) # Velocidad del punto del punto Q vQ # De manera similar se puede proceder para las velocidades de los puntos $M$ y $P$. # In[12]: beta = atan(h / b) tQ = th + beta # ángulo formado por el vector r_M/O con respecto a la horizontal tP = th + pi / 2 # ángulo formado por el vector r_P/O con respecto a la horizontal r = sqrt(h**2 + b**2) # diagonal del rectángulo rMO = Matrix([r * cos(tQ), r * sin(tQ), 0]) rPO = Matrix([r * cos(tP), r * sin(tP), 0]) vM = w.cross(rMO) # velocidad de M vP = w.cross(rPO) # velocidad de P
def intersection(self, o): """ The intersection with other geometrical entity. Parameters ========== Point, Point3D, LinearEntity, LinearEntity3D, Plane Returns ======= List Examples ======== >>> from sympy import Point, Point3D, Line, Line3D, Plane >>> a = Plane(Point3D(1, 2, 3), normal_vector=[1, 1, 1]) >>> b = Point3D(1, 2, 3) >>> a.intersection(b) Point3D(1, 2, 3) >>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2)) >>> a.intersection(c) Point3D(2, 2, 2) >>> d = Plane(Point3D(6, 0, 0), normal_vector=[2, -5, 3]) >>> e = Plane(Point3D(2, 0, 0), normal_vector=[3, 4, -3]) >>> d.intersection(e) [Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))] """ from sympy.geometry.line3d import LinearEntity3D from sympy.geometry.line import LinearEntity if isinstance(o, Point) or isinstance(o, Point3D): if o in self: return o else: return [] if isinstance(o, LinearEntity3D): t = symbols('t') x, y, z = symbols("x y z") if o in self: return [o] else: a = o.arbitrary_point(t) b = self.equation(x, y, z) c = solve(b.subs([(x, a.x), (y, a.y), (z, a.z)])) if c == []: return [] else: a = a.subs(t, c[0]) if a in o: return a else: return [] if isinstance(o, LinearEntity): t = symbols('t') x, y, z = symbols("x y z") if o in self: return [o] else: a = self.equation(x, y, z).subs(z,0) b = o.equation(x, y) c = solve((a,b),[x, y]) if c is {}: return [] else: return [Point(c[x], c[y])] if isinstance(o, Plane): if o == self: return self if self.is_parallel(o): return [] else: x, y, z = symbols("x y z") a, b= Matrix([self.normal_vector]), Matrix([o.normal_vector]) c = list(a.cross(b)) d = self.equation(x, y, z) e = o.equation(x, y, z) f = solve((d.subs(z,0), e.subs(z,0)),[x, y]) g = solve((d.subs(y,0), e.subs(y,0)),[x, z]) h = solve((d.subs(x,0), e.subs(x,0)),[y, z]) if len(f) == 2: return [Line3D(Point3D(f[x], f[y], 0), direction_ratio=c)] if len(g) == 2: return [Line3D(Point3D(g[x], 0, g[z]), direction_ratio=c)] if len(h) == 2: return [Line3D(Point3D(0, h[y], h[z]), direction_ratio=c)]
# Recuerde que si requiere ver las expresiones resultantes como fracciones decimales debe usar `evalf`. # # El producto escalar de dos vectores puede calcularlo utilizando el método `dot`, por # ejemplo $ \vec{u} \cdot \vec{v} $ lo puede especificar como: # In[51]: u.dot(v) # Para calcular el producto vectorial utilice el método `cross`, por ejemplo # $ \vec{u} \times \vec{v} $: # In[52]: u.cross(v) # Recuerde que el producto vectorial no es conmutativo, por tanto, $\vec{v} \times \vec{u} $ resultará en un vector # diferente al obtenido anteriormente, como puede verificar enseguida: # In[53]: v.cross(u) # ## Matrices # # Las matrices son arreglos rectangulares de números o cantidades simbólicas. En SymPy, # se definen utilizando la clase `Matrix`, pasándole como argumentos una lista de # listas, donde cada sublista corresponde a una fila de la matriz. # # Por ejemplo, vamos a definir las matrices $A$, $B$ y $C$, dadas por: