Transformation matrix of points in 2D coordinate system (translation, scaling, rotation, miscutting)

1. Translation

In 2D space, we often need to translate one point to another. Suppose a point P ( x , y ) P(x,y) in the space$P(x,y)$ ; direct it to$x,$Translatetx t_x$in\; the\; y\; direction$$t_x$，$t_y$, assuming that the coordinates of the point after translation are $(x^{\prime },y^{\prime })$, then the translation operation of the above points can be summarized as the following formula:
$$\begin{array}{rl}& x^{\prime }=x+t_x\\ & y^{\prime }=x+t_y\end{array}$$
Using a homogeneous matrix is ​​expressed as follows:
$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{ccc}1& b& t_x\\ 0& 1& t_y\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

import numpy as np


def translation():
    """
    原始数组a 三个点(1,1) (4,4) (7,7)
    构建齐次矩阵 P
    构建变换矩阵 T
    """
    a = np.array([[1, 1],
               [4, 4],
               [7, 7]])

    P = np.array([a[:, 0],
               a[:, 1],
               np.ones(len(a))])

    T = np.array([[1, 0, 2],
               [0, 1, 2],
               [0, 0, 1]])
    return np.dot(T, P)


print(translation())

"""
[[3. 6. 9.]
 [3. 6. 9.]
 [1. 1. 1.]]
"""

Animation Effect Demonstration

import matplotlib
import matplotlib.pyplot as plt
import numpy as np

X, Y = np.mgrid[0:1:5j, 0:1:5j]
x, y = X.ravel(), Y.ravel()

def trans_translate(x, y, tx, ty):
    T = [[1, 0, tx],
         [0, 1, ty],
         [0, 0, 1]]
    T = np.array(T)
    P = np.array([x, y, [1] * x.size])
    return np.dot(T, P)

fig, ax = plt.subplots(1, 4)
T_ = [[0, 0], [2.3, 0], [0, 1.7], [2, 2]]
for i in range(4):
    tx, ty = T_[i]
    x_, y_, _ = trans_translate(x, y, tx, ty)
    ax[i].scatter(x_, y_)
    ax[i].set_title(r'$t_x={0:.2f}$ , $t_y={1:.2f}$'.format(tx, ty))

    ax[i].set_xlim([-0.5, 4])
    ax[i].set_ylim([-0.5, 4])
    ax[i].grid(alpha=0.5)
    ax[i].axhline(y=0, color='k')
    ax[i].axvline(x=0, color='k')
plt.show()

2. Scaling

In 2D space, a point $(x,y)$ relative to another point$(p_x,p_y)$ for scaling operation, we might as well scale the factor in$The\; xandy$ directions are respectively:$s_x,s_y$, then the above scaling operation can be summarized as the following formula:
$$\begin{array}{rlr}& x^{\prime }=s_x(x-p_x)+p_x& =s_{x}x+p_x(1-s_x)\\ & y^{\prime }=s_y(y-p_y)+p_y& =s_{y}y+p_y(1-s_y)\end{array}$$
Using a homogeneous matrix is ​​expressed as follows:
$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{ccc}{s}_x& 0& p_x(1-s_x)\\ 0& s_y& p_y(1-s_y)\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

def trans_scale(x, y, px, py, sx, sy):
    T = [[sx, 0 , px*(1 - sx)],
         [0 , sy, py*(1 - sy)],
         [0 , 0 , 1          ]]
    T = np.array(T)
    P = np.array([x, y, [1]*x.size])
    return np.dot(T, P)

fig, ax = plt.subplots(1, 4)
S_ = [[1, 1], [1.8, 1], [1, 1.7], [2, 2]]
P_ = [[0, 0], [0, 0], [0.45, 0.45], [1.1, 1.1]]
for i in range(4):
    sx, sy = S_[i]; px, py = P_[i]
    x_, y_, _ = trans_scale(x, y, px, py, sx, sy)
    ax[i].scatter(x_, y_)
    ax[i].scatter(px, py)
    ax[i].set_title(r'$p_x={0:.2f}$ , $p_y={1:.2f}$'.format(px, py) + '\n'
                    r'$s_x={0:.2f}$ , $s_y={1:.2f}$'.format(sx, sy))
    
    ax[i].set_xlim([-2, 2])
    ax[i].set_ylim([-2, 2])
    ax[i].grid(alpha=0.5)
    ax[i].axhline(y=0, color='k')
    ax[i].axvline(x=0, color='k')

plt.show()

3. Rotation

In 2D space, for a point $(x,y)$ relative to another point$(p_x,p_y)$ for rotation operation, generally counterclockwise is positive, clockwise is negative, assuming the rotation angle is$\beta$ , then the above points$x,y$ relative to point$p_x,p_y$The rotation angle $Define\; the\; \beta$ function as a function:
$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{ccc}\cos b& -\sin b& p_x(1-\cos b)+p_y\sin b\\ \sin b& \cos b& p_y(1-\cos b)+p_x\sin b\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

def trans_rotate(x, y, px, py, beta):
    beta = np.deg2rad(beta)
    T = [[np.cos(beta), -np.sin(beta), px*(1 - np.cos(beta)) + py*np.sin(beta)],
         [np.sin(beta),  np.cos(beta), py*(1 - np.cos(beta)) - px*np.sin(beta)],
         [0           ,  0           , 1                                      ]]
    T = np.array(T)
    P = np.array([x, y, [1]*x.size])
    return np.dot(T, P)

fig, ax = plt.subplots(1, 4)

