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

GROUPS:
 Clayton Shonkwiler 5 Votes OsculatingConceptually 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: