def test_rational_derivative(self):
# testing the parametrization x(u,v) = [.5*u^3*(1-v)^3 / ((1-v)^3*(1-u)^3 + .5*u^3*(1-v)^3), 0]
# dx/du = (6*u^2*(u - 1)^2)/(u^3 - 6*u^2 + 6*u - 2)^2
# d2x/du2 = -(12*u*(u^5 - 3*u^4 + 2*u^3 + 4*u^2 - 6*u + 2))/(u^3 - 6*u^2 + 6*u - 2)^3
# d3x/du3 = (12*(3*u^8 - 12*u^7 + 10*u^6 + 48*u^5 - 156*u^4 + 176*u^3 - 72*u^2 + 4))/(u^3 - 6*u^2 + 6*u - 2)^4
# dx/dv = 0
controlpoints = [[0, 0, 1], [0, 0, 0], [0, 0, 0], [.5, 0, .5],
[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0],
[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0],
[0, 0, 0], [0, 0, 0]]
basis = BSplineBasis(4)
surf = Surface(basis, basis, controlpoints, rational=True)
def expect_derivative(u, v):
return (6 * u**2 * (u - 1)**2) / (u**3 - 6 * u**2 + 6 * u - 2)**2
def expect_derivative_2(u, v):
return -(12 * u *
(u**5 - 3 * u**4 + 2 * u**3 + 4 * u**2 - 6 * u + 2)) / (
u**3 - 6 * u**2 + 6 * u - 2)**3
def expect_derivative_3(u, v):
return (12 * (3 * u**8 - 12 * u**7 + 10 * u**6 + 48 * u**5 -
156 * u**4 + 176 * u**3 - 72 * u**2 + 4)) / (
u**3 - 6 * u**2 + 6 * u - 2)**4
# insert a few more knots to spice things up
surf.insert_knot([.3, .51], 0)
surf.insert_knot([.41, .53, .92], 1)
# test first derivatives
self.assertAlmostEqual(
surf.derivative(0.32, 0.22, d=(1, 0))[0],
expect_derivative(0.32, 0.22))
self.assertAlmostEqual(surf.derivative(0.32, 0.22, d=(1, 0))[1], 0)
self.assertAlmostEqual(
surf.derivative(0.71, 0.22, d=(1, 0))[0],
expect_derivative(0.71, 0.22))
self.assertAlmostEqual(
surf.derivative(0.71, 0.62, d=(1, 0))[0],
expect_derivative(0.71, 0.62))
# test second derivatives
self.assertAlmostEqual(
surf.derivative(0.32, 0.22, d=(2, 0))[0],
expect_derivative_2(0.32, 0.22))
self.assertAlmostEqual(
surf.derivative(0.71, 0.22, d=(2, 0))[0],
expect_derivative_2(0.71, 0.22))
self.assertAlmostEqual(
surf.derivative(0.71, 0.62, d=(2, 0))[0],
expect_derivative_2(0.71, 0.62))
# all cross derivatives vanish in this particular example
self.assertAlmostEqual(surf.derivative(0.32, 0.22, d=(1, 1))[0], 0)
self.assertAlmostEqual(surf.derivative(0.32, 0.22, d=(2, 1))[0], 0)
self.assertAlmostEqual(surf.derivative(0.32, 0.22, d=(1, 2))[0], 0)
# test third derivatives
self.assertAlmostEqual(
surf.derivative(0.32, 0.22, d=(3, 0))[0],
expect_derivative_3(0.32, 0.22))
self.assertAlmostEqual(
surf.derivative(0.71, 0.22, d=(3, 0))[0],
expect_derivative_3(0.71, 0.22))
self.assertAlmostEqual(
surf.derivative(0.71, 0.62, d=(3, 0))[0],
expect_derivative_3(0.71, 0.62))
# swapping u for v and symmetric logic
surf.swap()
self.assertAlmostEqual(
surf.derivative(0.22, 0.32, d=(0, 2))[0],
expect_derivative_2(0.32, 0.22))