R_ = [0, 225, 40, -10]
P_ = [[0, 0], [0, 0], [0.5, -0.5], [1.1, 1.1]]

for i in range(4):
    beta = R_[i]; px, py = P_[i]
    x_, y_, _ = trans_rotate(x, y, px, py, beta)
    ax[i].scatter(x_, y_)
    ax[i].scatter(px, py)
    ax[i].set_title(r'$\beta={0}°$ , $p_x={1:.2f}$ , $p_y={2:.2f}$'.format(beta, px, py))
    
    ax[i].set_xlim([-2, 2])
    ax[i].set_ylim([-2, 2])
    ax[i].grid(alpha=0.5)
    ax[i].axhline(y=0, color='k')
    ax[i].axvline(x=0, color='k')

plt.show（）

4. Shearing

In 2D space, for a point $(x,y)$ relative to another point$(p_x,p_y)$ for miscut operation, which is generally used for deformation processing of elastic objects. Assume that the miscutting parameters along the x direction and y direction are$l_x,l_y$, then the miscutting operation can be summarized and expressed as a homogeneous matrix as follows:

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{ccc}1& l_x& -l_x{p}_x\\ l_y& 1& -l_y{p}_y\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

import matplotlib.pyplot as plt
import numpy as np


X, Y = np.mgrid[0:1:5j, 0:1:5j]
x, y = X.ravel(), Y.ravel()

def trans_shear(x, y, px, py, lambdax, lambday):
    T = [[1      , lambdax, -lambdax*px],
         [lambday, 1      , -lambday*py],
         [0      , 0      ,  1         ]]
    T = np.array(T)
    P = np.array([x, y, [1]*x.size])
    return np.dot(T, P)

fig, ax = plt.subplots(1, 4)

L_ = [[0, 0], [2, 0], [0, -2], [-2, -2]]
P_ = [[0, 0], [0, 0], [0, 1.5], [1.1, 1.1]]

for i in range(4):
    lambdax, lambday = L_[i]; px, py = P_[i]
    x_, y_, _ = trans_shear(x, y, px, py, lambdax, lambday)
    ax[i].scatter(x_, y_)
    ax[i].scatter(px, py)
    ax[i].set_title(r'$p_x={0:.2f}$ , $p_y={1:.2f}$'.format(px, py) + '\n'
                    r'$\lambda_x={0:.2f}$ , $\lambda_y={1:.2f}$'.format(lambdax, lambday))

    ax[i].set_xlim([-3, 3])
    ax[i].set_ylim([-3, 3])
    ax[i].grid(alpha=0.5)
    ax[i].axhline(y=0, color='k')
    ax[i].axvline(x=0, color='k')

plt.show()

5. Reflection

For mirroring, the normal vector v of the axis of symmetry $v(v_x,v_y)$ , the mirror matrix$T_m$表示为：
$$\left[\begin{array}{ccc}1-2x{\_}_v{}^2& -2x\_{\_}_v{y}_v& 0\\ -2x\_{\_}_v{y}_v& 1-2y_v{}^2& 0\\ 0& 0& 1\end{array}\right]$$
In addition, a point is needed to represent the position of the symmetry axis (any point in the 2 points of the symmetry axis), expressed as $M\left(x_{\mathrm{m}},y_m\right)$ , transformation matrix$H=T_t\ast T_m\ast T_t^{-1}$:
$$H=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ x_{\mathrm{m}}& y_m& 1\end{array}\right]\left[\begin{array}{ccc}1-2x{\_}_v{}^2& -2x\_{\_}_v{y}_v& 0\\ -2x\_{\_}_v{y}_v& 1-2y_v{}^2& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -x_m& -y_m& 1\end{array}\right]$$
The coordinates after mirroring are $T_o\ast T_t\ast T_m\ast T_t^{-1}$。

Mirror matrix for mirroring along the X
$$\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 1& 0& 1\end{array}\right]$$
Mirror matrix for mirroring along the Y
$$\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 1& 1\end{array}\right]$$

import numpy as np


a = np.array([[1, 2],
              [2, 2],
              [3, 5],
              [4, 6]])
a = np.array([a[:,0], a[:,1], np.ones(len(a))])
print("\n",a)
print("--------------------------------")

T_x = np.array( [[ 1,  0,  0],
                 [ 0, -1,  0],
                 [ 1,  0,  1]])
print("\n",np.dot(T_x, a))
print("=================================")
T_y = np.array( [[-1,  0,  0],
                 [ 0,  1,  0],
                 [ 0,  1,  1]])
print("\n",np.dot(T_y, a))

Reference:
https://zhuanlan.zhihu.com/p/387578291
https://zhuanlan.zhihu.com/p/187411029
https://blog.csdn.net/Akiyama_sou/article/details/122144415