空间坐标变换(Python&C++实现)-Toy模板网

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空间坐标变换可看作坐标点乘以一个齐次矩阵，其中，齐次矩阵可表示为：
空间坐标转换,python,线性代数
其中：
①区域的3×3矩阵产生三维图形的比例、对称、旋转、错切等基本变换；
②区域产生图形的透视变换；
③区域产生沿X、Y、Z三个轴的平移变换；
④区域产生图形的总比例变换。

平移变换

平移变换可表示为：
$\begin{gathered} \quad \begin{bmatrix} x & y & z & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ l & m & n & 1 \end{bmatrix} = \begin{bmatrix} x+l & y+m & z+n & 1 \end{bmatrix} \end{gathered}$

绕X轴旋转

空间立体绕X轴旋转某个角度，实体上个点的X坐标不变，只有Y、Z坐标改变，可表示为：
$\begin{gathered} \quad \begin{bmatrix} x & y & z & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & cos(\theta) & sin(\theta) & 0 \\ 0 & -sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} x & y \times cos(\theta) - z \times sin(\theta) & y \times sin(\theta) + z \times cos(\theta) & 1 \end{bmatrix} \end{gathered}$

绕Y轴旋转

空间立体绕Y轴旋转某个角度，实体上个点的Y坐标不变，只有X、Z坐标改变，可表示为：
$\begin{gathered} \quad \begin{bmatrix} x & y & z & 1 \end{bmatrix} \begin{bmatrix} cos(\theta) & 0 & -sin(\theta) & 0 \\ 0 & 1 & 0 & 0 \\ sin(\theta) & 0 & cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} x \times cos(\theta) + z \times sin(\theta) & y & -x \times sin(\theta) + z \times cos(\theta) & 1 \end{bmatrix} \end{gathered}$

绕Z轴旋转

空间立体绕Z轴旋转某个角度，实体上个点的Z坐标不变，只有X、Y坐标改变，可表示为：
$\begin{gathered} \quad \begin{bmatrix} x & y & z & 1 \end{bmatrix} \begin{bmatrix} cos(\theta) & sin(\theta) & 0 & 0 \\ -sin(\theta) & cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} x \times cos(\theta) - y \times sin(\theta) & x \times sin(\theta) + y \times cos(\theta) & z & 1 \end{bmatrix} \end{gathered}$

python代码

绕X轴旋转

def rotate_X(x, y, z, alpha):
    alpha = alpha * (np.pi / 180)
    x_r = x
    y_r = np.cos(alpha) * y - np.sin(alpha) * z
    z_r = np.sin(alpha) * y + np.cos(alpha) * z
    return round(x_r, 4), round(y_r, 4), round(z_r, 4)

绕Y轴旋转

def rotate_Y(x, y, z, beta):
    beta = beta * (np.pi / 180)
    x_r = np.cos(beta) * x + np.sin(beta) * z
    y_r = y
    z_r = -np.sin(beta) * x + np.cos(beta) * z
    return round(x_r, 4), round(y_r, 4), round(z_r, 4)

绕Z轴旋转

def rotate_Z(x, y, z, gamma):
    gamma = gamma * (np.pi / 180)
    x_r = np.cos(gamma) * x - np.sin(gamma) * y
    y_r = np.sin(gamma) * x + np.cos(gamma) * y
    z_r = z
    return round(x_r, 4), round(y_r, 4), round(z_r, 4)

为了更直观的显示，使用matplotlib将旋转前后的点显示出来

def plot_3D(x, y, z, x1, y1, z1):
    a, b, c = [0, x, x1], \
              [0, y, y1], \
              [0, z, z1]
    d, e, f, g, h, i = [0], [0], [0], [0], [0], [0]
    d.append(x)
    e.append(y)
    f.append(z)
    g.append(x1)
    h.append(y1)
    i.append(z1)

    dd, ee, ff = [x, 0, 0, x, x, x], \
                 [y, y, 0, 0, y, y], \
                 [z, z, z, z, z, 0]

    dd1, ee1, ff1 = [0, x, x, 0, 0, 0], \
                    [0, 0, y, y, 0, 0], \
                    [0, 0, 0, 0, 0, z]
    dd2, ee2, ff2 = [x, x], \
                    [0, 0], \
                    [0, z]
    dd3, ee3, ff3 = [0, 0], \
                    [y, y], \
                    [0, z]

