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

Posted 1 year ago
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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[θ_] := Cos[3 θ + π/2];
 curvature[θ_] := (r[θ]^2 + 2 r'[θ]^2 - r[θ] r''[θ])/(r[θ]^2 + r'[θ]^2)^(3/2);
 parametrizedcurve[θ_] := {r[θ] Cos[θ], r[θ] Sin[θ]};
 Manipulate[
  Graphics[{Thickness[.005], Opacity[.8], 
    Table[{Blend[Append[cols[[;; 3]], cols[[1]]], Mod[θ, π]/π], 
      Circle[parametrizedcurve[θ] + 1/curvature[θ] Normalize[{-#[[2]], #[[1]]} 
        &[parametrizedcurve'[θ]]], 1/curvature[θ]]}, 
       {θ, t, π + t, 2 π/n}]},
    ImageSize -> 540, Background -> cols[[-1]], 
   PlotRange -> {{-3/2, 3/2}, {-9/8, 15/8}}],
  {t, 0, 2 π/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

2 Replies

I love this one quite a lot! Wonderfully confusing!

enter image description here - Congratulations! This post is now a Staff Pick as distinguished on your profile! Thank you, keep it coming!

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