2 lines
32 KiB
JavaScript
2 lines
32 KiB
JavaScript
import{_ as x,c,b as m,d as l,o as j}from"./app-BdNP3b8G.js";const a="/images/2015/01/ellipse01.gif",t="/images/2015/01/ellipse02.gif",e={};function n(i,s){return j(),c("div",null,s[0]||(s[0]=[m("h1",{id:"一道数学趣题",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#一道数学趣题"},[m("span",null,"一道数学趣题")])],-1),m("p",null,[l("在微博上看到一道很有意思的数学问题，原题是："),m("strong",null,"如果椭圆游泳池有一英尺宽的边缘，问：边缘外围是否仍为椭圆？"),l(" 这个问题背后还有个八卦故事，数学家"),m("a",{href:"http://en.wikipedia.org/wiki/Steven_Strogatz",target:"_blank",rel:"noopener noreferrer"},"Steven Strogatz"),l("是非线性动力学大师级人物，广为人知的是他和自己的高中数学老师Don Joffray有着深厚的友谊，这位老师是把他带入数学殿堂的引路人，一次他在接受采访时，Strogatz讲到自己和这位老师的故事，他们即使在毕业后也一直保持着联系，一开始是他写信请教老师问题，但转折点就是这个“elliptical pool”问题，老师第一次被问住了，反而是他给老师解释，这令他激动不已。"),m("br"),l(" 有趣的问题就是这样，看起来足够简单，却要费一番脑筋才能想清楚。这里先定义一个概念，“一英尺宽的边缘”的数学含义，是指在椭圆的每个点的法线方向上扩展一定的长度，微博上另一位博主给出了一个很相像的动态图来描述，这里直接借用一下：")],-1),m("figure",null,[m("img",{src:a,alt:"",tabindex:"0",loading:"lazy"}),m("figcaption")],-1),m("p",null,"这个问题有两种解决思路，一种是纯粹从数学公式入手，这里给出一个解法，首先对于任意一个椭圆，用参数函数表达：",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mi",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c73"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2061"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D703 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