    gg, hh, ii = [x1, 0, 0, x1, x1, x1], \
                 [y1, y1, 0, 0, y1, y1], \
                 [z1, z1, z1, z1, z1, 0]
    gg1, hh1, ii1 = [0, x1, x1, 0, 0, 0], \
                    [0, 0, y1, y1, 0, 0], \
                    [0, 0, 0, 0, 0, z1]
    gg2, hh2, ii2 = [x1, x1], \
                    [0, 0], \
                    [0, z1]
    gg3, hh3, ii3 = [0, 0], \
                    [y1, y1], \
                    [0, z1]

    ax = plt.axes(projection='3d')  # 创建一个三维的绘图工程
    ax.scatter3D(a, b, c, c='r')  # 绘制数据点 c: 'r'红色，'y'黄色，等颜色
    ax.text(x, y, z, (x, y, z), c='r')  # 显示点坐标
    ax.text(x1, y1, z1, (x1, y1, z1), c='r')
    ax.plot3D(d, e, f, c='b')
    ax.plot3D(dd, ee, ff, c='b', linestyle='--')
    ax.plot3D(dd1, ee1, ff1, c='b', linestyle='--')
    ax.plot3D(dd2, ee2, ff2, c='b', linestyle='--')
    ax.plot3D(dd3, ee3, ff3, c='b', linestyle='--')
    ax.plot3D(g, h, i, c='y')
    ax.plot3D(gg, hh, ii, c='y', linestyle='--')
    ax.plot3D(gg1, hh1, ii1, c='y', linestyle='--')
    ax.plot3D(gg2, hh2, ii2, c='y', linestyle='--')
    ax.plot3D(gg3, hh3, ii3, c='y', linestyle='--')

    ax.set_xlabel('X')  # 设置x坐标轴
    ax.set_ylabel('Y')  # 设置y坐标轴
    ax.set_zlabel('Z')  # 设置z坐标轴
    ax.grid(False)  # 关闭网格

    plt.show()

现设初始点为P0（0，0，6），P0先绕X轴旋转6°，再绕Y轴旋转45°得到P2
空间坐标转换,python,线性代数
完整代码：

import numpy as np
import matplotlib.pyplot as plt


# 1、绕Z周旋转gamma角
def rotate_Z(x, y, z, gamma):
    gamma = gamma * (np.pi / 180)
    x_r = np.cos(gamma) * x - np.sin(gamma) * y
    y_r = np.sin(gamma) * x + np.cos(gamma) * y
    z_r = z
    return round(x_r, 4), round(y_r, 4), round(z_r, 4)


# 2、绕Y轴旋转beta角
def rotate_Y(x, y, z, beta):
    beta = beta * (np.pi / 180)
    x_r = np.cos(beta) * x + np.sin(beta) * z
    y_r = y
    z_r = -np.sin(beta) * x + np.cos(beta) * z
    return round(x_r, 4), round(y_r, 4), round(z_r, 4)


# 3、绕X轴旋转alpha角
def rotate_X(x, y, z, alpha):
    alpha = alpha * (np.pi / 180)
    x_r = x
    y_r = np.cos(alpha) * y - np.sin(alpha) * z
    z_r = np.sin(alpha) * y + np.cos(alpha) * z
    return round(x_r, 4), round(y_r, 4), round(z_r, 4)


def plot_3D(x, y, z, x1, y1, z1):
    a, b, c = [0, x, x1], \
              [0, y, y1], \
              [0, z, z1]
    d, e, f, g, h, i = [0], [0], [0], [0], [0], [0]
    d.append(x)
    e.append(y)
    f.append(z)
    g.append(x1)
    h.append(y1)
    i.append(z1)

    dd, ee, ff = [x, 0, 0, x, x, x], \
                 [y, y, 0, 0, y, y], \
                 [z, z, z, z, z, 0]

    dd1, ee1, ff1 = [0, x, x, 0, 0, 0], \
                    [0, 0, y, y, 0, 0], \
                    [0, 0, 0, 0, 0, z]
    dd2, ee2, ff2 = [x, x], \
                    [0, 0], \
                    [0, z]
    dd3, ee3, ff3 = [0, 0], \
                    [y, y], \
                    [0, z]

