Sliced cylinder and Sine curve

Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 1995-2002 by Xah Lee. (http://xahlee.org/)

Packages and Initializations

Needs["RuledSurfacePlot`"]

Needs["Graphics`Shapes`"]	(* 0200-608 *)

$HistoryLength = 5;

definition

Development of a Cylinder

Sinusoid is the development of an elliptical section of a right circular cylinder.

Explaination:
Suppose you slice a cucumber slanted. Roll the cucumber on the table, the edge of the cutting traces a sinusoid. See animation below.

Animation of the developing cylinder tracing a sine curve

Clear[gp, step, n]; 
n = 20; step = ((2 π)/24); 
gp = 
        First[RuledSurfacePlot[{RotateSpaceCurve[{Sin[#1], 
                      Sqrt[2]* Cos[#1], 0} &, Pi/4, 0, 0][t], {Sin[t], 
                Cos[t], -2}}, {t, 0, 2 π}, 
            PlotPoints -> {n + 1, 2}, CheckerBoardPattern -> False, 
            DisplayFunction -> Identity]];
Do[
        Show[{RotateShape[Graphics3D[{EdgeForm[], gp}], 0, 0, i], 
            Graphics3D[
              First[Plot[Cos[t - i], {t, 0, 6.5}, PlotStyle -> {Hue[0]}, 
                    DisplayFunction -> 
                      Identity]] /. {x_?AtomQ, 
                    y_?AtomQ} -> {x, 1, y}], 
            Graphics3D[
              Table[Line[{{i + j, 1, -2}, {i + j, 1, 
                      Cos[j]}}], {j, -i, 2 π, (2*Pi)/n}]]}, 
          Boxed -> True, Ticks -> None, Axes -> False, 
          DisplayFunction -> $DisplayFunction, 
          PlotRange -> {{-1, 2 π}, {-2, 1}, {-2, 
                  2}} 1.03, 
          LightSources -> {{{0.48, 0.55, 0.51}, 
                Hue[0.65, 1. , 0.68]}, {{0.99, 0.89, 0.126}, 
                Hue[0.36, 0.62, 0.82]}, {{0.78, 0.78, 0.96}, 
                Hue[0.208, 0.97, 0.66]}}], {i, 0, 2 π, 
          step}];
Clear[gp, step, n]