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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)),T[43]||(T[43]=Q("figure",null,[Q("img",{src:d,alt:"",tabindex:"0",loading:"lazy"}),Q("figcaption")],-1)),T[44]||(T[44]=Q("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)),Q("mjx-container",s,[(m(),t("svg",o,T[0]||(T[0]=[l('',1)]))),T[1]||(T[1]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"x"),Q("mo",null,"="),Q("mi",null,"a"),Q("mi",null,"cos"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mi",null,"θ"),Q("mo",null,","),Q("mstyle",{scriptlevel:"0"},[Q("mspace",{width:"1em"})]),Q("mi",null,"y"),Q("mo",null,"="),Q("mi",null,"b"),Q("mi",null,"sin"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mi",null,"θ"),Q("mo",null,","),Q("mstyle",{scriptlevel:"0"},[Q("mspace",{width:"1em"})]),Q("mn",null,"0"),Q("mo",null,"≤"),Q("mi",null,"θ"),Q("mo",null,"≤"),Q("mn",null,"2"),Q("mi",null,"π")])],-1))]),Q("p",null,[T[6]||(T[6]=a("利用微分知识可知，对于平面上任意一个连续的曲线函数，如果其参数方程表达为")),Q("mjx-container",g,[(m(),t("svg",H,T[2]||(T[2]=[l('',1)]))),T[3]||(T[3]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"x"),Q("mo",null,"="),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,","),Q("mi",null,"y"),Q("mo",null,"="),Q("mi",null,"y"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),T[7]||(T[7]=a(",那么在任意点")),Q("mjx-container",h,[(m(),t("svg",p,T[4]||(T[4]=[l('',1)]))),T[5]||(T[5]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msub",null,[Q("mi",null,"P"),Q("mn",null,"0")])])],-1))]),T[8]||(T[8]=a("点处的法线方程可以表达为"))]),Q("mjx-container",i,[(m(),t("svg",L,T[9]||(T[9]=[l('',1)]))),T[10]||(T[10]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mstyle",{displaystyle:"true",scriptlevel:"0"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mfrac",null,[Q("mrow",null,[Q("mi",null,"y"),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"y"),Q("mn",null,"0")])]),Q("mrow",null,[Q("mi",null,"x"),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"x"),Q("mn",null,"0")])])]),Q("mo",null,"="),Q("mo",null,"−"),Q("mfrac",null,[Q("mrow",null,[Q("msup",null,[Q("mi",null,"x"),Q("mo",{"data-mjx-alternate":"1"},"′")]),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"t"),Q("mn",null,"0")]),Q("mo",{stretchy:"false"},")")]),Q("mrow",null,[Q("msup",null,[Q("mi",null,"y"),Q("mo",{"data-mjx-alternate":"1"},"′")]),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"t"),Q("mn",null,"0")]),Q("mo",{stretchy:"false"},")")])])])])])],-1))]),Q("p",null,[T[13]||(T[13]=a("所以对于椭圆上任意一点")),Q("mjx-container",u,[(m(),t("svg",c,T[11]||(T[11]=[l('',1)]))),T[12]||(T[12]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"x"),Q("mn",null,"0")]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"y"),Q("mn",null,"0")]),Q("mo",{stretchy:"false"},")")])],-1))]),T[14]||(T[14]=a(",它的法线方程是"))]),Q("mjx-container",f,[(m(),t("svg",M,T[15]||(T[15]=[l('',1)]))),T[16]||(T[16]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mstyle",{displaystyle:"true",scriptlevel:"0"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mfrac",null,[Q("mrow",null,[Q("mi",null,"y"),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"y"),Q("mn",null,"0")])]),Q("mrow",null,[Q("mi",null,"x"),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"x"),Q("mn",null,"0")])])]),Q("mo",null,"="),Q("mfrac",null,[Q("mrow",null,[Q("mi",null,"a"),Q("mi",null,"sin"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mi",null,"θ")]),Q("mrow",null,[Q("mi",null,"b"),Q("mi",null,"cos"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mi",null,"θ")])])])])])],-1))]),Q("p",null,[T[23]||(T[23]=a("假设轮廓的宽度为")),Q("mjx-container",V,[(m(),t("svg",Z,T[17]||(T[17]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D43E",d:"M285 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当然，如果只是想知道答案的话，可以不用这么麻烦，还有一种思路就是利用极端情况。设想一下这个椭圆极其“扁”，那么它的形状像是两条紧紧贴在一起的线段，不难想象，如果在这个椭圆外扩展一个宽度，那么新的轮廓的的上下边缘类似于两条分开的平行线段，但两端却无法平滑连在一起，这个形状类似于一个圆角的矩形，显然不是椭圆，下面是一个用Mathematica模拟的动态图"))]),T[45]||(T[45]=Q("figure",null,[Q("img",{src:e,alt:"",tabindex:"0",loading:"lazy"}),Q("figcaption")],-1))])}const R=n(r,[["render",O],["__file","Ellipse.html.vue"]]),J=JSON.parse('{"path":"/blog/2025/02/Ellipse.html","title":"一道数学趣题","lang":"zh-CN","frontmatter":{"title":"一道数学趣题","tags":"数学"},"headers":[],"git":{},"filePathRelative":"blog/2025/02/Ellipse.md"}');export{R as comp,J as data};