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Pretty Graphics of Astroid by envolope

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

Little pretty graphics to share with you.

(*begin mma code-----------------------------*)
astgp1 =
        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 =
        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}]);
(*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
 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 
O Menaechmus, but you can always travel by plane. Ever heard of Mathematica?
-- A Mathematician of WasaMata U.

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