如何绘制多项式函数绕y轴旋转的立体?

问题描述

函数 y = x**2 绕 y 轴旋转的实体可以使用以下代码绘制:

A solid of revolution of a function y = x**2 around the y-axis can be plotted using the code below:

import numpy as np
import matplotlib.pyplot as plt
import mpl_toolkits.mplot3d.axes3d as axes3d
from matplotlib import cm

np.seterr(divide='ignore', invalid='ignore')

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')


ll, ul = 0, 1
u = np.linspace(ll, ul, 60)
v = np.linspace(0, 2*np.pi, 60)
U, V = np.meshgrid(u, v)

Z = U

X = np.sqrt(Z)*np.cos(V)
Y = np.sqrt(Z)*np.sin(V)

ax.set_xlabel('Y axis')
ax.set_ylabel('X axis')
ax.set_zlabel('Z axis')

ax.plot_surface(X, Y, Z, cmap=plt.cm.YlGnBu_r)

plt.show()

这段代码只是绘制了函数 x = sqrt(y) 的旋转实体，它是 y = x**2 的倒数.但是如何绘制函数的旋转体为:

This code just plots the solid of revolution of the function x = sqrt(y) which is the inverse of y = x**2. But how to draw the solid of revolution for a function as:

y = x**5 + x**4 + x**3 + x**2 + x

?

推荐答案

(U,V)参数空间到(X,Y,Z)坐标的映射可以非常灵活.通常 U 被选择为类似于 np.linspace(ll, ul, 100) 并且被认为等于 Y(如果 y 是旋转轴).但是您不必那样使用 U.相反，U 可以表示半径:

The mapping from (U,V) parameter space to (X,Y,Z) coordinates can be very flexible.Often U is chosen to be something like np.linspace(ll, ul, 100) and is taken to be equal to Y (if y is the axis of rotation). But you don't have to use U that way.Instead U could represent the radius:

ll, ul = 0, 1
u = np.linspace(ll, ul, 100)
v = np.linspace(0, 2*np.pi, 60)
U, V = np.meshgrid(u, v)

然后X、Y、Z可以根据半径来定义，U:

and then X, Y, Z can be defined in terms of the radius, U:

Y = U**5 + U**4 + U**3 + U**2 + U
X = U*np.cos(V)
Z = U*np.sin(V)

import numpy as np
import matplotlib.pyplot as plt
import mpl_toolkits.mplot3d.axes3d as axes3d
from matplotlib import cm

np.seterr(divide='ignore', invalid='ignore')

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

ll, ul = 0, 1
u = np.linspace(ll, ul, 100)
v = np.linspace(0, 2*np.pi, 60)
U, V = np.meshgrid(u, v)

Y = U**5 + U**4 + U**3 + U**2 + U
X = U*np.cos(V)
Z = U*np.sin(V)

ax.set_xlabel('Y axis')
ax.set_ylabel('X axis')
ax.set_zlabel('Z axis')

ax.plot_surface(X, Y, Z, cmap=plt.cm.YlGnBu_r)

plt.show()

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