趣题:下一根枕木应该画在哪儿?

一位画家正在画画。画布上是一望无际的平原,一条笔直的铁路向无限远的地方延伸。画家画了铁路上的两根相邻的枕木,它们在画面上呈两条平行的线段,并且都与地平线平行。这时,画家突然犯难了:根据透视的原理,下一根枕木应该画在哪儿呢?你能帮他确定出下一根枕木的位置吗?

这里,我们假设陆地是一个无限大的平面,并且铁路上的相邻枕木之间的间距相等。

 

 

 

 

 

 

 

 

 

假设两根枕木分别是 AB 和 CD 。容易想到,如果地面上的两条直线交于地平线处,就说明这两条直线是平行的。另外,注意到相邻枕木之间构成了一个个全等的矩形,它们的对角线应该是平行的。于是,我们就有了这样一个方案:延长 BC 与地平线交于点 P ,然后连接 PD ,与左边的铁轨线交于 E 。现在,只需要作一条与已有枕木平行的线段 EF 。线段 EF 就是铁路中的下一根枕木。

这个问题来自 Ross Honsberger 的 More Mathematical Morsels 一书。

36 条评论

  • beautifultable

    前排围观

  • lushl9301

    ji zhi + qiao miao

  • resty

    透视法基本原理啊

  • Dr. What

    画那么长的线误差容易变很大……
    一个简单的办法是,选取CD中点O,延长BO,AO,分别交两射线于E、F……不知是不是省事点。

    • Dr. What

      相对而言,原文所使用的方法在枕木与地平线不平行的时候更方便些,因为不平行的时候中点在透视图里就不在中点的位置了。只不过原方法需要加上枕木的消失点,多了一倍的辅助线。
      测试一下html语法……

    • 喵小黑

      画画肯定都是这么画的。。只要俩铁轨是平行线- –

  • 野鹤

    又一道好题
    Dr. What 的方法也不错哦

  • Pegasus

    透视学里简单的东西..没接触过的人初看有点转不过弯的很正常..接触过的就好多了..
    其实很多东西都是酱紫….

  • !!!帅呆了一天一更

  • 太好了 是几何- –

  • 维尔

    失踪回来以后要日更嘛?hhh
    最近试着画了画埃舍尔球,Curvilinear perspective 什么的……
    现在觉得一点透视简直是天使啊!

  • godultimate24

    你一日一更,我天天好心情,坚持啊!大神

  • Erin

    终于更了!前排⊙▽⊙

  • 辰尘

    学设计的表示对于这种简单的透视原理,看到的第一眼就想到答案了。。。。

    • dr what

      既然这题目太简单了,那换个难点的:
      假如纸面太小了,地平线在画面之外,画面里只能容下ABCDEF几个点,现在已知ABCD并且AB和CD并不与地平线平行,那么怎样只在纸面内作图确定E与F的位置?

  • moonshiningo

    BE与CD的交点就是CD的中点

  • ruosen

    abcd打个叉子交点G,轨道尽头H,GH与CD相交点就是中点

  • 辰尘

    求解,我们这种情况一般都是靠感觉到。。。。

  • Spectrum

    一楼竟然不是localhost~

  • 方程

    我把问题想复杂了……

  • woody

    【这里,我们假设陆地是一个无限大的平面】无限大的平面还存在地平线,我感觉挺别扭。地球的地平线,那里确实有个明确的界限,因为地面是凸出来的,类似的有山与天的界限,但这个无限大平面怎么理解……

    • woody

      当然我知道,假如站在无限大平面上,低头看见地,抬头看见天,没别的东西,于是两者一定“相接”。但我还是难以想象地平线的样子。

  • Kingium

    如果考虑实际作图的误差的话……假设我只有非常精准的量角器和不太精准的直尺,那怎么画误差更小呢?

    • Latesha

      All things coedisnred, this is a first class post

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  • Ian

    回樓上, 這題只需要一把沒有刻度的直尺就可以完成

  • TFGIRLS队长

    Soga

  • Hello math

    那两根木头不是平行吧

  • williamszhu

    假设平行直线交与地平线处,那么木枕之间作平行线不就是违背了假设吗

  • Bourbon Fu

    有问题 有问题 有问题
    我请教了学美术的同学
    这个题及其不严谨 一句话解释就是
    此图为俯视图!!!!!
    而且图没有给出严格的灭点 消失点2 3
    如果你再从左右两边同时确定枕木 多画几条 会发现 图中的枕木根本不平行 好吗
    与枕木是相互平行的的事实 相悖
    所以 题目是错的 希望博主可以好好请教一下专业同学 民科害死人

  • Bourbon Fu

    希望博主可以及时回应我

  • ppppp

    这道题的本质是第四调和点吧
    楼上的这个是可以从数学的角度证明都是平行的 应该是没有问题的吧

  • Antonio

    这很显然,因为透视不改变这四点成调和点列

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