Problem 43648. Arc length of points interpolation

Created by Kwin

Solve Later

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Calculate the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Arc_length\"><w:r><w:t>arc length</w:t></w:r></w:hyperlink><w:r><w:t> of the closed loop curve going though these points in the order that they are given. The parametric curve,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c(t)</w:t></w:r><w:r><w:t> , between points</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>p(k)</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>p(k+1)</w:t></w:r><w:r><w:t> is defined as,</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c(t) = p(k-1) * (-t/2+t^2-t^3/2) + p(k) * (1-5/2*t^2+3/2*t^3) + p(k+1) * (t/2+2*t^2-3/2*t^3) + p(k+2) * (-t^2/2+t^3/2)</w:t></w:r><w:r><w:t>,</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>t</w:t></w:r><w:r><w:t> goes from 0 to 1. These interpolation polynomials can also be found using the constraints</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c(0)=p(k)</w:t></w:r><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c(1)=p(k+1)</w:t></w:r><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c'(0)=(p(k+1)-p(k-1))/2</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>c'(1)=(p(k+2)-p(k))/2</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For example for the points</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[points = [[1; 0] [0; 1] [-1; 0] [0; -1]];]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>would yield to following curve:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:customXml 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"}]}

Solve

Solution Stats

12 Solutions
5 Solvers

Last Solution submitted on Mar 07, 2021

Last 200 Solutions

Problem Comments

3 Comments

3 Comments

Tim on 1 Nov 2016

How confident are you that your answers are correct to the tolerances you have specified?

Alfonso Nieto-Castanon on 1 Nov 2016

Have not tried the problem yet but tolerances seem below eps(dist_correct) in several cases, could you perhaps just relax these a bit?

goc3 on 14 Dec 2016

I relaxed the tolerances in the test suite.

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