章节大纲

  • 常规

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  • 翻译和反思顺序

    Rigid Motion in Robotics
    ::机器人的僵硬动作

    Every day, robots are becoming more ingrained in our society. Robots  are used in assembly processes, in law enforcement, and even to explore other planets. To create robots that can perform the same real-world tasks as humans, the computer scientists and engineers who design these robots need to have a strong understanding of rigid motion. You may not realize it, but every second your brain is processing and executing translations, reflections , and rotations . While you don't need a deep knowledge of mathematics to personally take a step in a given direction or to turn around,  you do need a deep knowledge of mathematics to program robots to those things.
    ::机器人在我们的社会中越来越根深蒂固。机器人被用于组装过程、执法甚至探索其他行星。为了创造能够像人类一样执行真实世界任务的机器人,设计这些机器人的计算机科学家和工程师需要深刻理解僵硬运动。你可能没有意识到这一点,但你的大脑每秒钟都在处理和执行翻译、反射和旋转。虽然你不需要深奥的数学知识来亲自向某个特定方向迈出一步,或者转过身,但你确实需要深入的数学知识来将机器人编程到那些东西上。

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    Mars Rover Curiosity

    One such robot is  NASA's Curiosity rover, which explores the surface of Mars transmitting videos and geological data. One of the biggest challenges faced by NASA is the delay when sending a signal from Earth to Mars.  I t can take  the Curiosity team up to about 24 minutes to tell the rover where to go, and it takes up to 20 hours for the rover to transmit the video of its movement back to Earth. Every day the curiosity team members have to program the rover to move around the Martian terrain , without being able to see the movement until the next day . Due to this delay, they need to carefully calculate sequences of translations and rotations in advance that will allow the rover to cover greater distances given the limited communication window . 
    ::其中一个机器人就是美国航天局的Curiosity Rover, 它探索火星的表面, 传送视频和地质数据。 美国航天局面临的最大挑战之一是从地球向火星发送信号的延迟。 它可以花大约24分钟的时间告诉巡游者到哪里去, 巡游者需要多达20小时的时间才能将巡游的视频传送回地球。 好奇小组成员每天必须让巡游者在火星的地形上移动, 无法在第二天之前看到行踪。 由于这一延误, 他们需要事先仔细计算翻译和轮换的顺序, 以便让巡游者在有限的通信窗口中能够覆盖更远的距离 。

     

    Sequences of Translations
    ::翻译顺序

    You've  explored  translations (sometimes called slides), that move an image a certain distance in a specific direction without changing  its size or shape. There is no limit to the number of times an image can translate. However, no matter how many times an image is translated, there is always one equivalent translation that could have been performed to move it from the starting location straight to the ending location.
    ::您已经探索过翻译( 有时称为幻灯片) , 它将图像移动到特定方向而不改变其大小或形状 。 一个图像可以翻译的次数没有限制 。 但是, 无论一个图像翻译多少次, 总是可以进行一个相当的翻译, 才能将图像从起始位置直接移到结束位置 。

    Every time you perform a translation, the letters assigned to the vertices in new image will get an apostrophe. When A B C  is translated, the new image will be labeled A ′ B ′ C ′ . If it is translated a second time, the new image will be labeled A ″ B ″ C ″  .
    ::每次您执行翻译时, 新图像中指定给顶端的字母将获得一个代号。 当 ABC 翻译时, 新图像将被标为 A_B_C_. 如果第二次翻译, 新图像将被标为 A_B_C_ 。

    The interactive below will allow you to perform a sequence of translations on the pre-image.  You can then click a button to see the equivalent translation to go from A B C  straight to  A ″ B ″ C ″ .
    ::下面的交互效果将允许您在预映像上进行一系列翻译。 您可以单击一个按钮查看相应的翻译, 从ABC直接到 ABC。

     

     

    What is the equivalent translation when you place  A ″ B ″ C ″   directly   on top of  A B C ?
    ::当您直接将 ABC直接放在ABC 顶端时, 对应的翻译是什么 ?

     

    Congruence Confirmation
    ::一致性确认

    You already know  that when a translation takes place, the image and pre-image are congruent (the same size and shape). Is the same true when two translations take place? What about two different rigid motions, such as a translation and a reflection ?
    ::您已经知道, 当翻译发生时, 图像和图像预映是相同的( 大小和形状相同 ) 。 当两次翻译发生时, 图像和图像预映是否都一样 ? 那么两种不同的僵硬动作, 比如翻译和反射, 怎么样 ?

    The  interactive below shows a shape which has undergone a translation and a reflection.
    ::下面的交互显示是经过翻译和反射的形状。

     

     

    Discussion Question
    ::讨论问题

    Is the image produced by a series of rigid motions congruent to its pre-image? How do you know? for more examples of translations and reflections!
    ::由一系列的僵硬动议产生的图像是否与其预视图像一致? 你怎么知道? 更多翻译和反思的例子!

    Use the interactive below to determine whether the image and pre-image are congruent by using a sequence of translations and reflections to map one shape onto another.
    ::使用下面的交互式图像来确定图像和预映像是否一致,方法是使用一系列翻译和反射来将一个形状映射到另一个形状。

     

     

     

    Glide Reflections
    ::倾角反射

    A glide reflection is  a sequence with a translation and a reflection.  Glide reflections are commonly used to create decorative patterns in architecture, woodworking, art, and many other fields. These patterns are often created by using translations to repeat the glide reflections. When a glide reflection repeats along a straight line , it is called a frieze pattern.
    ::滑翔反射是一个带有翻译和反射的序列。 滑翔反射通常用于在建筑、木工、艺术和其他许多字段中创建装饰性模式。 这些模式通常是通过翻译来重复滑翔反射而形成的。 当滑翔反射沿直线重复时, 它被称为碎屑模式。

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    F rieze patterns  are commonly used in architecture  where the term 'frieze' refers to  a painted or sculpted pattern that runs along the base of a ceiling. Friezes were an important part of ancient Greek architecture where they were used to decorate temples.
    ::Friezes是古希腊建筑的重要部分,用来装饰寺庙。

    Use the interactive below to create your frieze pattern which could run along the base of a ceiling.
    ::使用下面的交互 来创建你的碎裂模式 它可以沿着天花板的基底运行

     

     

        Summary
    ::摘要

    • With each transformation in a sequence, the letters assigned to the vertices in new image will get an apostrophe.
      ::随着每个转换的顺序, 分配给新图像中的顶端的字母将会有一个误差 。
    • A n image and its pre-image are always congruent after a rigid motion,  therefore they are also congruent after a sequence of rigid motions.
      ::刻板动议之后,图像及其预视图像总是一致的,因此,在一连串的僵硬动议之后,图像和预视图像也是一致的。
    • A glide reflection is a rigid motion sequence with a translation and a reflection.
      ::滑翔反射是一种僵硬的动作顺序,有翻译和反射。