       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.

```

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