平面或空间中任意点的旋转

自己琢磨出来的。若有错误请指出。谢谢！

rotate

1. 旋转2D

假设平面上有一个点\(P(x, y)\)，旋转任意角度\(\beta\)，求旋转后的点\(P'(x',y')\)。

设平面坐标系上有一个半径为\(r\)的圆，圆心位于原点\(O\)。圆与\(x\)轴正坐标的交点为\(P_0(x_0, y_0)\)。

\[P_0 = \begin{cases} x_0 = r\\ y_0 = 0\\ \end{cases} \]

将点\(P_0\)旋转任意角度\(\alpha\)弧度得任意点\(P_1(x_1, y_1)\)，那么点\(P_1(x_1, y_1)\)的坐标该怎么通过计算获得？

\[P_1=\begin{cases} x_1 = r \times cos(\alpha)\\ y_1 = r \times sin(\alpha)\\ \end{cases} \]

现在将另一任意点\(P_2(x_2, y_2)\)相对于点\(P_1(x_1, y_1)\)旋转任意角度\(\beta\)，那么点\(P_2(x_2, y_2)\)的坐标该怎么通过计算获得？

\[P_2 = \begin{cases} x_2 = r \times cos(\alpha+\beta)\\ y_2 = r \times sin(\alpha+\beta)\\ \end{cases} \]

使用三角函数的和角公式把点\(P_2\)的坐标公式展开得：

\[P_2=\begin{cases} x_2=r \times cos(\alpha+\beta) = r \times cos\alpha \times cos\beta - r \times sin\alpha \times sin\beta\\ y_2=r \times sin(\alpha+\beta) = r \times sin\alpha \times cos\beta + r \times cos\alpha \times sin\beta \end{cases} \]

把\(P_1\)得坐标带入上面\(P_2\)的坐标展开式，则有：

\[P_2=\begin{cases} x_2=x_1 \times cos\beta - y_1 \times sin\beta\\ y_2=y_1 \times cos\beta + x_1 \times sin\beta\\ \end{cases} \]

总结：对于平面上任意点\(P(x, y)\)，旋转任意角度\(\beta\)后，得到的点\(P'(x', y')\)为：

\[P'=\begin{cases} x'=x \times cos\beta - y \times sin\beta\\ y'=y \times cos\beta + x \times sin\beta\\ \end{cases} \]

三角函数和差公式

\[sin(\alpha+\beta)=sin\alpha \cdot cos\beta + cos\alpha \cdot sin\beta\\ sin(\alpha-\beta)=sin\alpha \cdot cos\beta - cos\alpha \cdot sin\beta\\ cos(\alpha+\beta)=cos\alpha \cdot cos\beta - sin\alpha \cdot sin\beta\\ cos(\alpha-\beta)=cos\alpha \cdot cos\beta + sin\alpha \cdot sin\beta\\ \]

2. 旋转3D

假设平面上有一个点\(P(x, y, z)\)，绕\(x\)轴、\(y\)轴和\(z\)轴相对于原点旋转任意角度\(\alpha\)、\(β\)和\(\gamma\)，求旋转后的点\(P'(x',y', z')\)。

按照类似于2D的情况来考虑此问题。该问题可以分解为分别在\(yz\)平面，\(xz\)平面和\(xy\)平面内相对于原点旋转。那么，有以下结论：

在\(yz\)平面内绕\(x\)轴相对于原点旋转\(\alpha\)角度旋转：

\[\begin{align} &x'=x\\ &y'=y \times cos\alpha-z \times sin\alpha\\ &z'=z \times cos\alpha+z \times sin\alpha\\ \end{align} \]

在\(xz\)平面内绕\(y\)轴相对于原点旋转\(\beta\)角度旋转：

\[\begin{align} &x'=x \times cos\beta-z \times sin\beta\\ &y'=y \\ &z'=z \times cos\beta+x \times sin\beta\\ \end{align} \]

在\(xy\)平面内绕\(z\)轴相对于原点旋转\(\gamma\)角度旋转：

\[\begin{align} &x'=x \times cos\gamma-y \times sin\gamma\\ &y'=y \times cos\gamma+x \times sin\gamma\\ &z'=z\\ \end{align} \]

注：绕\(z\)轴旋转，理解为点\(P\)在\(xy\)平面内的投影，并绕原点旋转。其余的情况类似。

3. 代码

这里使用Python语言来编写上面两种旋转的代码，如下：

import math


def rotate_2d(x: int | float,
              y: int | float,
              rad: int | float) -> (float, float):
    """
    Rotate a point around orientation at any given radian.

    :param x: X scale of point.
    :param y: Y scale of point.
    :param rad: Radian value to be rotated.
    :return: A new point has been rotated by rad radian.
    :raise: None.

    """
    rad = float(rad)

    r_x: float = (x * math.cos(rad)) - (y * math.sin(rad))
    r_y: float = (y * math.cos(rad)) + (x * math.sin(rad))

    return r_x, r_y


def rotate_3d(x: int | float,
              y: int | float,
              z: int | float,
              ax: float,
              ay: float,
              az: float) -> (float, float, float):
    """
    Rotate a point in 3d space around orientation at any given radian.

    :param x: X scale of point.
    :param y: Y scale of point.
    :param z: Z scale of point.
    :param ax: Radian value to be rotated around orientation x.
    :param ay: Radian value to be rotated around orientation y.
    :param az: Radian value to be rotated around orientation z.
    :return: A new point has been rotated by rad radian.
    :raise: None.

    """
    ax = float(ax)
    ay = float(ay)
    az = float(az)

    # rotate around axis x
    r_x = x
    r_y = y * math.cos(ax) - z * math.sin(ax)
    r_z = z * math.cos(ax) + y * math.sin(ax)
    x, y, z = r_x, r_y, r_z

    # rotate around axis y
    r_y = y
    r_x = x * math.cos(ay) - z * math.sin(ay)
    r_z = z * math.cos(ay) + x * math.sin(ay)
    x, y, z = r_x, r_y, r_z

    # rotate around axis z
    r_z = z
    r_x = x * math.cos(az) - y * math.sin(az)
    r_y = y * math.cos(az) + x * math.sin(az)

    return r_x, r_y, r_z


def radian(degree: int | float):
    """
    Convert radian value in to radian value.

    :param degree: Degree value to be converted.
    :return: Radian value.
    :raise: None.
    """
    return degree * math.pi / 180.0


if __name__ == "__main__":

    def main():
        x = 1.0
        y = 0.0

        for i in range(360+1):
            p = rotate_2d(x, y, radian(i))
            print("{:>3d} degree, ".format(i), '(x: {:+.3f}, y:{:+.3f})'.format(*p))

        return


    main()