self.assertAlmostEqual(
surf.derivative(0.22, 0.71, d=(0, 2))[0],
expect_derivative_2(0.71, 0.22))
self.assertAlmostEqual(
surf.derivative(0.62, 0.71, d=(0, 2))[0],
expect_derivative_2(0.71, 0.62))
self.assertAlmostEqual(
surf.derivative(0.22, 0.32, d=(0, 3))[0],
expect_derivative_3(0.32, 0.22))
self.assertAlmostEqual(
surf.derivative(0.22, 0.71, d=(0, 3))[0],
expect_derivative_3(0.71, 0.22))
self.assertAlmostEqual(
surf.derivative(0.62, 0.71, d=(0, 3))[0],
expect_derivative_3(0.71, 0.62))
def test_derivative(self):
# knot vector [t_1, t_2, ... t_{n+p+1}]
# polynomial degree p (order-1)
# n basis functions N_i(t), for i=1...n
# the power basis {1,t,t^2,t^3,...} can be expressed as:
# 1 = sum N_i(t)
# t = sum ts_i * N_i(t)
# t^2 = sum t2s_i * N_i(t)
# ts_i = sum_{j=i+1}^{i+p} t_j / p
# t2s_i = sum_{j=i+1}^{i+p-1} sum_{k=j+1}^{i+p} t_j*t_k / (p 2)
# (p 2) = binomial coefficient
# creating the mapping:
# x(u,v) = u^2*v + u(1-v)
# y(u,v) = v
controlpoints = [[0, 0], [1.0 / 4, 0], [3.0 / 4, 0], [.75, 0], [0, 1],
[0, 1], [.5, 1], [1, 1]]
basis1 = BSplineBasis(3, [0, 0, 0, .5, 1, 1, 1])
basis2 = BSplineBasis(2, [0, 0, 1, 1])
surf = Surface(basis1, basis2, controlpoints)
# call evaluation at a 5x4 grid of points
val = surf.derivative([0, .2, .5, .6, 1], [0, .2, .4, 1], d=(1, 0))
self.assertEqual(len(val.shape),
3) # result should be wrapped in 3-index tensor
self.assertEqual(val.shape[0], 5) # 5 evaluation points in u-direction
self.assertEqual(val.shape[1], 4) # 4 evaluation points in v-direction
self.assertEqual(val.shape[2], 2) # 2 coordinates (x,y)
self.assertAlmostEqual(surf.derivative(.2, .2, d=(1, 0))[0],
.88) # dx/du=2uv+(1-v)
self.assertAlmostEqual(surf.derivative(.2, .2, d=(1, 0))[1],
0) # dy/du=0
self.assertAlmostEqual(surf.derivative(.2, .2, d=(0, 1))[0],
-.16) # dx/dv=u^2-u
self.assertAlmostEqual(surf.derivative(.2, .2, d=(0, 1))[1],
1) # dy/dv=1
self.assertAlmostEqual(surf.derivative(.2, .2, d=(1, 1))[0],
-.60) # d2x/dudv=2u-1
self.assertAlmostEqual(surf.derivative(.2, .2, d=(2, 0))[0],
0.40) # d2x/dudu=2v
self.assertAlmostEqual(surf.derivative(.2, .2, d=(3, 0))[0],
0.00) # d3x/du3=0
self.assertAlmostEqual(surf.derivative(.2, .2, d=(0, 2))[0],
0.00) # d2y/dv2=0
# test errors and exceptions
with self.assertRaises(ValueError):
val = surf.derivative(-10,
.5) # evalaute outside parametric domain
with self.assertRaises(ValueError):
val = surf.derivative(+10,
.3) # evalaute outside parametric domain
with self.assertRaises(ValueError):
val = surf.derivative(.5,
-10) # evalaute outside parametric domain
with self.assertRaises(ValueError):
val = surf.derivative(.5,
+10) # evalaute outside parametric domain