三维曲面

# ===== 示例：可视化偏导数（沿x和y方向切线） =====
import numpy as np
from sympy import lambdify, diff, exp
from sympy.abc import x, y
from matplotlib import pyplot as plt

# 创建网格数据
num = 301
x_array = np.linspace(-3, 3, num)
y_array = np.linspace(-3, 3, num)
xx, yy = np.meshgrid(x_array, y_array)

# 定义二元函数（MATLAB经典peaks函数）
f_xy = 3*(1-x)**2*exp(-(x**2) - (y+1)**2) \
    - 10*(x/5 - x**3 - y**5)*exp(-x**2-y**2) \
    - 1/3*exp(-(x+1)**2 - y**2)

f_xy_fcn = lambdify([x, y], f_xy)
f_xy_zz = f_xy_fcn(xx, yy)

# 计算偏导数（符号计算）
df_dx = f_xy.diff(x)  # 对x求偏导
df_dy = f_xy.diff(y)  # 对y求偏导

df_dx_fcn = lambdify([x, y], df_dx)
df_dy_fcn = lambdify([x, y], df_dy)

# 定义切点网格
x_s = np.linspace(-2.4, 2.4, 9)
y_s = np.linspace(-2.4, 2.4, 9)
xx_s, yy_s = np.meshgrid(x_s, y_s)


# ===== 沿x方向切线 =====
def plot_d_x_tangent(x_t, y_t, df_dx_fcn, f_xy_fcn, color, ax):
    # 计算切线斜率（偏导数）
    k = df_dx_fcn(x_t, y_t)
    # 切点z坐标
    z_t = f_xy_fcn(x_t, y_t)
    # 切线x轴数组
    x_array = np.linspace(x_t-0.6, x_t+0.6, 10)
    # 切线函数：z = k*(x - x_t) + z_t
    z_array = k*(x_array - x_t) + z_t
    
    # 绘制切线
    ax.plot(x_array, x_array*0 + y_t, z_array, 
           color=color, lw=0.2)
    # 绘制切点
    ax.plot(x_t, y_t, z_t, color=color, 
           marker='.', markersize=5)

fig, ax = plt.subplots(subplot_kw={'projection': '3d'})

# 绘制曲面（只显示沿x方向的网格线）
ax.plot_wireframe(xx, yy, f_xy_zz,
                 color=[0.5, 0.5, 0.5],
                 rstride=15, cstride=0,
                 linewidth=0.25)

# 绘制多条沿x方向的切线
colors = plt.cm.rainbow(np.linspace(0, 1, len(xx_s.ravel())))

for i in range(len(xx_s.ravel())):
    x_t = xx_s.ravel()[i]
    y_t = yy_s.ravel()[i]
    color = colors[i, :]
    plot_d_x_tangent(x_t, y_t, df_dx_fcn, f_xy_fcn, color, ax)

ax.set_proj_type('ortho')
ax.view_init(azim=-135, elev=30)
ax.grid(False)
plt.show()


# ===== 沿y方向切线 =====
def plot_d_y_tangent(x_t, y_t, df_dy_fcn, f_xy_fcn, color, ax):
    k = df_dy_fcn(x_t, y_t)
    z_t = f_xy_fcn(x_t, y_t)
    
    y_array = np.linspace(y_t-0.6, y_t+0.6, 10)
    z_array = k*(y_array - y_t) + z_t
    
    ax.plot(y_array*0 + x_t, y_array, z_array, 
           color=color, lw=0.2)
    ax.plot(x_t, y_t, z_t, color=color, 
           marker='.', markersize=5)

fig, ax = plt.subplots(subplot_kw={'projection': '3d'})

# 绘制曲面（只显示沿y方向的网格线）
ax.plot_wireframe(xx, yy, f_xy_zz,
                 color=[0.5, 0.5, 0.5],
                 rstride=0, cstride=15,
                 linewidth=0.25)

colors = plt.cm.rainbow(np.linspace(0, 1, len(yy_s.ravel())))

for i in range(len(yy_s.ravel())):
    x_t = xx_s.ravel()[i]
    y_t = yy_s.ravel()[i]
    color = colors[i, :]
    plot_d_y_tangent(x_t, y_t, df_dy_fcn, f_xy_fcn, color, ax)

ax.set_proj_type('ortho')
ax.view_init(azim=-135, elev=30)
ax.grid(False)
plt.show()


# 💡 关键学习点：
# 1. sympy.diff()：符号求导，计算偏导数
# 2. lambdify()：将符号表达式转为数值函数
# 3. 偏导数几何意义：曲面沿某方向的切线斜率
# 4. ∂f/∂x：固定y，沿x方向的变化率
# 5. ∂f/∂y：固定x，沿y方向的变化率