I'd like to make an animation showing how we wrap a rectangle to form a cylinder (by joining a pair of parallel sides).
Here's my first pass:
Manipulate[
ParametricPlot3D[
t {Cos[theta], Sin[theta], rho} + (1 - t) {rho, -1, theta},
{theta, -Pi, Pi}, {rho, 0, 2},
PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}, {-1.5, 1.5}},
PerformanceGoal -> "Quality", ViewPoint -> {-2.3, 0.77, -2},
ViewVertical -> {-0.08, 1, -0.06}],
{t, 0, 1}]
I'd much prefer the simpler looking transformation (without the twists this animation shows) which we'd do by simply rotating the pair of parallel sides until they touch to form the cylinder.
With[{t = Pi/2}, ParametricPlot3D[{Piecewise[{{Cos[u], t <= u}}, Cos[t] + u Sin[t]], Piecewise[{{Sin[t] - u Cos[t], t > u}}, Sin[u]], z}, {u, 0, 2 Pi}, {z, -1, 1}, Exclusions -> None]]
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