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 ( $\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ )，其经过缩放后变成一个新向量 ( $\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$ )。如果缩放因子是 ( $a$ )，那么我们得到：

$y = a x$

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

$\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}$

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 ( $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ )，如果缩放因子 ( $a = 2$ )，那么：

$\times \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$

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 ( $\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ )，并且膨胀矩阵 ( $D$ ) 是一个对角矩阵：

$\begin{pmatrix} d_1 & 0 \\ 0 & d_2 \end{pmatrix}$

则变换后的向量是：

$\begin{pmatrix} d_1 x_1 \\ d_2 x_2 \end{pmatrix}$

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 ( $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ )，膨胀矩阵是 ( $\begin{pmatrix} 2 & 0 \\ 0 & 0.5 \end{pmatrix}$ )，那么变换后的向量是：

$\begin{pmatrix} 2 & 0 \\ 0 & 0.5 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \\ 1.5 \end{pmatrix}$

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 ( $\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ )，旋转后的向量 ( $y$ ) 可以通过以下矩阵与向量相乘得到：

$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$

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 ( $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ )，我们将其逆时针旋转 90 度，即 ( $\theta = \frac{\pi}{2}$ )。那么旋转矩阵为：

$\begin{pmatrix} \cos\frac{\pi}{2} & -\sin\frac{\pi}{2} \\ \sin\frac{\pi}{2} & \cos\frac{\pi}{2} \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

旋转后的向量是：

$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

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

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：

$\begin{pmatrix} \cos(2\theta) & \sin(2\theta) \\ \sin(2\theta) & -\cos(2\theta) \end{pmatrix} x$

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 ( $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ )，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：

$\begin{pmatrix} \cos\frac{\pi}{2} & \sin\frac{\pi}{2} \\ \sin\frac{\pi}{2} & -\cos\frac{\pi}{2} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

反射后的向量是：

$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$

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 ( $\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ ), and after scaling, it becomes a new vector ( $\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$ ). If the scaling factor is ( $a$ ), then the transformation is:

$y = a x$

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

$\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}$

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 ( $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ ), and the scaling factor ( $a = 2$ ). Then:

$\times \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$

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 ( $\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ ) undergoes dilation by a diagonal matrix ( $D$ ):

$\begin{pmatrix} d_1 & 0 \\ 0 & d_2 \end{pmatrix}$

Then the transformed vector ( $y$ ) is:

$\begin{pmatrix} d_1 x_1 \\ d_2 x_2 \end{pmatrix}$

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 ( $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ ), and the dilation matrix is ( $\begin{pmatrix} 2 & 0 \\ 0 & 0.5 \end{pmatrix}$ ). Then:

$\begin{pmatrix} 2 & 0 \\ 0 & 0.5 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \\ 1.5 \end{pmatrix}$

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 ( $\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ ). After rotating by ( $\theta$ ) radians counterclockwise, the transformed vector ( $y$ ) is given by:

$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$

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 ( $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ ), and we want to rotate it counterclockwise by 90 degrees, or ( $\theta = \frac{\pi}{2}$ ). The rotation matrix is:

$\begin{pmatrix} \cos\frac{\pi}{2} & -\sin\frac{\pi}{2} \\ \sin\frac{\pi}{2} & \cos\frac{\pi}{2} \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

The rotated vector is:

$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

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

4. Reflection

Reflection is the transformation that mirrors a vector across a line through the origin. Suppose we reflect the vector ( $\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ ) across a line inclined at an angle ( $\theta$ ) with respect to the horizontal axis. The reflection matrix is:

$\begin{pmatrix} \cos(2\theta) & \sin(2\theta) \\ \sin(2\theta) & -\cos(2\theta) \end{pmatrix} x$

Effects of Reflection:

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

Example:

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

$\begin{pmatrix} \cos\frac{\pi}{2} & \sin\frac{\pi}{2} \\ \sin\frac{\pi}{2} & -\cos\frac{\pi}{2} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

The reflected vector is:

$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$

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!