2D坐标系下的点的转换矩阵(平移、缩放、旋转、错切)

文章目录

        • 1. 平移 (Translation)
        • 2. 缩放 (Scaling)
        • 3. 旋转 (Rotation)
        • 4. 错切 (Shearing)
        • 5. 镜像 (Reflection)

1. 平移 (Translation)

在2D空间中,我们经常需要将一个点平移到另一个位置。假设空间中的一点 P ( x , y ) P(x,y) P(x,y);将其向 x , y x, y x,y方向分别平移 t x t_x tx t y t_y ty, 假设平移后点的坐标为 ( x ′ , y ′ ) (x',y') (x,y),则上述点的平移操作可以归纳为如下公式:
x ′ = x + t x y ′ = x + t y \begin{alignat}{2} &x'=x + t_x\\ &y'=x + t_y \end{alignat} x=x+txy=x+ty
使用齐次矩阵表示如下:
[ x ′ y ′ 1 ] = [ 1 b t x 0 1 t y 0 0 1 ] [ x y 1 ] \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & b & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} xy1 = 100b10txty1 xy1

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.]]
"""

动画效果演示

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. 缩放 (Scaling)

在2D空间中,点 ( x , y ) (x,y) (x,y)相对于另一点 ( p x , p y ) (p_x,p_y) (px,py)进行缩放操作,我们不妨缩放因子在 x , y x,y xy方向分别为: s x , s y s_x, s_y sx,sy, 则上述的缩放操作可以归纳为如下公式:
x ′ = s x ( x − p x ) + p x = s x x + p x ( 1 − s x ) y ′ = s y ( y − p y ) + p y = s y y + p y ( 1 − s y ) \begin{alignat}{2} &x'=s_x(x-p_x) + p_x &=s_xx + p_x(1-s_x)\\ &y'=s_y(y-p_y) + p_y &=s_yy + p_y(1-s_y) \end{alignat} x=sx(xpx)+pxy=sy(ypy)+py=sxx+px(1sx)=syy+py(1sy)
使用齐次矩阵表示如下:
[ x ′ y ′ 1 ] = [ s x 0 p x ( 1 − s x ) 0 s y p y ( 1 − s y ) 0 0 1 ] [ x y 1 ] \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} s_x& 0 & p_x(1-s_x) \\ 0 & s_y& p_y(1-s_y)\\ 0 & 0& 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} xy1 = sx000sy0px(1sx)py(1sy)1 xy1

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()

2D坐标系下的点的转换矩阵(平移、缩放、旋转、错切)_第1张图片

3. 旋转 (Rotation)

在2D空间中,对点 ( x , y ) (x,y) (x,y)相对于另一点 ( p x , p y ) (p_x,p_y) (px,py)进行旋转操作,一般来说逆时针为正,顺时针为负,假设旋转角度为 β \beta β, 则上述点 x , y x,y x,y 相对于点 p x , p y p_x,p_y px,py的旋转角度 β \beta β 的操作使用齐次矩阵表示如下:
[ x ′ y ′ 1 ] = [ cos ⁡ β − sin ⁡ β p x ( 1 − cos ⁡ β ) + p y sin ⁡ β sin ⁡ β cos ⁡ β p y ( 1 − cos ⁡ β ) + p x sin ⁡ β 0 0 1 ] [ x y 1 ] \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} \cos \beta& -\sin \beta & p_x(1-\cos \beta) + p_y \sin \beta \\ \sin \beta & \cos \beta& p_y(1-\cos \beta) + p_x \sin \beta \\ 0 & 0& 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} xy1 = cosβsinβ0sinβcosβ0px(1cosβ)+pysinβpy(1cosβ)+pxsinβ1 xy1

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. 错切 (Shearing)

在2D空间中,对点 ( x , y ) (x,y) (x,y) 相对于另一点 ( p x , p y ) (p_x,p_y) (px,py)进行错切操作,错切一般用于弹性物体的变形处理。 假设沿x 方向与y方向错切参数分别为 λ x , λ y \lambda _x,\lambda _y λxλy, 则的错切操作可以归纳使用齐次矩阵表示如下:

[ x ′ y ′ 1 ] = [ 1 λ x − λ x p x λ y 1 − λ y p y 0 0 1 ] [ x y 1 ] \begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1& \lambda _x & -\lambda _x p_x \\ \lambda _y & 1& -\lambda _y p_y \\ 0 & 0& 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} xy1 = 1λy0λx10λxpxλypy1 xy1

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()

2D坐标系下的点的转换矩阵(平移、缩放、旋转、错切)_第2张图片

5. 镜像 (Reflection)

对于镜像,对称轴的法向量 v ( v x , v y ) v(v_x,v_y) v(vx,vy), 镜像矩阵 T m T_{m} Tm表示为:
[ 1 − 2 x v 2 − 2 x v y v 0 − 2 x v y v 1 − 2 y v 2 0 0 0 1 ] \left[ \begin{array}{ccc} 1-2 x_{v}{ }^{2} & -2 x_{v} y_{v} & 0 \\ -2 x_{v} y_{v} & 1-2 y_{v}{ }^{2} & 0 \\ 0 & 0 & 1 \end{array} \right] 12xv22xvyv02xvyv12yv20001
另外就是需要一个表示对称轴位置的点(对称轴2点中任意一点),表示为 M ( x m , y m ) M\left(x_{\mathrm{m}}, y_{m}\right) M(xm,ym),变换矩阵 H = T t ∗ T m ∗ T t − 1 H = T_{t}*T_{m}*T_{t}^{-1} H=TtTmTt1,即:
H = [ 1 0 0 0 1 0 x m y m 1 ] [ 1 − 2 x v 2 − 2 x v y v 0 − 2 x v y v 1 − 2 y v 2 0 0 0 1 ] [ 1 0 0 0 1 0 − x m − y m 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} 1-2 x_{v}{ }^{2} & -2 x_{v} y_{v} & 0 \\ -2 x_{v} y_{v} & 1-2 y_{v}{ }^{2} & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -x_{\mathrm{m}} & -y_{m} & 1 \end{array} \right] H= 10xm01ym001 12xv22xvyv02xvyv12yv20001 10xm01ym001
镜像后的坐标即为 T o ∗ T t ∗ T m ∗ T t − 1 T_{o}*T_{t}*T_{m}*T_{t}^{-1} ToTtTmTt1

沿 X 轴 镜像的镜像矩阵:
[ 1 0 0 0 − 1 0 1 0 1 ] \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 1 & 0 & 1 \end{array} \right] 101010001
沿 Y 轴 镜像的镜像矩阵:
[ − 1 0 0 0 1 0 0 1 1 ] \left[ \begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{array} \right] 100011001

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))

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

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