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Staff Picks Mathematics Visual Arts Dynamic Interactivity Geometry Graphics and Visualization Wolfram Language

[GIF] Osculating (Osculating circles to a polar curve)

Clayton Shonkwiler

Clayton Shonkwiler, Colorado State University

Posted 8 years ago

Osculating Conceptually fairly simple, though it took a while to find the right number of circles to include without causing unpleasant visual artifacts. Anyway, the animation shows 100 osculating circles to the curve $r = \cos (3\theta+\pi/2)$ as they traverse along the curve. As in Bessel and Tangents, I think it's interesting how the form of the curve emerges from a family of approximations even though the curve itself is never explicitly drawn. Here's the code: DynamicModule[{r, curvature, parametrizedcurve, n = 100, cols = RGBColor /@ {"#08D9D6", "#EAEAEA", "#FF2E63", "#252A34"}}, r[?_] := Cos[3 ? + ?/2]; curvature[?_] := (r[?]^2 + 2 r'[?]^2 - r[?] r''[?])/(r[?]^2 + r'[?]^2)^(3/2); parametrizedcurve[?_] := {r[?] Cos[?], r[?] Sin[?]}; Manipulate[ Graphics[{Thickness[.005], Opacity[.8], Table[{Blend[Append[cols[[;; 3]], cols[[1]]], Mod[?, ?]/?], Circle[parametrizedcurve[?] + 1/curvature[?] Normalize[{-#[[2]], #[[1]]} &[parametrizedcurve'[?]]], 1/curvature[?]]}, {?, t, ? + t, 2 ?/n}]}, ImageSize -> 540, Background -> cols[[-1]], PlotRange -> {{-3/2, 3/2}, {-9/8, 15/8}}], {t, 0, 2 ?/n}] ] And a still with 700 osculating circles which (to my eyes) doesn't work as an animation because of the aforementioned visual artifacts:

Osculating circles to a polar curve

Osculating

Conceptually fairly simple, though it took a while to find the right number of circles to include without causing unpleasant visual artifacts.

Anyway, the animation shows 100 osculating circles to the curve $r = \cos (3\theta+\pi/2)$ as they traverse along the curve.

As in Bessel and Tangents, I think it's interesting how the form of the curve emerges from a family of approximations even though the curve itself is never explicitly drawn.

Here's the code:

DynamicModule[{r, curvature, parametrizedcurve, n = 100, 
  cols = RGBColor /@ {"#08D9D6", "#EAEAEA", "#FF2E63", "#252A34"}},
 r[?_] := Cos[3 ? + ?/2];
 curvature[?_] := (r[?]^2 + 2 r'[?]^2 - r[?] r''[?])/(r[?]^2 + r'[?]^2)^(3/2);
 parametrizedcurve[?_] := {r[?] Cos[?], r[?] Sin[?]};
 Manipulate[
  Graphics[{Thickness[.005], Opacity[.8], 
    Table[{Blend[Append[cols[[;; 3]], cols[[1]]], Mod[?, ?]/?], 
      Circle[parametrizedcurve[?] + 1/curvature[?] Normalize[{-#[[2]], #[[1]]} 
        &[parametrizedcurve'[?]]], 1/curvature[?]]}, 
       {?, t, ? + t, 2 ?/n}]},
    ImageSize -> 540, Background -> cols[[-1]], 
   PlotRange -> {{-3/2, 3/2}, {-9/8, 15/8}}],
  {t, 0, 2 ?/n}]
 ]

And a still with 700 osculating circles which (to my eyes) doesn't work as an animation because of the aforementioned visual artifacts:

enter image description here