空間坐標(biāo)變換可看作坐標(biāo)點(diǎn)乘以一個(gè)齊次矩陣，其中，齊次矩陣可表示為：

其中：
①區(qū)域的3×3矩陣產(chǎn)生三維圖形的比例、對(duì)稱、旋轉(zhuǎn)、錯(cuò)切等基本變換；
②區(qū)域產(chǎn)生圖形的透視變換；
③區(qū)域產(chǎn)生沿X、Y、Z三個(gè)軸的平移變換；
④區(qū)域產(chǎn)生圖形的總比例變換。

平移變換

平移變換可表示為：
[ x y z 1 ] [ 1 0 0 0 0 1 0 0 0 0 1 0 l m n 1 ] = [ x + l y + m z + n 1 ] \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} [x?y?z?1?] ?100l?010m?001n?0001? ?=[x+l?y+m?z+n?1?]?

繞X軸旋轉(zhuǎn)

空間立體繞X軸旋轉(zhuǎn)某個(gè)角度，實(shí)體上個(gè)點(diǎn)的X坐標(biāo)不變，只有Y、Z坐標(biāo)改變，可表示為：
[ x y z 1 ] [ 1 0 0 0 0 c o s ( θ ) s i n ( θ ) 0 0 ? s i n ( θ ) c o s ( θ ) 0 0 0 0 1 ] = [ x y × c o s ( θ ) ? z × s i n ( θ ) y × s i n ( θ ) + z × c o s ( θ ) 1 ] \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} [x?y?z?1?] ?1000?0cos(θ)?sin(θ)0?0sin(θ)cos(θ)0?0001? ?=[x?y×cos(θ)?z×sin(θ)?y×sin(θ)+z×cos(θ)?1?]?

繞Y軸旋轉(zhuǎn)

空間立體繞Y軸旋轉(zhuǎn)某個(gè)角度，實(shí)體上個(gè)點(diǎn)的Y坐標(biāo)不變，只有X、Z坐標(biāo)改變，可表示為：
[ x y z 1 ] [ c o s ( θ ) 0 ? s i n ( θ ) 0 0 1 0 0 s i n ( θ ) 0 c o s ( θ ) 0 0 0 0 1 ] = [ x × c o s ( θ ) + z × s i n ( θ ) y ? x × s i n ( θ ) + z × c o s ( θ ) 1 ] \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} [x?y?z?1?] ?cos(θ)0sin(θ)0?0100??sin(θ)0cos(θ)0?0001? ?=[x×cos(θ)+z×sin(θ)?y??x×sin(θ)+z×cos(θ)?1?]?

繞Z軸旋轉(zhuǎn)

空間立體繞Z軸旋轉(zhuǎn)某個(gè)角度，實(shí)體上個(gè)點(diǎn)的Z坐標(biāo)不變，只有X、Y坐標(biāo)改變，可表示為：
[ x y z 1 ] [ c o s ( θ ) s i n ( θ ) 0 0 ? s i n ( θ ) c o s ( θ ) 0 0 0 0 1 0 0 0 0 1 ] = [ x × c o s ( θ ) ? y × s i n ( θ ) x × s i n ( θ ) + y × c o s ( θ ) z 1 ] \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} [x?y?z?1?] ?cos(θ)?sin(θ)00?sin(θ)cos(θ)00?0010?0001? ?=[x×cos(θ)?y×sin(θ)?x×sin(θ)+y×cos(θ)?z?1?]?

python代碼

繞X軸旋轉(zhuǎn)

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)

繞Y軸旋轉(zhuǎn)

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)

繞Z軸旋轉(zhuǎn)

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)

為了更直觀的顯示，使用matplotlib將旋轉(zhuǎn)前后的點(diǎn)顯示出來

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')  # 創(chuàng)建一個(gè)三維的繪圖工程
    ax.scatter3D(a, b, c, c='r')  # 繪制數(shù)據(jù)點(diǎn) c: 'r'紅色，'y'黃色，等顏色
    ax.text(x, y, z, (x, y, z), c='r')  # 顯示點(diǎn)坐標(biāo)
    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')  # 設(shè)置x坐標(biāo)軸
    ax.set_ylabel('Y')  # 設(shè)置y坐標(biāo)軸
    ax.set_zlabel('Z')  # 設(shè)置z坐標(biāo)軸
    ax.grid(False)  # 關(guān)閉網(wǎng)格

    plt.show()

現(xiàn)設(shè)初始點(diǎn)為P0（0，0，6），P0先繞X軸旋轉(zhuǎn)6°，再繞Y軸旋轉(zhuǎn)45°得到P2

完整代碼：

import numpy as np
import matplotlib.pyplot as plt


# 1、繞Z周旋轉(zhuǎn)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、繞Y軸旋轉(zhuǎn)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、繞X軸旋轉(zhuǎn)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')  # 創(chuàng)建一個(gè)三維的繪圖工程
    ax.scatter3D(a, b, c, c='r')  # 繪制數(shù)據(jù)點(diǎn) c: 'r'紅色，'y'黃色，等顏色
    ax.text(x, y, z, (x, y, z), c='r')  # 顯示點(diǎn)坐標(biāo)
    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')  # 設(shè)置x坐標(biāo)軸
    ax.set_ylabel('Y')  # 設(shè)置y坐標(biāo)軸
    ax.set_zlabel('Z')  # 設(shè)置z坐標(biāo)軸
    ax.grid(False)  # 關(guān)閉網(wǎng)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++代碼：文章來源地址http://www.zghlxwxcb.cn/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;
 } 

到了這里，關(guān)于空間坐標(biāo)變換(Python&C++實(shí)現(xiàn))的文章就介紹完了。如果您還想了解更多內(nèi)容，請(qǐng)?jiān)谟疑辖撬阉鱐OY模板網(wǎng)以前的文章或繼續(xù)瀏覽下面的相關(guān)文章，希望大家以后多多支持TOY模板網(wǎng)！