矩阵变换：Scaling、Dilation、Rotation 和 Reflection对应的中文是什么？（中英双语）

中文版

矩阵变换：Scaling、Dilation、Rotation 和 Reflection

在二维或三维空间中，矩阵变换是一种通过矩阵与向量相乘，来实现从一个点到另一个点的映射过程。今天，我们将深入探讨四种常见的几何变换：Scaling（缩放）、Dilation（膨胀）、Rotation（旋转） 和 Reflection（反射），并通过矩阵和简单的例子来理解这些变换。

1. Scaling（缩放）

缩放是指通过一个标量因子对向量进行放大或缩小的操作。假设我们有一个二维向量 ( $x=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$ )，其经过缩放后变成一个新向量 ( $y=\left(\begin{array}{c}{y}_1\\ y_2\end{array}\right)$ )。如果缩放因子是 ( $a$ )，那么我们得到：

$$y=ax$$

这里，矩阵 ( $A$ ) 就是 ( $aI$ )，其中 ( $I$ ) 是单位矩阵，表示：

$$A=\left(\begin{array}{cc}a& 0\\ 0& a\end{array}\right)$$

缩放的效果：

• 如果 ( $|a|>1$ )，那么向量会被放大。
• 如果 ( $|a|<1$ )，那么向量会被缩小。
• 如果 ( $a<0$ )，向量的方向会反转。

举个例子：

假设原始向量是 ( $x=\left(\begin{array}{c}2\\ 3\end{array}\right)$ )，如果缩放因子 ( $a=2$ )，那么：

$$y=2\times \left(\begin{array}{c}2\\ 3\end{array}\right)=\left(\begin{array}{c}4\\ 6\end{array}\right)$$

可以看到，原来的向量被放大了。

2. Dilation（膨胀）

膨胀变换是一种沿着不同轴分别伸缩的操作。假设我们有一个二维向量 ( $x=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$ )，并且膨胀矩阵 ( $D$ ) 是一个对角矩阵：

$$D=\left(\begin{array}{cc}{d}_1& 0\\ 0& d_2\end{array}\right)$$

则变换后的向量是：

$$y=Dx=\left(\begin{array}{c}{d}_1{x}_1\\ d_2{x}_2\end{array}\right)$$

膨胀的效果：

• 如果 ( $|d_1|>1$ )，则沿 ( $x_1$ ) 轴放大，反之缩小。
• 如果 ( $|d_2|>1$ )，则沿 ( $x_2$ ) 轴放大，反之缩小。
• 如果 ( $d_1$ ) 或 ( $d_2$ ) 为负数，向量会在相应的轴上翻转。

举个例子：

假设原始向量是 ( $x=\left(\begin{array}{c}2\\ 3\end{array}\right)$ )，膨胀矩阵是 ( $D=\left(\begin{array}{cc}2& 0\\ 0& 0.5\end{array}\right)$ )，那么变换后的向量是：

$$y=\left(\begin{array}{cc}2& 0\\ 0& 0.5\end{array}\right)\left(\begin{array}{c}2\\ 3\end{array}\right)=\left(\begin{array}{c}4\\ 1.5\end{array}\right)$$

可以看到，沿 ( $x_1$ ) 轴放大了，而沿 ( $x_2$ ) 轴缩小了。

3. Rotation（旋转）

旋转变换是指将向量绕原点旋转一个角度 ( $\theta$ )。假设原始向量是 ( $x=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$ )，旋转后的向量 ( $y$ ) 可以通过以下矩阵与向量相乘得到：

$$y=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$$

这是一个旋转矩阵，它表示向量绕原点旋转 ( $\theta$ ) 弧度。

旋转的效果：

• 向量 ( $x$ ) 被绕原点顺时针或逆时针旋转。
• 旋转矩阵可以通过不同的角度 ( $\theta$ ) 来调整旋转的方向和幅度。

举个例子：

假设原始向量是 ( $x=\left(\begin{array}{c}1\\ 0\end{array}\right)$ )，我们将其逆时针旋转 90 度，即 ( $\theta =\frac{\pi }{2}$ )。那么旋转矩阵为：

$$\left(\begin{array}{cc}\cos \frac{\pi }{2}& -\sin \frac{\pi }{2}\\ \sin \frac{\pi }{2}& \cos \frac{\pi }{2}\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$$

旋转后的向量是：

$$y=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 1\end{array}\right)$$

向量成功地绕原点旋转了 90 度，变成了 ( $\left(\begin{array}{c}0\\ 1\end{array}\right)$ )。

4. Reflection（反射）

反射变换是指将向量通过某个指定的直线反射。假设我们要将向量 ( $x$ ) 反射到一个通过原点的直线上，该直线与水平轴的夹角为 ( $\theta$ )。反射矩阵为：

$$y=\left(\begin{array}{cc}\cos (2\theta )& \sin (2\theta )\\ \sin (2\theta )& -\cos (2\theta )\end{array}\right)x$$

反射的效果：

• 向量会沿着指定的直线反射，反射后的方向与原来的方向对称。

举个例子：

假设原始向量是 ( $x=\left(\begin{array}{c}1\\ 1\end{array}\right)$ )，我们希望通过与水平轴成 45 度角的直线反射，即 ( $\theta =\frac{\pi }{4}$ )，则反射矩阵为：

$$\left(\begin{array}{cc}\cos \frac{\pi }{2}& \sin \frac{\pi }{2}\\ \sin \frac{\pi }{2}& -\cos \frac{\pi }{2}\end{array}\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

反射后的向量是：

$$y=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}1\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ 1\end{array}\right)$$

虽然反射的结果在此例中与原始向量相同，但如果角度不同，反射后向量将改变。

总结

通过矩阵与向量相乘，我们可以实现多种几何变换。理解这些变换有助于我们在计算机图形学、机器人学等领域进行更复杂的空间变换。

英文版

Matrix Transformations: Scaling, Dilation, Rotation, and Reflection

In 2D or 3D space, matrix transformations provide a way to map points from one position to another by multiplying a matrix with a vector. In this post, we’ll explore four common geometric transformations: Scaling, Dilation, Rotation, and Reflection, and explain each concept with simple examples.

