11.18刚性变换

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  刚性变换

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  [Figure 1]

  
  ::[图1]

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  Newton is planning to build a tiny house as a music studio in his back yard. But when he shows his dad the site plans, his dad points out a few problems with the design. As it's drawn, the tiny house will get all the run-off from the house roof, and it's oriented with the long axis with most of the windows facing East-West, which means that the house will almost always be too hot. How should Newton change his design to fix these problems?
  ::牛顿计划在其后院建造一栋小房子作为音乐工作室。但是当他向父亲展示现场计划时,他父亲指出设计上的一些问题。正如所画的那样,小房子会从房顶上得到所有径流,而且它与东-西面大部分窗户的长轴相对应,这意味着房子几乎总是太热。牛顿应该如何改变设计来解决这些问题呢?

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  In this concept, you will learn how to recognize different transformations of rigid shapes on a coordinate  plane .
  ::在这个概念中,你会学会如何识别 坐标平面上硬形的不同变形。

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  Recognizing Rigid Transformations
  ::硬硬变换

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  The  x-axis  is the bold  line  running from left to right on the coordinate plane, and it is usually labeled with an "x". The x-coordinate of an ordered pair is found with relation to it.
  ::X 轴是坐标平面上从左向右运行的粗线,通常贴有“x”的标签。

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  The  y-axis  is the central line that runs up-down and is labeled with a "y". Y-coordinates are plotted in reference to this axis. 
  ::Y 轴是自上而下并标有“y”标签的中心线。 Y 坐标是参照此轴绘制的。

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  The   vertex  of a shape is the place where two sides of the shape come together. 
  ::形状的顶端是形状的两面聚集在一起的地方。

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  A  transformation  is the movement of a figure on the coordinate grid. There are three different types of transformations: translation,  reflection , and  rotation .
  ::变换是指坐标网格上数字的移动。变换有三种不同的变换类型:翻译、反射和轮换。

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  A  translation  is when a figure appears to slide from one place to another, but its orientation remains the same. 
  ::翻译是指一个数字似乎从一个地方滑到另一个地方,但其方向保持不变。

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  Here is a translation.
  ::这是翻译。

  

  [Figure 2]

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  You can see here that the figure was not changed at all. It simply slid from one  point  to another point.
  ::您可以在这里看到这个数字根本没有改变。 它只是从一个点滑到另一个点。

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  A  reflection  is a flip of a figure. A mirror image is one common example. Another is the image you see in the lake or a spoon. When a figure is reflected on the coordinate grid, it is flipped across an axis or a  line of symmetry . Here is a reflection.
  ::反射是图的翻转。镜像图像是一个常见的例子。另一个是您在湖中或勺子中看到的图像。当一个图在坐标网格上被反射时,它会翻过一个轴或对称线。这里是一个反射。

  

  [Figure 3]

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  In this example, the y-axis is the line of symmetry because the shape is a mirror image on the other side of that line.
  ::在此示例中, Y 轴是对称线, 因为形状是线另一侧的镜像图像 。

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  A  rotation  is a turn on the coordinate grid. Here is an example.
  ::旋转是坐标网格的转弯。这里举一个例子。

  

  [Figure 4]

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  The easiest way to identify what kind of transformation has taken place is to draw both the original and the transformed shapes on a grid and to compare them. If the transformed shape has the same orientation as the original, then it is a translation. If the orientation has changed and it's a mirror image, then it's a reflection. And if it has turned on its side, then it is a rotation.
  ::最容易确定发生了何种转变的方法就是在网格上同时绘制原始形状和变形形状并进行比较。 如果变形形状与原始形状具有相同的方向, 那么它就是一个翻译。 如果方向已经改变, 它是一个镜像图像, 那么它就是一个反射。 如果它转过来了, 那么它就是一个旋转。

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  Here is an example.
  ::举一个例子。

  

  [Figure 5]

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  In this picture,  triangle   $\begin{array}{r}ABC\end{array}$  is the original figure. What kind of transformation is  $\begin{array}{r}{A}^{\prime }{B}^{\prime }{C}^{\prime }\end{array}$ ?
  ::在这张图中,三角形A B C是原始数字。什么类型的转换是 A = B = C ?

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  First, determine whether the shape has changed its orientation.
  ::首先,确定形状是否改变了方向。

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  In this case, it has not. So this is an example of a translation.
  ::在这种情况下,它没有。所以这是翻译的例子。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Newton and his back yard tiny house.
  ::早些时候,你得到一个问题 关于牛顿和他的后院小房子。

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  His father told him that his house would get leaked on and would overheat if he put it where he planned to. So he has to figure out how to move the house so that it isn't under the house eaves and so that the long axis of the house isn't facing East-West.
  ::他父亲告诉他,如果他把房子放在他计划的地方,他的房子就会被漏掉,而且会过热。所以他必须想出如何把房子搬走,以免房子在屋梯下面,以免房子的长轴向东西方。

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  Newton breaks his problem into two parts: the water problem and the orientation problem.
  ::牛顿将问题分成两部分:水问题和定向问题。

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  First, he focuses on the water problem. In order to get it out from under the eaves of the big house roof, he just needs to move the tiny house away from the building. That is easy. He just translates the tiny house several feet away from the big house on his site plans.
  ::首先,他专注于水问题。为了从大屋顶的地下取出水,他只需要把小房子搬离大楼。这很简单。他只是将小房子翻译为离他工地计划上的大房子几英尺远。

