Pretty Graphics of Astroid by envolope

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg1137] Pretty Graphics of Astroid by envolope*From*: Xah Y Lee <xyl10060 at fhda.edu>*Date*: Wed, 17 May 1995 03:39:19 -0400

Little pretty graphics to share with you. (*begin mma code-----------------------------*) astgp1 = Table[ Line[{{Cos[i],0}, {0,Sin[i]}}], {i,0, Pi/2, (Pi/2)/11} ]; astgp2 = (astgp1 /. Line[{p1_,p2_}]->Line[{-p1,p2}]); astgp3 = (astgp1 /. Line[{p1_,p2_}]->Line[{-p1,-p2}]); whatgp1 = Table[ Line[{{i,0},{0,1-i}}], {i,0,1,.1} ]; whatgp2 = (whatgp1 /. Line[{p1_,p2_}]->Line[{-p1,p2}]); whatgp3 = (whatgp1 /. Line[{p1_,p2_}]->Line[{-p1,-p2}]); whatgp4 = (whatgp1 /. Line[{p1_,p2_}]->Line[{p1,-p2}]); Show[ Graphics[ {Hue[.0],astgp1,astgp3, Hue[.7],whatgp2,whatgp4} ], AspectRatio->Automatic, Axes->False ] (*end mma code------------------------------*) The red lines are all equal length. The outline they formed (their envelope) is the Astroid. Astroid can be generated by rolling a circle inside a fixed circle 4 times its radius. Generated much the same way as cycloids. A parametric form for astroid is {(3*Cos[t])/4 + Cos[3*t]/4, (3*Sin[t])/4 - Sin[3*t]/4}. I wrote a package TrochoidPlot.m (mathsource #0207-234) that will show animation of Astroid. The blue lines are such that the distance between neighboring end points are constant. I don't know what curve they form. This an an example from Tom Wickham-Jones' book Mma Graphics, p.118. Xah Lee Permemant email: 74631.731 at compuserve.com Quote of the day: O King, for traveling over the country, there are royal roads and roads for common citizens; but in geometry there is one road for all -- Menaechmus. When his pupil Alexander the Great asked for a shortcut to geometry. O Menaechmus, but you can always travel by plane. Ever heard of Mathematica? -- A Mathematician of WasaMata U.