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From: Tim Rentsch <tr.17687@z991.linuxsc.com>
Newsgroups: comp.lang.c
Subject: Re: quick & easy random dir vector, how?
Date: Wed, 05 Jun 2019 08:05:49 -0700
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richard@cogsci.ed.ac.uk (Richard Tobin) writes:

> In article <87r2891i5d.fsf@bsb.me.uk>,
> Ben Bacarisse  <ben.usenet@bsb.me.uk> wrote:
>
>>> I will say this with regard to my solution... as I see it, I'm
>>> trying to get random values on the surface of a sphere.
>>
>> That's easy.  Pick x, y and z in [-1, 1] and normalise.  You don't get a
>> uniform distribution though.
>
> Pick x, y and z in [-1, 1].  If they're outside the unit sphere, try
> again until they are inside - this will take 2 x 3 / pi attempts
> average.  You now have a point with uniform distribution within the
> unit sphere.  Normalize to get a point on the surface.

Slight refinement, assuming a 64-bit integer RNG is used:  generate
one 64-bit random number, divide into three 21-bit values mapped to
the range [-1048575,1048575] (discarding samples that would produce
the value -1048576), testing x*x + y*y + z*z <= 1048575*1048575.
When the test passes normalize appropriately.

> Apologies if this has already been suggested - I haven't read the
> whole thread.  And it's not particularly efficient.

But probably more efficient than a "more direct" method using trig
functions, especially since the rejection tests can be done using
integer-only calculations.