Random numbers within a sphere?

[Stavros Christoforou]

Hello everyone,

I was wondering if someone could help me with an issue I have in C++. I
want to select random points within the volume of a sphere. I know how
to get random numbers using srand() and rand(), but have no idea how to
do that within a more complicated geometry. Any help would be greatly
appreciated..

Regards
Stavros

[Fabio Rossi]

Stavros Christoforou wrote:

Hello everyone,

I was wondering if someone could help me with an issue I have in C++. I
want to select random points within the volume of a sphere. I know how
to get random numbers using srand() and rand(), but have no idea how to
do that within a more complicated geometry. Any help would be greatly
appreciated..

You could represent any point of the sphere in spherical coordinates: a
radius and two angles. A coordinate is a uniform variable (using rand()) in
its [min,max] interval.

[Victor Bazarov]

Stavros Christoforou wrote:

I was wondering if someone could help me with an issue I have in C++. I
want to select random points within the volume of a sphere. I know how
to get random numbers using srand() and rand(), but have no idea how to
do that within a more complicated geometry. Any help would be greatly
appreciated..

Select a random radius (between 0 and the sphere radius), a randon azimuth
(0 through 2*Pi) and a random elevation (-Pi/2 through Pi/2) and then
convert the three values from spherical coordinates into Cartesian (if
that's your desired result). And please don't post off-topic questions
here. Your question has nothing to do with C++ language and ought to be
posted in comp.graphics.a lgorithms or comp.programmin g.

V

[Phlip]

Fabio Rossi wrote:

Stavros Christoforou wrote:

Hello everyone,

I was wondering if someone could help me with an issue I have in C++. I
want to select random points within the volume of a sphere. I know how
to get random numbers using srand() and rand(), but have no idea how to
do that within a more complicated geometry. Any help would be greatly
appreciated..
You could represent any point of the sphere in spherical coordinates: a
radius and two angles. A coordinate is a uniform variable (using rand())

in its [min,max] interval.

Stavros might need evenly distributed numbers. A polar coordinate scheme
would bunch numbers up around the center, at least.

The fix is to bound the sphere with a cube, tangent on all faces, and find
random points in the cube by finding random xyz points, each >= -r and < r,
where r is the radius of the sphere. Then filter out all points laying
outside the sphere, by comparing x^2 + y^2 + z^2 to r^2.

--
Phlip
http://industrialxp.org/community/bi...UserInterfaces

[Ron Natalie]

Stavros might need evenly distributed numbers. A polar coordinate scheme
would bunch numbers up around the center, at least. How so? If the [0, sphere-radius] random number generator is uniform, so would
the density with respect to the center.

The fix is to bound the sphere with a cube, tangent on all faces, and find
random points in the cube by finding random xyz points, each >= -r and < r,
where r is the radius of the sphere. Then filter out all points laying
outside the sphere, by comparing x^2 + y^2 + z^2 to r^2.

That wouldn't be any better and a lot slower as a lot of samples would have
to be generated, tested, and discarded.

[Victor Bazarov]

Ron Natalie wrote:

Stavros might need evenly distributed numbers. A polar coordinate scheme
would bunch numbers up around the center, at least.

How so? If the [0, sphere-radius] random number generator is uniform,
so would
the density with respect to the center.

The fix is to bound the sphere with a cube, tangent on all faces, and
find
random points in the cube by finding random xyz points, each >= -r and
< r,
where r is the radius of the sphere. Then filter out all points laying
outside the sphere, by comparing x^2 + y^2 + z^2 to r^2.

That wouldn't be any better and a lot slower as a lot of samples would have
to be generated, tested, and discarded.

"A lot of samples"? The sphere is about half the volume of its
circumscribing cube. So, less than half of all samples are going to
be discarded. Doesn't seem like "a lot" to me. Of course, everybody
has their own definition of "a lot". Is there a better way to produce
points evenly distributed in the volume of a sphere?

V

[Shezan Baig]

Ron Natalie wrote:

Stavros might need evenly distributed numbers. A polar coordinate scheme would bunch numbers up around the center, at least. How so? If the [0, sphere-radius] random number generator is

uniform, so would the density with respect to the center.

No, the probability of being close to the center is higher than the
probability of being around the edges.

[Stavros Christoforou]

No, the probability of being close to the center is higher than the
probability of being around the edges.

May I ask why?
Stavros might need evenly distributed numbers. A polar coordinate
scheme would bunch numbers up around the center, at least.

How so? If the [0, sphere-radius] random number generator is uniform, so would the density with respect to the center.

The density would be selected from a pdf, eg

s = - (log ((double) (rand()+ 1.0)/RAND_MAX))/sigmat

(just a random function)

Therefore my problem focuses more on how to select the points within the
sphere, and my idea so far was similar to Philip's. However, I am sure
something faster and more "code-correct" exists.

Also, sorry if I posted this on the wrong group, but as I am creating
the program on C++ I thought that this would be the appropriate place to
ask questions.
Stavros

[Shezan Baig]

Stavros Christoforou wrote:

No, the probability of being close to the center is higher than the
probability of being around the edges.

May I ask why?

It might be a little hard to explain without graphical aid :)

But basically, to be around the center, you need to have a small
radius. The angles don't really matter here - as long as the radius is
small, then the point will be near the center.

To make the point closer to a particular edge, you need a bigger radius
*and* the correct angle (or direction, or whatever). So the
probability of this happening is less.

Hope this helps,
-shez-

[Karl Heinz Buchegger]

Ron Natalie wrote:

Stavros might need evenly distributed numbers. A polar coordinate scheme
would bunch numbers up around the center, at least. How so?

The problem is that the mean distance between 2 neighboring points is
smaller near the center then it is at the circumference. So the number
of points near the center is packed tighter -> the point density (nr of points
per partial volume) is higher. Depending on the application this might
or might not be acceptable.
If the [0, sphere-radius] random number generator is uniform, so would
the density with respect to the center.

The fix is to bound the sphere with a cube, tangent on all faces, and find
random points in the cube by finding random xyz points, each >= -r and < r,
where r is the radius of the sphere. Then filter out all points laying
outside the sphere, by comparing x^2 + y^2 + z^2 to r^2.

That wouldn't be any better and a lot slower as a lot of samples would have
to be generated, tested, and discarded.

.... Yes, but it generates a uniform volume sampling of the sphere. Something
the radius/angle method doesn't do.

--
Karl Heinz Buchegger
kb******@gascad .at