# [GIF] Wrap: wrapping lines around a circle

Posted 7 months ago
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 WrapThis started as a visualization of the exponential map, which for the circle corresponds to the map $t \mapsto e^{i t}$, where it's useful to visualize the real line (where $t$ lives) as being the vertical line $\Re(z)=1$ (i.e., parallel to the imaginary axis) which is tangent to the unit circle at the point $z=1$. To wrap the line around the circle I defined the following function, which is just a straight line outside the interval $[-s,s]$, and sends $t$ to $e^{it}$ for $t$ in the interval: wrap[s_, t_] := Which[t < -s, E^(-I s) + I E^(-I s) (t + s), -s <= t <= s, E^(I t), t > s, E^(I s) + I E^(I s) (t - s)]; With the help of the smootherstep function smootherstep[t_] := 6 t^5 - 15 t^4 + 10 t^3; and some rotation, we get the first half of the animation: DynamicModule[{u, cols = RGBColor /@ {"#ffa323", "#ff6337", "#004a2f"}}, Manipulate[ u = \[Pi] smootherstep[s]; Show[ Table[ ParametricPlot[ ReIm[-I E^(I u) r wrap[u, t]], {t, -\[Pi], \[Pi]}, PlotRange -> 2.5, PlotStyle -> Directive[Thickness[.01], Blend[cols, 2/3 r], CapForm["Round"], JoinForm[None]]], {r, 0, 1.5, .1}], Axes -> False, ImageSize -> 540, Background -> cols[[-1]]], {s, 0, 1}] ] And the second half: DynamicModule[{u, cols = RGBColor /@ {"#ffa323", "#ff6337", "#004a2f"}}, Manipulate[ u = \[Pi] smootherstep[s]; Show[ Table[ ParametricPlot[ ReIm[Conjugate[I E^(I u) r wrap[u, t]]], {t, -\[Pi], \[Pi]}, PlotRange -> 2.5, PlotStyle -> Directive[Thickness[.01], Blend[cols, 2/3 r], CapForm["Round"], JoinForm[None]]], {r, 0, 1.5, .1}], Axes -> False, ImageSize -> 540, Background -> cols[[-1]]], {s, 1, 0}] ] 
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Posted 7 months ago
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Posted 7 months ago
 good question, is there any copyright on the GIF or the code, or could just anybody take this and use it for reposting and commercialization, without even giving credit?