File:Roulette examples.gif

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Roulette_examples.gif ‎(360 × 376 pixels, file size: 4.24 MB, MIME type: image/gif, looped, 567 frames)

Captions

Captions

If you track the position of a point on a curve while it rolls on a second curve, the result is known as a "roulette".

Summary

[edit]
Description
English: If you track the position of a point on a curve while it rolls on a second curve, the result is known as a "roulette". This GIF shows a few famous and simple examples (but there are many other).
Date
Source https://twitter.com/j_bertolotti/status/1086254369501253632
Author Jacopo Bertolotti
Permission
(Reusing this file)		 https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 11.0 code

[edit]
r = 1;
y[t_] := -r Sin[t + \[Pi]/2];
x[t_] := r Cos[t + \[Pi]/2] + r t;
p1 = Table[
   Show[
    ParametricPlot[{x[\[Tau]], y[\[Tau]]}, {\[Tau], 0, t}, 
     PlotRange -> {{-r, 4 \[Pi] + r}, {-2 \[Pi] - r, 2 \[Pi] + r}}, 
     Axes -> False, PlotStyle -> {Red}]
    ,
    Graphics[{Black, Thick, Line[{{-r, -r}, {4 \[Pi] + r, -r}}], 
      Circle[{r t, 0}, r], Red, Disk[{x[t], y[t]}, 0.2]}, 
     PlotRange -> {{-r, 4 \[Pi] + r}, {-2 \[Pi] - r, 2 \[Pi] + r}}]
    , PlotLabel -> "Circle on line \[Rule] Cycloid", 
    LabelStyle -> {Black, Bold}
    ]
   , {t, 0.001, 4 \[Pi], 0.2}];
r = 1;
R = 4;
x[t_] := (R - r) Cos[t] + r Cos[(R - r)/r (t)];
y[t_] := (R - r) Sin[t] - r Sin[(R - r)/r (t)];
p2 = Table[
   Show[
    ParametricPlot[{x[\[Tau]], y[\[Tau]]}, {\[Tau], 0, t}, 
     PlotRange -> {{-6, 6}, {-6, 6}}, Axes -> False, 
     PlotStyle -> {Red}]
    ,
    Graphics[{Black, Thick, Circle[{0, 0}, R], 
      Circle[{(R - r) Cos[t], (R - r) Sin[t]}, r], Red, 
      Disk[{x[t], y[t]}, 0.2]}, PlotRange -> {{-6, 6}, {-6, 6}}]
    , PlotLabel -> "Circle inside a circle \[Rule] Hypocycloid", 
    LabelStyle -> {Black, Bold}
    ]
   , {t, 0.001, 4 \[Pi], 0.1}];
r = 2;
R = 4;
x[t_] := (R - r) Cos[t] + r Cos[(R - r)/r (t)];
y[t_] := (R - r) Sin[t] - r Sin[(R - r)/r (t)];
p21 = Table[
   Show[
    ParametricPlot[{x[\[Tau]], y[\[Tau]]}, {\[Tau], 0, t}, 
     PlotRange -> {{-6, 6}, {-6, 6}}, Axes -> False, 
     PlotStyle -> {Red}]
    ,
    Graphics[{Black, Thick, Circle[{0, 0}, R], 
      Circle[{(R - r) Cos[t], (R - r) Sin[t]}, r], Red, 
      Disk[{x[t], y[t]}, 0.2]}, PlotRange -> {{-6, 6}, {-6, 6}}]
    , PlotLabel -> 
     "Circle inside a circle (R=2r) \[Rule] Straight line", 
    LabelStyle -> {Black, Bold}
    ]
   , {t, 0.001, 4 \[Pi], 0.1}];
R = 4;
x[t_] := (R + r) Cos[t] - r Cos[(R + r)/r (t)];
y[t_] := (R + r) Sin[t] - r Sin[(R + r)/r (t)];
p3 = Table[
   Show[
    ParametricPlot[{x[\[Tau]], y[\[Tau]]}, {\[Tau], 0, t}, 
     PlotRange -> {{-6, 6}, {-6, 6}}, Axes -> False, 
     PlotStyle -> {Red}]
    ,
    Graphics[{Black, Thick, Circle[{0, 0}, R], 
      Circle[{(R + r) Cos[t], (R + r) Sin[t]}, r], Red, 
      Disk[{x[t], y[t]}, 0.2]}, PlotRange -> {{-6, 6}, {-6, 6}}]
    , PlotLabel -> "Circle outside a circle \[Rule] Epicycloid", 
    LabelStyle -> {Black, Bold}
    ]
   , {t, 0.001, 4 \[Pi], 0.1}];
r = 2;
R = 2;
x[t_] := (R + r) Cos[t] - r Cos[(R + r)/r (t)];
y[t_] := (R + r) Sin[t] - r Sin[(R + r)/r (t)];
p4 = Table[
   Show[
    ParametricPlot[{x[\[Tau]], y[\[Tau]]}, {\[Tau], 0, t}, 
     PlotRange -> {{-6, 6}, {-6, 6}}, Axes -> False, 
     PlotStyle -> {Red}]
    ,
    Graphics[{Black, Thick, Circle[{0, 0}, R], 
      Circle[{(R + r) Cos[t], (R + r) Sin[t]}, r], Red, 
      Disk[{x[t], y[t]}, 0.2]}, PlotRange -> {{-6, 6}, {-6, 6}}]
    , PlotLabel -> "Circle outside a circle (R=r) \[Rule] Cardioid", 
    LabelStyle -> {Black, Bold}
    ]
   , {t, 0.001, 4 \[Pi], 0.1}];
ListAnimate[Join[p1, p2, p21, p3, p4]]

Licensing

[edit]
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/1030470604418428929, was reviewed on 20 January 2019 by reviewer Ronhjones, who confirmed that it was available there under the stated license on that date.

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Date/TimeThumbnailDimensionsUserComment
current15:11, 19 January 2019Thumbnail for version as of 15:11, 19 January 2019360 × 376 (4.24 MB)Berto (talk | contribs)User created page with UploadWizard

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