    gg, hh, ii = [x1, 0, 0, x1, x1, x1], \
                 [y1, y1, 0, 0, y1, y1], \
                 [z1, z1, z1, z1, z1, 0]
    gg1, hh1, ii1 = [0, x1, x1, 0, 0, 0], \
                    [0, 0, y1, y1, 0, 0], \
                    [0, 0, 0, 0, 0, z1]
    gg2, hh2, ii2 = [x1, x1], \
                    [0, 0], \
                    [0, z1]
    gg3, hh3, ii3 = [0, 0], \
                    [y1, y1], \
                    [0, z1]

    ax = plt.axes(projection='3d')  # 创建一个三维的绘图工程
    ax.scatter3D(a, b, c, c='r')  # 绘制数据点 c: 'r'红色，'y'黄色，等颜色
    ax.text(x, y, z, (x, y, z), c='r')  # 显示点坐标
    ax.text(x1, y1, z1, (x1, y1, z1), c='r')
    ax.plot3D(d, e, f, c='b')
    ax.plot3D(dd, ee, ff, c='b', linestyle='--')
    ax.plot3D(dd1, ee1, ff1, c='b', linestyle='--')
    ax.plot3D(dd2, ee2, ff2, c='b', linestyle='--')
    ax.plot3D(dd3, ee3, ff3, c='b', linestyle='--')
    ax.plot3D(g, h, i, c='y')
    ax.plot3D(gg, hh, ii, c='y', linestyle='--')
    ax.plot3D(gg1, hh1, ii1, c='y', linestyle='--')
    ax.plot3D(gg2, hh2, ii2, c='y', linestyle='--')
    ax.plot3D(gg3, hh3, ii3, c='y', linestyle='--')

    ax.set_xlabel('X')  # 设置x坐标轴
    ax.set_ylabel('Y')  # 设置y坐标轴
    ax.set_zlabel('Z')  # 设置z坐标轴
    ax.grid(False)  # 关闭网格

    plt.show()


if __name__ == '__main__':
    x, y, z = 0, 0, 6
    x1, y1, z1 = rotate_X(x, y, z, 6)
    x2, y2, z2 = rotate_Y(x1, y1, z1, 45)
    print(x2, y2, z2)
    plot_3D(x, y, z, x2, y2, z2)

C++代码：文章来源地址https://www.toymoban.com/news/detail-645235.html

#include<iostream>
#include <iomanip>
#include<math.h>
#define PAI acos(-1)

using namespace std;
void  rotate_Z(double x, double y, double z, double theta);
void  rotate_Y(double x, double y, double z, double theta);
void  rotate_X(double x, double y, double z, double theta);

 
int main() 
{
	double x = 0;
	double y = 0;
	double z = 6;
	rotate_Z(x,y,z,45);
	return 0;
}

void  rotate_Z(double x, double y, double z, double theta)
{
	double theta_r = theta * (PAI/180);
	double x_r = cos(theta_r) * x - sin(theta_r) * y;
	double y_r = sin(theta_r) * x + sin(theta_r) * y;
	double z_r = z;
	
	cout.setf(ios::fixed);
	cout << "X:" << setprecision(4) << x_r << endl;
	cout << "Y:" << setprecision(4) << y_r << endl;
	cout << "Z:" << setprecision(4) << z_r << endl;
 } 
 
void  rotate_Y(double x, double y, double z, double theta)
{
	double theta_r = theta * (PAI/180);
	double x_r = cos(theta_r) * x + sin(theta_r) * z;
	double y_r = y;
	double z_r = -sin(theta_r) * x + cos(theta_r) * z;
	
	cout.setf(ios::fixed);
	cout << "X:" << setprecision(4) << x_r << endl;
	cout << "Y:" << setprecision(4) << y_r << endl;
	cout << "Z:" << setprecision(4) << z_r << endl;
 } 
 
void  rotate_X(double x, double y, double z, double theta)
{
	double theta_r = theta * (PAI/180);
	double x_r = x;
	double y_r = cos(theta_r) * y - sin(theta_r) * z;
	double z_r = sin(theta_r) * y + cos(theta_r) * z;
	
	cout.setf(ios::fixed);
	cout << "X:" << setprecision(4) << x_r << endl;
	cout << "Y:" << setprecision(4) << y_r << endl;
	cout << "Z:" << setprecision(4) << z_r << endl;
 }