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  Next, he turns to the overheating problem. He concludes that this is a problem of orientation. As long as the house is oriented East-West, it will get too much sun, so he has to change the orientation. But if he were to just flip the house, it would still be oriented East-West. So the only option is for him to rotate the plan until the long axis of his house is oriented North-South.
  ::接下来,他转而谈论过热的问题。他的结论是,这是一个方向问题。只要房子面向东西,太阳就会过热,所以他必须改变方向。但如果他只是翻转房子,它仍然会面向东西。因此,唯一的选择是让他旋转计划,直到他房子的长轴面向南北。

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  The answer is Newton must first translate and then rotate his tiny house.
  ::答案是牛顿必须首先翻译 然后旋转他的小房子

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  Example 2
  ::例2

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  [Figure 6]

  
  ::[图6]

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  In the photo above, tell what kind of transformation this is, and explain why.
  ::在以上照片中,请说明这是什么样的转变,并解释原因。

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  First, describe what the picture is about.
  ::首先,描述一下画面的本质。

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  In this picture a group of carved chess pieces are seen standing up and then, again, upside down in the glass tabletop..
  ::在这张照片中,看到一群雕刻的象棋碎片站起来,然后又倒在玻璃桌上。

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  Next, determine whether the orientation of anything in the photo has changed.
  ::接下来,确定照片中任何内容的取向是否已经改变。

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  The pieces are upside down, so their orientation has changed.
  ::碎片颠倒了,所以它们的方向改变了。

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  Then, determine whether there is a line of symmetry across which the pieces are flipped.
  ::然后,确定是否有一行对称线, 将碎片翻转。

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  In this case, there is. The line of symmetry is the line between the board and the water. The pieces appear the same on the other side but flipped.
  ::在这种情况下,有。对称线是板和水之间的线。碎片在另一侧看起来相同,但被翻了。

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  The answer is reflection.
  ::答案是反省。

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  Identify each of the following examples as a rotation, translation or reflection.
  ::将下列各例作为轮流、翻译或反思的实例。

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  Example 3
  ::例3

  

  [Figure 7]

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  First, determine whether the orientation of the shape has changed.
  ::首先,确定形状的方向是否已经改变。

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  In this case, it has, so it is not a translation.
  ::在这种情况下,它有,所以它不是翻译。

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  Next, determine how the orientation has changed. Is the shape on its side or upside down?
  ::接下来,决定方向是如何变化的。是形状在它的侧面还是颠倒的?

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  In this case, it is on its side. So it is not a reflection.
  ::在这种情况下,它是站在它一边的,所以不是倒影。

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  The answer is rotation.
  ::答案是轮换。

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  Example 4
  ::例4

  

  [Figure 8]

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  First, determine whether the shape has changed its orientation.
  ::首先,确定形状是否改变了方向。

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  It has not. So it is a translation.
  ::它没有,所以它是一个翻译。

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  The answer is translation.
  ::答案是翻译。

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  Example 5
  ::例5

  

  [Figure 9]

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  First, determine what the  congruent  shape in the image is.
  ::首先,确定图像的相容形状。

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  In this case, the shape in question is the wing of the butterfly.
  ::在这种情况下,有关形状是蝴蝶的翅膀。

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  Next, determine if both butterfly wings are the same orientation.
  ::接下来,确定两个蝴蝶翅膀是否方向相同

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  They are not.
  ::他们不是。

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  Then, determine how the orientation has changed. Is the other wing flipped or is it laid on its side?
  ::然后,确定方向是如何改变的。另一翼是翻转的呢?还是侧着的呢?

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  In this case, it is flipped across an axis of symmetry--the butterfly body--so it is reflected.
  ::在此情况下,它会翻过对称轴-蝴蝶体-的轴, 所以它就会被反射出来。

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  The answer is reflection.
  
  ::答案是反省。

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  Review
  ::回顾

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  Identify each image as a translation, rotation or reflection.
  ::将每个图像识别为翻译、旋转或反射。

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  1. 

     [Figure 10]
     
     ::[图10]
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  2. 

     [Figure 11]
     
     ::[表11]
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  3. 

     [Figure 12]
     
     ::[图12]
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  4. 

     [Figure 13]
     
     ::[图13]
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  5. 

     [Figure 14]
     
     ::[表14]

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  For numbers 6 – 12 Draw your own figures, and demonstrate three different translations, three different rotations and three different reflections.
  ::对于数字 6 - 12 绘制您自己的数字, 并显示三种不同的翻译, 三种不同的旋转和三种不同的反省 。

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  Complete the following reflections according to the directions by drawing your answers to questions 13 and 14.
  ::回答问题13和14时,根据方向完成以下思考。

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  13. Reflect this image across the  $\begin{array}{r}y\end{array}$  axis.
      ::将图像反射到 Y 轴上。

  

  [Figure 15]

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  14. Rotate the following image.
      ::旋转以下图像。

  

  [Figure 16]

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  15 – 20 Go on a scavenger hunt and find examples of each of the different types of transformations in real world contexts.
  ::15 - 20 继续寻找寻宝者, 寻找真实世界中各种变化的例子。

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  Review (Answers) 
  ::回顾(答复)

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  Click   to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源