1. Elliptical integral

Elliptic integral is a general term for a type of integral, and here only the integral used to calculate the ellipse perimeter. This kind of integral cannot be expressed by basic functions, but can only be expressed in integral form. To calculate the perimeter of an ellipse using integrals, you have to use curve integrals. L is the circumference of the ellipse. Calculate the length of 90 degrees and multiply by 4.

And in Mathenatica there is also elliptic integral EllipticE, where EllipticE[θ,m] can be used for any angle of elliptic arc length.

2. Elliptical cycloid cluster

Originally, the starting point was placed on the end point of the long axis, and then I thought I could put the starting point on the end point of the short axis.

It can be seen that the two types of cycloid are completely different, and then think about it, because the curve from each point on the circle is the same, and the ellipse is different, so the Each point on the 90 degree arc creates a cycloid, forming a cycloid cluster.



You can also modify the color, or the other half of the cycloid.

Increase the length of the scroll, but the computer conditions are not allowed, and the angle is relatively small.

Code above:

f[x_] := x - Quotient[x, 2 Pi]*2 Pi;
h[x_] := Piecewise[{{ArcTan[4 Tan[x]], f[x] <= 0.5 Pi},
    {ArcTan[4 Tan[x]] + Pi, 0.5 Pi < f[x] <= Pi},
    {ArcTan[4 Tan[x]] + Pi, Pi < f[x] <= 1.5 Pi},
    {ArcTan[4 Tan[x]] + 2 Pi, 1.5 Pi < f[x] <= 2 Pi}}];
a = Table[
  Show[{
    Plot[x^2, {x, 1, 3}, Axes -> True, AxesOrigin -> {0, 0}, 
     AspectRatio -> 1/4, PlotStyle -> {Thick, White}, 
     PlotRange -> {{-3, 24}, {-3, 5}}],	
    Table[ParametricPlot[{2 EllipticE[t, 0.75] + 
        Cos[0.5 Pi - h[t] + t]*2/Sqrt[3*Sin[t]^2 + 1] - 
        2*Cos[0.5 Pi - h[t] - k]/Sqrt[3*Sin[k]^2 + 1], 
       Sin[0.5 Pi - h[t] + t]*2/Sqrt[3*Sin[t]^2 + 1] - 
        2 Sin[0.5 Pi - h[t] - k]/Sqrt[3*Sin[k]^2 + 1]}, {t, 0.0001, 
       t}, PlotStyle -> ColorData["VisibleSpectrum"][400 + 130 k/1.5]],
     {k, 0.001, Pi, 0.05 Pi}],
    Graphics[
     Rotate[{Red, Thick, 
       Circle[{2 EllipticE[t, 0.75] + 
          Cos[0.5 Pi - h[t] + t]*2/Sqrt[3*Sin[t]^2 + 1],
         Sin[0.5 Pi - h[t] + t]*2/Sqrt[3*Sin[t]^2 + 1]},
        {1, 2}
        ], AxesStyle -> {Thick, Black, Black}},
      -h[t]]
     ]
    }], {t, 0.001, 4 Pi, 0.04 Pi}] (*plot coordinate axis is set in the front, graphic can be displayed together after the back *)
 Export["e:/Program Files/math files/Elliptical scroll 88.gif", a, 
  ImageResolution -> 100];