====== Random Constructions ======

===== Random points on the unit sphere =====

The easiest way to randomly construct a polytope is by sampling points on the unit sphere. The following chooses 100 points on the unit sphere in 3-space.

<code perl>
> $p1=rand_sphere(3,100);
> print $p1->SIMPLICIAL;
true
</code>
===== Random points sampled from a normal distribution =====

A second way to randomly construct a polytope is by sampling points by a standard normal distribution. The following chooses 100 points sampled from the standard normal distribution in 3-space.

<code perl>
> $p2 = rand_normal(3, 100);
> print $p2 -> SIMPLICIAL, "\n", $p2 -> F_VECTOR;
true
24 66 44
</code>
With probability one such polytopes under either distribution are simplicial.

===== Random polytopes with are neither simplicial nor simple =====

<code perl>
> ($d,$m,$n) = (4,50,30);
> $p1=rand_sphere($d,$m);
> $p2=polarize($p1);
> $p3=new Polytope(POINTS=>rand_vert($p2->VERTICES,$n));
> print $p3->SIMPLICIAL, " ", $p3->SIMPLE, "\n", $p3->F_VECTOR;
false false
30 162 251 119
</code>