def test_derivative(self):
# testing the parametrization x(t) = [.5*x^3 / ((1-x)^3+.5*x^3), 0]
cp = [[0, 0, 1], [0, 0, 0], [0, 0, 0], [.5, 0, .5]]
crv = Curve(BSplineBasis(4), cp, rational=True)
def expect_derivative(x):
return 6 * (1 - x)**2 * x**2 / (x**3 - 6 * x**2 + 6 * x - 2)**2
def expect_derivative_2(x):
return -12 * x * (x**5 - 3 * x**4 + 2 * x**3 + 4 * x**2 - 6 * x +
2) / (x**3 - 6 * x**2 + 6 * x - 2)**3
# insert a few more knots to spice things up
crv.insert_knot([.2, .71])
self.assertAlmostEqual(
crv.derivative(0.32, 1)[0], expect_derivative(0.32))
self.assertAlmostEqual(crv.derivative(0.32, 1)[1], 0)
self.assertAlmostEqual(
crv.derivative(0.71, 1)[0], expect_derivative(0.71))
self.assertAlmostEqual(
crv.derivative(0.22, 2)[0], expect_derivative_2(0.22))
self.assertAlmostEqual(crv.derivative(0.22, 2)[1], 0)
self.assertAlmostEqual(
crv.derivative(0.86, 2)[0], expect_derivative_2(0.86))
def test_derivative(self):
# testing the parametrization x(t) = [.5*x^3 / ((1-x)^3+.5*x^3), 0]
cp = [[0,0,1], [0,0,0], [0,0,0], [.5,0,.5]]
crv = Curve(BSplineBasis(4), cp, rational=True)
def expect_derivative(x):
return 6*(1-x)**2*x**2/(x**3 - 6*x**2 + 6*x - 2)**2
def expect_derivative_2(x):
return -12*x*(x**5 - 3*x**4 + 2*x**3 + 4*x**2 - 6*x + 2)/(x**3 - 6*x**2 + 6*x - 2)**3
# insert a few more knots to spice things up
crv.insert_knot([.2, .71])
self.assertAlmostEqual(crv.tangent(0.32)[0], expect_derivative(0.32))
self.assertAlmostEqual(crv.tangent(0.32)[1], 0)
self.assertAlmostEqual(crv.tangent(0.71)[0], expect_derivative(0.71))
self.assertAlmostEqual(crv.derivative(0.22, 2)[0], expect_derivative_2(0.22))
self.assertAlmostEqual(crv.derivative(0.22, 2)[1], 0)
self.assertAlmostEqual(crv.derivative(0.86, 2)[0], expect_derivative_2(0.86))
def test_append(self):
crv = Curve(BSplineBasis(3), [[0,0], [1,0], [0,1]])
crv2 = Curve(BSplineBasis(4), [[0,1,0], [0,1,1], [0,2,1], [0,2,2]])
crv2.insert_knot(0.5)
crv3 = crv.clone()
crv3.append(crv2)
expected_knot = [0,0,0,0,1,1,1,1.5,2,2,2,2]
self.assertEqual(crv3.order(direction=0), 4)
self.assertEqual(crv3.rational, False)
self.assertEqual(np.linalg.norm(crv3.knots(0, True)-expected_knot), 0.0)
t = np.linspace(0,1,11)
pt = np.zeros((11,3))
pt[:,:-1] = crv(t)
pt2 = crv3(t)
self.assertAlmostEqual(np.linalg.norm(pt-pt2), 0.0)
pt = crv2(t)
pt2 = crv3(t+1.0)
self.assertAlmostEqual(np.linalg.norm(pt-pt2), 0.0)
def test_append(self):
crv = Curve(BSplineBasis(3), [[0, 0], [1, 0], [0, 1]])
crv2 = Curve(BSplineBasis(4),
[[0, 1, 0], [0, 1, 1], [0, 2, 1], [0, 2, 2]])
crv2.insert_knot(0.5)
crv3 = crv.clone()
crv3.append(crv2)
expected_knot = [0, 0, 0, 0, 1, 1, 1, 1.5, 2, 2, 2, 2]
self.assertEqual(crv3.order(direction=0), 4)
self.assertEqual(crv3.rational, False)
self.assertAlmostEqual(
np.linalg.norm(crv3.knots(0, True) - expected_knot), 0.0)
t = np.linspace(0, 1, 11)
pt = np.zeros((11, 3))
pt[:, :-1] = crv(t)
pt2 = crv3(t)
self.assertAlmostEqual(np.linalg.norm(pt - pt2), 0.0)
pt = crv2(t)
pt2 = crv3(t + 1.0)
self.assertAlmostEqual(np.linalg.norm(pt - pt2), 0.0)
def test_insert_knot(self):
# non-uniform knot vector of a squiggly quadratic n=5 curve in 3D
controlpoints = [[0, 0, 0], [1, 1, 1], [2, -1, 0], [3, 0, -1], [0, 0, -5]]
crv = Curve(BSplineBasis(3, [0, 0, 0, .3, .4, 1, 1, 1]), controlpoints)
evaluation_point1 = crv(0.37)
crv.insert_knot(.2)
crv.insert_knot(.3)
crv.insert_knot(.9)
evaluation_point2 = crv(0.37)
# ensure that curve has not chcanged, by comparing evaluation of it
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
self.assertEqual(len(crv.knots(0, with_multiplicities=True)), 11)
# test knot insertion on single knot span
crv = Curve(BSplineBasis(5), [[0, 0, 0], [1, 1, 1], [2, -1, 0], [3, 0, -1], [0, 0, -5]])
evaluation_point1 = crv(0.27)
crv.insert_knot(.2)
evaluation_point2 = crv(0.27)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
# test knot insertion on first of two-knot span
crv = Curve(
BSplineBasis(3, [0, 0, 0, .5, 1, 1, 1]), [[0, 0, 0], [2, -1, 0], [3, 0, -1], [0, 0, -5]
])
evaluation_point1 = crv(0.27)
crv.insert_knot(.2)
evaluation_point2 = crv(0.27)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
# test knot insertion on last of two-knot span
crv = Curve(
