平板的变换

Transformation is a process that changes the shape, size or position of a figure to create a new image. It is a function that takes points in the plane as inputs and gives other points as outputs. You can think of a transformation as a rule that tells you how to create new points.
::转换是一个改变图的形状、 大小或位置以创建新图像的过程。 它是一个函数, 将平面中的点作为输入点, 并将其他点作为输出点。 您可以将转换视为一条规则, 告诉您如何创建新点 。

Suppose you have a transformation $\begin{align*}F\end{align*}$ that applies a horizontal  stretch factor of two   to each point . Below, this transformation is applied to triangle $\begin{align*}S\end{align*}$ to create triangle $\begin{align*}S^\prime.\end{align*}$
::假设您有一个变换 F, 将水平伸展因子为 2 应用到每个点。 下面, 此变换将应用到三角 S 上, 以创建三角 S 。

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• $\begin{align*}S^\prime\end{align*}$ is considered the image of $\begin{align*}S\end{align*}$ by $\begin{align*}F.\end{align*}$  
  ::S ' 被视为F的S 图像。
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• It is also correct to  say that $\begin{align*}F\end{align*}$ maps $\begin{align*}S\end{align*}$ to $\begin{align*}S'.\end{align*}$  
  ::说F向S-S-地图也是正确的。
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• Each of the points in the image are labeled with a  $\begin{align*}'\end{align*}$  symbol, which is read as "prime."
  ::图像中的每一个点都贴上符号的标签, 符号被读为“ 原始 ” 。

This helps to show how points on $\begin{align*}S\end{align*}$ correspond to points on $\begin{align*}S^\prime.\end{align*}$  For example, you could say that "point $\begin{align*}A\end{align*}$ maps to point $\begin{align*}A\end{align*}$ -prime."
::这有助于显示 S 上的点与 S 的点对应。 例如, 您可以说“ A 点地图 指向 A 点 ” 。

Some transformations preserve length and angles . Preserving length means that if a line segment is 3 units, its image will also be 3 units. Similarly, preserving angles means if an angle is $\begin{align*}60^{\text{o}}\!,\end{align*}$  its image will also be $\begin{align*}60^{\text{o}}\!.\end{align*}$
::有些变形保留长度和角度。 保留长度意味着如果一个线段为3个单位, 其图像也将是3个单位。 同样, 保存角度意味着如果一个角度为60度, 其图像也将是60度 。

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• A transformation that preserves length and angles is called a rigid transformation.
  ::保持长度和角度的变换 被称为僵硬变换

Recognizing Rigid Transformations
::硬硬变换

1. Is a horizontal stretch an example of a rigid transformation?
::1. 横向伸展是否是僵硬转变的一个例子?

No. You can prove this using the picture above by showing that length is not preserved .
::否。您可以通过显示长度未保留来用以上图片来证明这一点。

 2. A transformation reflects points in shape $\begin{align*}K\end{align*}$ across $\begin{align*}\overleftrightarrow{AB}\end{align*}$ to create shape $\begin{align*}K^\prime\end{align*}$ . Is this reflection a rigid transformation?
::2. 转换反映了横跨AB的K型形状,以形成K形形状。

Yes, reflections are rigid transformations. You can verify that the distances between the points are preserved.
::是的,反射是僵硬的变形。你可以验证点之间的距离是否保持。

3. A transformation translates points in shape $\begin{align*}K\end{align*}$ along vector $\begin{align*}\vec{v}\end{align*}$ to create shape $\begin{align*}K^\prime.\end{align*}$  Is this translation a rigid transformation?
::3. 转换将矢量 v 上的 K 形状的点转换为形状, 以创建 K 形状 。 这是否翻译为刻板的转换 ?

Yes, translations are rigid transformations. You can verify that the distances between the points are preserved.
::是的,翻译是硬质转换。 您可以验证点之间的距离是否被保存 。

Examples
::实例实例实例实例

Example 1
::例1

You slide a book across your desk. You pour soda from a can into a big glass.  Describe these actions as  transformations.
::你把一本书滑过桌子。你把苏打水从罐子倒到一个大杯子。把这些动作描述为转换。

Sliding a book across your desk is a rigid transformation because a book is a rigid object that does not change shape. The distances and angles that make up the book don't change once the book is in a new location. Pouring soda, on the other hand, is not a rigid transformation. Liquid is not a rigid object and it can change shape depending on its surroundings. The overall shape of the soda in the can will be different from the overall shape of the soda in the glass.
::在您的办公桌上滑动一本书是一个僵硬的转换, 因为一本书是一个不改变形状的僵硬对象。 构成这本书的距离和角度一旦在新位置, 就不会改变。 另一方面, 投注苏打水并不是一个僵硬的转换。 液体不是一个僵硬的物体, 它可以根据周围环境改变形状。 罐中苏打水的整体形状将与玻璃中苏打水的整体形状不同 。

A transformation rotates points in shape $\begin{align*}K\end{align*}$ around point $\begin{align*}D\end{align*}$ to create shape $\begin{align*}K^\prime.\end{align*}$  Does this rotation look like a rigid transformation? Use algebra to prove your answer.
::转换旋转点在 D 点周围的形状 K 旋转点以创建形状 K 。 此旋转看起来像是僵硬的变换吗? 使用代数来证明您的回答 。

This appears to be a rigid transformation.
::这似乎是一种僵硬的转变。

With the Pythagorean Theorem , you can show that the corresponding sides are the same length . For example:
::Pytagorean Theorem, 您可以显示对应的两边长度相同。 例如 :

$\begin{align*}& \text{The length of } \overline{AC} \\ & = \sqrt{3^2 + 2^2} \\ & = \sqrt{9 +4} \\ & = \sqrt{13} \ \text{units} \end{align*}$
::AC =32+22=9+4=13 单位的长度

$\begin{align*}& \text{The length of } \overline{A'C'} \\ & = \sqrt{3^2 + 2^2} \\ & = \sqrt{9 + 4} \\ & = \sqrt{13} \ \text{units}\end{align*}$
::A{C}32+22=9+4=13单位的长度

Thus it is indeed a rigid transformation.
::因此,这确实是一种僵硬的转变。

Example 3
::例3

What makes a transformation a rigid transformation?
::改变是什么使僵硬的转变?

Rigid transformations preserve distance and angles. All corresponding sides will be the same length and all will be the same measure .
::硬质变换保持距离和角度。 所有对应的侧面的长度都相同, 大小都相同 。