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File:Steiner chain animation-50dpi.gif

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Steiner_chain_animation-50dpi.gif (250 × 251 pixels, file size: 1.35 MB, MIME type: image/gif, looped, 126 frames)

Summary

Description
English: Animation of a Steiner chain of 9 circles, showing that the locus of the position of the circles' centres is an ellipse (in red), and that the locus of the contact points between the circles is a circle itself (in orange)
Date
Source https://twitter.com/j_bertolotti/status/1039543306433249280
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 11.0 code

n = 9;
\[Theta] = \[Pi]/n;
\[CapitalDelta]\[Theta] = (2 \[Pi])/n;
R = 0.2;
\[Rho] = R/(1 + 1/Sin[\[Theta]]);
r = \[Rho] (1/Sin[\[Theta]] - 1);
center = {.3, 0};
plots = Reap[For[start = 0, start <= 2 \[Pi], start = start + 0.05,
      CandR = 
       Table[{center[[1]] + (\[Rho] + r) Cos[i], 
         center[[2]] + (\[Rho] + r) Sin[i], \[Rho]}, {i, start, 
         start + (n - 
             1)*\[CapitalDelta]\[Theta], \[CapitalDelta]\[Theta]}];
      createcoord[{x_, y_, 
         z_}] := {(x )/(x^2 + y^2 - z^2), (y)/(x^2 + y^2 - z^2), 
        z/(x^2 + y^2 - z^2)};
      
      innerc = createcoord[Flatten[{center, r}]];
      outherc = createcoord[Flatten[{center, R}]];
      ellipseCenter = {(outherc[[1]] + innerc[[1]])/2, 0};
      ellipseA = 
       Abs[createcoord[{center[[1]] + \[Rho] + r, 0, \[Rho]}][[1]] - 
         ellipseCenter[[1]]];
      ellipseC = (outherc[[1]] - innerc[[1]])/2;
      ellipseB = Sqrt[ellipseA^2 - ellipseC^2];
      tmp = Map[createcoord, CandR];
      p1 = {x, y} /. 
        Solve[{(x - tmp[[1, 1]])^2 + (y - tmp[[1, 2]])^2 == 
             tmp[[1, 
              3]]^2 && (x - tmp[[2, 1]])^2 + (y - tmp[[2, 2]])^2 == 
             tmp[[2, 3]]^2}, {x, y}][[1]];
      p2 = {x, y} /. 
        Solve[{(x - tmp[[3, 1]])^2 + (y - tmp[[3, 2]])^2 == 
             tmp[[3, 
              3]]^2 && (x - tmp[[2, 1]])^2 + (y - tmp[[2, 2]])^2 == 
             tmp[[2, 3]]^2}, {x, y}][[1]];
      p3 = {x, y} /. 
        Solve[{(x - tmp[[3, 1]])^2 + (y - tmp[[3, 2]])^2 == 
             tmp[[3, 
              3]]^2 && (x - tmp[[4, 1]])^2 + (y - tmp[[4, 2]])^2 == 
             tmp[[4, 3]]^2}, {x, y}][[1]];
      c = 
       Abs[{xc, yc, rc} /. 
         Solve[((#1 - xc)^2 + (#2 - yc)^2 == rc^2) & @@@ {p1, p2, 
             p3}, {xc, yc, rc}][[1]] ];
      Sow@Show[
        Graphics[{Orange, Thick, 
          Evaluate[Circle[{c[[1]], c[[2]]}, c[[3]]] ]}],
        Graphics[{Black, Thick, 
          Circle[{#1, #2}, #3] & @@@ Map[createcoord, CandR] }],
        Graphics[{Blue, Thick, 
          Circle[{#1, #2}, #3] & @@@ {innerc, outherc} }],
        Graphics[{Red, Thick, 
          Circle[ellipseCenter, {ellipseA, ellipseB}]}],
        Graphics[{PointSize[0.02], 
          Point[Map[createcoord, CandR][[All, 1 ;; 2]]] }]
        ];
      ];][[2, 1]];
ListAnimate[plots]

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

This file, which was originally posted to https://twitter.com/j_bertolotti/status/1039543306433249280, was reviewed on 19 October 2018 by reviewer Ronhjones, who confirmed that it was available there under the stated license on that date.

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11 September 2018

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Date/TimeThumbnailDimensionsUserComment
current08:29, 12 September 2018Thumbnail for version as of 08:29, 12 September 2018250 × 251 (1.35 MB)BertoUser created page with UploadWizard

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