1. Scaling

Scaling refers to resizing a vector by a scalar factor. Suppose we have a 2D vector ( $x=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$ ), and after scaling, it becomes a new vector ( $y=\left(\begin{array}{c}{y}_1\\ y_2\end{array}\right)$ ). If the scaling factor is ( $a$ ), then the transformation is:

$$y=ax$$

In matrix form, this is represented by ( $A=aI$ ), where ( $I$ ) is the identity matrix:

$$A=\left(\begin{array}{cc}a& 0\\ 0& a\end{array}\right)$$

Effects of Scaling:

• If ( $|a|>1$ ), the vector is stretched (scaled up).
• If ( $|a|<1$ ), the vector is shrunk (scaled down).
• If ($a<0$ ), the direction of the vector is reversed (flipped).

Example:

Suppose the original vector is ( $x=\left(\begin{array}{c}2\\ 3\end{array}\right)$ ), and the scaling factor ( $a=2$ ). Then:

$$y=2\times \left(\begin{array}{c}2\\ 3\end{array}\right)=\left(\begin{array}{c}4\\ 6\end{array}\right)$$

The vector has been scaled up by a factor of 2.

2. Dilation

Dilation is a transformation that stretches the vector by different factors along each axis. Suppose the vector ( $x=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$ ) undergoes dilation by a diagonal matrix ( $D$ ):

$$D=\left(\begin{array}{cc}{d}_1& 0\\ 0& d_2\end{array}\right)$$

Then the transformed vector ( $y$ ) is:

$$y=Dx=\left(\begin{array}{c}{d}_1{x}_1\\ d_2{x}_2\end{array}\right)$$

Effects of Dilation:

• If ( $|d_1|>1$ ), the vector is stretched along the ( $x_1$ )-axis.
• If ( $|d_2|>1$ ), the vector is stretched along the ( $x_2$ )-axis.
• If ( $d_1$ ) or ( $d_2$ ) is negative, the vector is flipped along the corresponding axis.

Example:

Suppose the original vector is ( $x=\left(\begin{array}{c}2\\ 3\end{array}\right)$ ), and the dilation matrix is ( $D=\left(\begin{array}{cc}2& 0\\ 0& 0.5\end{array}\right)$ ). Then:

$$y=\left(\begin{array}{cc}2& 0\\ 0& 0.5\end{array}\right)\left(\begin{array}{c}2\\ 3\end{array}\right)=\left(\begin{array}{c}4\\ 1.5\end{array}\right)$$

Here, the vector is stretched along the ( $x_1$ )-axis and shrunk along the ( $x_2$ )-axis.

3. Rotation

Rotation refers to rotating a vector around the origin by an angle ( $\theta$ ). Suppose the original vector is ( $x=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$ ). After rotating by ( $\theta$ ) radians counterclockwise, the transformed vector ( $y$ ) is given by:

$$y=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$$

This is called the rotation matrix.

Effects of Rotation:

• The vector ( $x$ ) is rotated counterclockwise by the angle ( $\theta$ ) around the origin.

Example:

Suppose the original vector is ( $x=\left(\begin{array}{c}1\\ 0\end{array}\right)$ ), and we want to rotate it counterclockwise by 90 degrees, or ( $\theta =\frac{\pi }{2}$ ). The rotation matrix is:

$$\left(\begin{array}{cc}\cos \frac{\pi }{2}& -\sin \frac{\pi }{2}\\ \sin \frac{\pi }{2}& \cos \frac{\pi }{2}\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$$

The rotated vector is:

$$y=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 1\end{array}\right)$$

The vector has been rotated 90 degrees counterclockwise, resulting in ( $\left(\begin{array}{c}0\\ 1\end{array}\right)$ ).

4. Reflection

Reflection is the transformation that mirrors a vector across a line through the origin. Suppose we reflect the vector ( $x=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$ ) across a line inclined at an angle ( $\theta$ ) with respect to the horizontal axis. The reflection matrix is:

$$y=\left(\begin{array}{cc}\cos (2\theta )& \sin (2\theta )\\ \sin (2\theta )& -\cos (2\theta )\end{array}\right)x$$

Effects of Reflection:

• The vector is reflected across the line at an angle ( $\theta$ ), creating a mirror image.

Example:

Suppose the original vector is ( $x=\left(\begin{array}{c}1\\ 1\end{array}\right)$ ), and we want to reflect it across a line at ( $\theta =\frac{\pi }{4}$ ). The reflection matrix is:

$$\left(\begin{array}{cc}\cos \frac{\pi }{2}& \sin \frac{\pi }{2}\\ \sin \frac{\pi }{2}& -\cos \frac{\pi }{2}\end{array}\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

The reflected vector is:

$$y=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}1\\ 1\end{array}\right)=\left(\begin{array}{c}1\\ 1\end{array}\right)$$

In this specific example, the reflected vector is the same as the original one, but for other angles, the reflection would result in a new direction.

Conclusion

By multiplying matrices with vectors, we can perform a variety of geometric transformations. Understanding these transformations is essential for fields like computer graphics, robotics, and physics, where spatial transformations are commonly used. With this post, you should have a clearer understanding of Scaling, Dilation, Rotation, and Reflection matrices and how they can be applied to manipulate vectors in space!

后记

2024年12月18日22点52分于上海，在GPT4o大模型辅助下完成。