Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 1995-2000 by Xah Lee. (http://xahlee.org/)

Packages and Initializations

Needs["ParaPlot`"]
(*MathSource Item:0207-986*)

Needs["PlaneCurveFormulas`"]


Needs["PlaneCurveGenerator`"]

Needs["Graphics`ImplicitPlot`"]
(*MathSource Item:0200-529*)

string art animation

Fun Animation

Here is a fun animation based on the tangent properties of parabolas. The envelope of these lines are parabolas.

Clear[autoLinesGP, n]; 
n = 15; 
autoLinesGP[a1_, a2_, b1_, b2_, n_] := 
      Module[{v1, v2, length1, length2, v1Points, v2Points}, 
        v1 = N[unitV[a2 - a1]]; 
        length1 = N[(Sqrt[(a2 - a1) . (a2 - a1)])]; 
        v2 = N[unitV[b2 - b1]]; 
        length2 = N[(Sqrt[(b2 - b1) . (b2 - b1)])]; 
        v1Points = Table[v1 i + N[a1], {i, 0, length1, (length1/(n - 1))}]; 
        v2Points = Table[v2 i + N[b1], {i, 0, length2, (length2/(n - 1))}]; 
        MapThread[Line[{#1, #2}] &, {v1Points, v2Points}]]
Do[Show[
        Graphics[{{Hue[ .65], Thickness[ .007], 
              Line[{{-1, 0}, {0, -1}, {1, 0}, {0, 1}, {-1, 0}}]}, 
            Hue[ .35, 1,  .5], Thickness[ .004], 
            autoLinesGP[{Cos[i], Sin[i]}, -{Cos[i], Sin[i]}, {-1, 
                0}, {0, -1}, n], 
            autoLinesGP[{Cos[i], 
                Sin[i]}, -{Cos[i], Sin[i]}, {0, -1}, {1, 0}, n], 
            autoLinesGP[{Cos[i], Sin[i]}, -{Cos[i], Sin[i]}, {1, 0}, {0, 
                1}, n], 
            autoLinesGP[{Cos[i], Sin[i]}, -{Cos[i], Sin[i]}, {0, 
                1}, {-1, 0}, n]}], 
        PlotRange -> {{-1, 1}, {-1, 1}} 1.05, 
        AspectRatio -> Automatic, Axes -> False], {i, 0, 
        2 π, ((2 π)/(4 8))}]; 
Clear[autoLinesGP, n];