具有方向导数的贝塞尔曲面矩阵形式
问题描述 投票：0回答：1
我之前问过一个关于贝塞尔曲面的问题并得到了很好的答案，但进一步研究它们的构造，我决定修改代码并添加沿 u 和 w 方向导数的计算。
我在构造最终的贝塞尔曲面时遇到困难。我不知道如何合并我计算过的方向导数
QuQw我将变换后的控制点添加到曲面图中。我所有的计算和数组都是正确的。我只需要知道如何绘制它。我当前的代码（在 Jupyter 笔记本中）显示没有表面：
import numpy as np
import matplotlib.pyplot as plt
def bernstein_poly(i, n, t):
return comb(n, i) * (t**i) * ((1 - t)**(n - i))
def bernstein_matrix(n, t):
return np.array([bernstein_poly(i, n, t) for i in range(n + 1)])
B = np.array([
[[-15, 0, 15], [-15, 5, 5], [-15, 5, -5], [-15, 0, -15]],
[[-5, 5, 15], [-5, 5, 5], [-5, 5, -5], [-5, 5, -15]],
[[5, 5, 15], [5, 5, 5], [5, 5, -5], [5, 5, -15]],
[[15, 0, 15], [15, 5, 5], [15, 5, -5], [15, 0, -15]]
])
N = np.array([[-1, 3, -3, 1],
[3, -6, 3, 0],
[-3, 3, 0, 0],
[1, 0, 0, 0]])
Nt = N.T
B_transformed = np.zeros((4, 4, 3))
for i in range(3):
B_transformed[:, :, i] = N @ B[:, :, i] @ Nt
print("Transformed control points matrix B_transformed:")
print(B_transformed)
u = 0.5
w = 0.5
U = np.array([u**3, u**2, u, 1])
W = np.array([[w**3], [w**2], [w], [1]])
print(U)
print(W)
Q = np.array([U @ B_transformed[:, :, i] @ W for i in range(3)])
print(" Q(0.5, 0.5):")
print(Q)
# Derivative formulas for u and w
U1 = np.array([3*u**2, 2*u, 1, 0])
W1 = np.array([[3*w**2], [2*w], [1], [0]])
print(U1)
print(W1)
# Directional derivative u
Qu = np.array([U1 @ B_transformed[:, :, i] @ W for i in range(3)])
print("Qu(0.5, 0.5):")
print(Qu)
# Directional derivative w
Qw = np.array([U @ B_transformed[:, :, i] @ W1 for i in range(3)])
print("Qw(0.5, 0.5):")
print(Qw)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(Qw, Qu, Q, rstride=1, cstride=1, color='b', alpha=0.6, edgecolor='w')
ax.scatter(B[:, :, 0], B[:, :, 1], B[:, :, 2], color='r', s=50)
1个回答
1
投票
投票
要计算导数，您需要使用与我之前的答案相同的逻辑：
import numpy as np
import matplotlib.pyplot as plt
from scipy.special import comb
def bernstein_poly(i, n, t):
return comb(n, i) * (t**i) * ((1 - t)**(n - i))
def bernstein_matrix(n, t):
return np.array([bernstein_poly(i, n, t) for i in range(n + 1)])
B = np.array([
[[-15, 0, 15], [-15, 5, 5], [-15, 5, -5], [-15, 0, -15]],
[[-5, 5, 15], [-5, 5, 5], [-5, 5, -5], [-5, 5, -15]],
[[5, 5, 15], [5, 5, 5], [5, 5, -5], [5, 5, -15]],
[[15, 0, 15], [15, 5, 5], [15, 5, -5], [15, 0, -15]]
])
N = np.array([[-1, 3, -3, 1],
[3, -6, 3, 0],
[-3, 3, 0, 0],
[1, 0, 0, 0]])
Nt = N.T
B_transformed = np.zeros((4, 4, 3))
for i in range(3):
B_transformed[:, :, i] = N @ B[:, :, i] @ Nt
print("Transformed control points matrix B_transformed:")
print(B_transformed)
u_values = np.linspace(0, 1, 50)
w_values = np.linspace(0, 1, 50)
u_grid, w_grid = np.meshgrid(u_values, w_values)
Q = np.zeros((u_grid.shape[0], u_grid.shape[1], 3))
Qu = np.zeros((u_grid.shape[0], u_grid.shape[1], 3))
Qw = np.zeros((u_grid.shape[0], u_grid.shape[1], 3))
for i in range(u_grid.shape[0]):
for j in range(u_grid.shape[1]):
u = u_grid[i, j]
w = w_grid[i, j]
U = np.array([u**3, u**2, u, 1])
W = np.array([[w**3], [w**2], [w], [1]])
U1 = np.array([3*u**2, 2*u, 1, 0])
W1 = np.array([[3*w**2], [2*w], [1], [0]])
Q[i, j, :] = np.array([U @ B_transformed[:, :, k] @ W for k in range(3)]).flatten()
Qu[i, j, :] = np.array([U1 @ B_transformed[:, :, k] @ W for k in range(3)]).flatten()
Qw[i, j, :] = np.array([U @ B_transformed[:, :, k] @ W1 for k in range(3)]).flatten()
print(" Q(0.5, 0.5):")
print(Q)
print("Qu(0.5, 0.5):")
print(Qu)
print("Qw(0.5, 0.5):")
print(Qw)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(Q[:, :, 0], Q[:, :, 1], Q[:, :, 2], rstride=1, cstride=1, color='b', alpha=0.6, edgecolor='w')
ax.scatter(B[:, :, 0], B[:, :, 1], B[:, :, 2], color='r', s=50)
plt.show()
返回：
Transformed control points matrix B_transformed:
[[[ 0. 0. 0.]
[ 0. 0. 0.]
[ 0. 0. 0.]