BSplineBasis(3, [0, 0, 0, .5, 1, 1, 1]), [[0, 0, 0], [2, -1, 0], [3, 0, -1], [0, 0, -5]
])
evaluation_point1 = crv(0.27)
crv.insert_knot(.9)
evaluation_point2 = crv(0.27)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
# test knot insertion down to C0 basis
crv = Curve(BSplineBasis(3), [[0, 0, 0], [2, -1, 0], [0, 0, -5]])
evaluation_point1 = crv(0.27)
crv.insert_knot(.4)
crv.insert_knot(.4)
evaluation_point2 = crv(0.27)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
# test rational curves, here a perfect circle represented as n=9, p=2-curve
s = 1.0 / sqrt(2)
controlpoints = [[1, 0, 1], [s, s, s], [0, 1, 1], [-s, s, s], [-1, 0, 1], [-s, -s, s],
[0, -1, 1], [s, -s, s], [1, 0, 1]]
crv = Curve(BSplineBasis(3, [-1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5]), controlpoints, True)
evaluation_point1 = crv(0.37)
crv.insert_knot(.2)
crv.insert_knot(.3)
crv.insert_knot(.9)
evaluation_point2 = crv(0.37)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertEqual(len(crv.knots(0, with_multiplicities=True)), 15)
# test errors and exceptions
with self.assertRaises(TypeError):
crv.insert_knot(1, 2, 3) # too many arguments
with self.assertRaises(ValueError):
crv.insert_knot(1, 2) # direction=2 is illegal for curves
with self.assertRaises(TypeError):
crv.insert_knot() # too few arguments
with self.assertRaises(ValueError):
crv.insert_knot(-0.2) # Outside-domain error
with self.assertRaises(ValueError):
crv.insert_knot(4.4) # Outside-domain error
def test_insert_knot(self):
# non-uniform knot vector of a squiggly quadratic n=5 curve in 3D
controlpoints = [[0, 0, 0], [1, 1, 1], [2, -1, 0], [3, 0, -1],
[0, 0, -5]]
crv = Curve(BSplineBasis(3, [0, 0, 0, .3, .4, 1, 1, 1]), controlpoints)
evaluation_point1 = crv(0.37)
crv.insert_knot(.2)
crv.insert_knot(.3)
crv.insert_knot(.9)
evaluation_point2 = crv(0.37)
# ensure that curve has not chcanged, by comparing evaluation of it
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
self.assertEqual(len(crv.knots(0, with_multiplicities=True)), 11)
# test knot insertion on single knot span
crv = Curve(BSplineBasis(5),
[[0, 0, 0], [1, 1, 1], [2, -1, 0], [3, 0, -1], [0, 0, -5]])
evaluation_point1 = crv(0.27)
crv.insert_knot(.2)
evaluation_point2 = crv(0.27)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
# test knot insertion on first of two-knot span
crv = Curve(BSplineBasis(3, [0, 0, 0, .5, 1, 1, 1]),
[[0, 0, 0], [2, -1, 0], [3, 0, -1], [0, 0, -5]])
evaluation_point1 = crv(0.27)
crv.insert_knot(.2)
evaluation_point2 = crv(0.27)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
# test knot insertion on last of two-knot span
crv = Curve(BSplineBasis(3, [0, 0, 0, .5, 1, 1, 1]),
[[0, 0, 0], [2, -1, 0], [3, 0, -1], [0, 0, -5]])
evaluation_point1 = crv(0.27)
crv.insert_knot(.9)
evaluation_point2 = crv(0.27)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
# test knot insertion down to C0 basis
crv = Curve(BSplineBasis(3), [[0, 0, 0], [2, -1, 0], [0, 0, -5]])
evaluation_point1 = crv(0.27)
crv.insert_knot(.4)
crv.insert_knot(.4)
evaluation_point2 = crv(0.27)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertAlmostEqual(evaluation_point1[2], evaluation_point2[2])
# test rational curves, here a perfect circle represented as n=9, p=2-curve
s = 1.0 / sqrt(2)
controlpoints = [[1, 0, 1], [s, s, s], [0, 1, 1], [-s, s, s],
[-1, 0, 1], [-s, -s, s], [0, -1, 1], [s, -s, s],
[1, 0, 1]]
crv = Curve(BSplineBasis(3, [-1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5]),
controlpoints, True)
evaluation_point1 = crv(0.37)
crv.insert_knot(.2)
crv.insert_knot(.3)
crv.insert_knot(.9)
evaluation_point2 = crv(0.37)
self.assertAlmostEqual(evaluation_point1[0], evaluation_point2[0])
self.assertAlmostEqual(evaluation_point1[1], evaluation_point2[1])
self.assertEqual(len(crv.knots(0, with_multiplicities=True)), 15)
# test errors and exceptions
with self.assertRaises(TypeError):
crv.insert_knot(1, 2, 3) # too many arguments
with self.assertRaises(ValueError):
crv.insert_knot(1, 2) # direction=2 is illegal for curves
with self.assertRaises(TypeError):
crv.insert_knot() # too few arguments
with self.assertRaises(ValueError):
crv.insert_knot(-0.2) # Outside-domain error
with self.assertRaises(ValueError):
crv.insert_knot(4.4) # Outside-domain error