[ 0. 0. 0.]]
[[ 0. 0. 0.]
[ 0. -45. 0.]
[ 0. 45. 0.]
[ 0. -15. 0.]]
[[ 0. 0. 0.]
[ 0. 45. 0.]
[ 0. -45. 0.]
[ 30. 15. 0.]]
[[ 0. 0. 0.]
[ 0. -15. 0.]
[ 0. 15. -30.]
[-15. 0. 15.]]]
Q(0.5, 0.5):
[[[-15. 0. 15. ]
[-14.3877551 0.29987505 15. ]
[-13.7755102 0.58725531 15. ]
...
[ 13.7755102 0.58725531 15. ]
[ 14.3877551 0.29987505 15. ]
[ 15. 0. 15. ]]
[[-15. 0.29987505 14.3877551 ]
[-14.3877551 0.58176509 14.3877551 ]
[-13.7755102 0.85190972 14.3877551 ]
...
[ 13.7755102 0.85190972 14.3877551 ]
[ 14.3877551 0.58176509 14.3877551 ]
[ 15. 0.29987505 14.3877551 ]]
[[-15. 0.58725531 13.7755102 ]
[-14.3877551 0.85190972 13.7755102 ]
[-13.7755102 1.10553686 13.7755102 ]
...
[ 13.7755102 1.10553686 13.7755102 ]
[ 14.3877551 0.85190972 13.7755102 ]
[ 15. 0.58725531 13.7755102 ]]
...
[[-15. 0.58725531 -13.7755102 ]
[-14.3877551 0.85190972 -13.7755102 ]
[-13.7755102 1.10553686 -13.7755102 ]
...
[ 13.7755102 1.10553686 -13.7755102 ]
[ 14.3877551 0.85190972 -13.7755102 ]
[ 15. 0.58725531 -13.7755102 ]]
[[-15. 0.29987505 -14.3877551 ]
[-14.3877551 0.58176509 -14.3877551 ]
[-13.7755102 0.85190972 -14.3877551 ]
...
[ 13.7755102 0.85190972 -14.3877551 ]
[ 14.3877551 0.58176509 -14.3877551 ]
[ 15. 0.29987505 -14.3877551 ]]
[[-15. 0. -15. ]
[-14.3877551 0.29987505 -15. ]
[-13.7755102 0.58725531 -15. ]
...
[ 13.7755102 0.58725531 -15. ]
[ 14.3877551 0.29987505 -15. ]
[ 15. 0. -15. ]]]
Qu(0.5, 0.5):
[[[ 30. 15. 0. ]
[ 30. 14.3877551 0. ]
[ 30. 13.7755102 0. ]
...
[ 30. -13.7755102 0. ]
[ 30. -14.3877551 0. ]
[ 30. -15. 0. ]]
[[ 30. 14.10037484 0. ]
[ 30. 13.52484934 0. ]
[ 30. 12.94932384 0. ]
...
[ 30. -12.94932384 0. ]
[ 30. -13.52484934 0. ]
[ 30. -14.10037484 0. ]]
[[ 30. 13.23823407 0. ]
[ 30. 12.69789798 0. ]
[ 30. 12.1575619 0. ]
...
[ 30. -12.1575619 0. ]
[ 30. -12.69789798 0. ]
[ 30. -13.23823407 0. ]]
...
[[ 30. 13.23823407 0. ]
[ 30. 12.69789798 0. ]
[ 30. 12.1575619 0. ]
...
[ 30. -12.1575619 0. ]
[ 30. -12.69789798 0. ]
[ 30. -13.23823407 0. ]]
[[ 30. 14.10037484 0. ]
[ 30. 13.52484934 0. ]
[ 30. 12.94932384 0. ]
...
[ 30. -12.94932384 0. ]
[ 30. -13.52484934 0. ]
[ 30. -14.10037484 0. ]]
[[ 30. 15. 0. ]
[ 30. 14.3877551 0. ]
[ 30. 13.7755102 0. ]
...
[ 30. -13.7755102 0. ]
[ 30. -14.3877551 0. ]
[ 30. -15. 0. ]]]
Qw(0.5, 0.5):
[[[ 0. 15. -30. ]
[ 0. 14.10037484 -30. ]
[ 0. 13.23823407 -30. ]
...
[ 0. 13.23823407 -30. ]
[ 0. 14.10037484 -30. ]
[ 0. 15. -30. ]]
[[ 0. 14.3877551 -30. ]
[ 0. 13.52484934 -30. ]
[ 0. 12.69789798 -30. ]
...
[ 0. 12.69789798 -30. ]
[ 0. 13.52484934 -30. ]
[ 0. 14.3877551 -30. ]]
[[ 0. 13.7755102 -30. ]
[ 0. 12.94932384 -30. ]
[ 0. 12.1575619 -30. ]
...
[ 0. 12.1575619 -30. ]
[ 0. 12.94932384 -30. ]
[ 0. 13.7755102 -30. ]]
...
[[ 0. -13.7755102 -30. ]
[ 0. -12.94932384 -30. ]
[ 0. -12.1575619 -30. ]
...
[ 0. -12.1575619 -30. ]
[ 0. -12.94932384 -30. ]
[ 0. -13.7755102 -30. ]]
[[ 0. -14.3877551 -30. ]
[ 0. -13.52484934 -30. ]
[ 0. -12.69789798 -30. ]
...
[ 0. -12.69789798 -30. ]
[ 0. -13.52484934 -30. ]
[ 0. -14.3877551 -30. ]]
[[ 0. -15. -30. ]
[ 0. -14.10037484 -30. ]
[ 0. -13.23823407 -30. ]
...
[ 0. -13.23823407 -30. ]
[ 0. -14.10037484 -30. ]
[ 0. -15. -30. ]]]
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