3D Theory - Programming Particle Systems - forum discussion

By: nobody ( Nobody/Anonymous )
Random distribution of points / vectors  
2004-02-14 18:04

When building particle engines you want the emitter to expell particles with a certain range of velocity. Restraints on the direction ther are expeled are given by a direction and an angle.
The direction is the general direction of particle expulsion and the angle defines an angular variance. So what you get is particles expelled in a cone volume oriented on the direction axis. What is the best way for generating uniformly distributed direction vectors for these particles.
Initially I thought I needed only a random angle and random direction in (360 degrees) so I could build my new particle velocity.
This does not give uniform distribution over the cone. There is a more concentrated distribution torwards the direction axis. Intuitively I can explain this to myself by generating 100 random points on a circumference. Take any 100 random points on a circle of circumference of 5 and project then outward to a circle ten times the circumference. Obviously in that case there is more space between points.
Some may argue that this is actually a better particle emitter since it echos a tendency for particles to deviate less and less as they get further from the source. A sort of bell curve distribution.
Anyways I am now implementing a version where I pick a random point on a disk using methods described on the Wolfram math website. ( You have probably run across this by now and it is a useful reasources for all kinds of general and isoteric math often providing several different approachs or ways of looking at the same situation ). I am firing my direction vector through that point to the hull of the sphere. Which actually gives me a different but also biased mapping.
So what I am looking for is a way to generate points uniformly distributed on the intersection between a cone and a sphere.

By: nobody ( Nobody/Anonymous )
RE: Random distribution of points / vectors  
2004-02-15 09:50

Exactly as I thought of it. Anyways here is my solutions which may be flawed but appears to work well. I took functions for uniformly ditributing points on a sphere from wolfram site and started fooling with them.
Here is code for generating a point on the sphere:

vec3 Gen_Rand_Point_on_Unit_Sphere(){
float u,v;
float alpha, omega;
vec3 orient( 0, 1, 0 );

u = RANDOM( 1 );
v = RANDOM( 2 ) - 1;

omega = 2*Q_PI*u;
alpha = acos( v );

orient = orient.Rot_z( alpha );
orient = orient.Rot_y( omega );

return orient;
}

you can see my orientation vector is always pointed along the y axis. To base everything on a different direction just rotate all generated points.
On this first function it doesn't make a difference but in the following 2 it will.
Anyways I looked at the results and it looked pretty even to me.

I changed the constraints on the v value in the next to get 2 variations. The first one generates points in the cone around the y axis. The second one generates points in a ring between 2 angles passed to the function:

/*******************************************************
Given an angle in radians generate a random vector
around the y axis
*******************************************************/
vec3 Gen_Rand_Point_in_angle_on_Unit_Sphere( float ang ){
float u,v;
float alpha, omega;
double ht;
vec3 orient( 0, 1, 0 );

if( (ang<0) || (ang>Q_PI) ) return vec3( 0, 0, 1 );

ht = cos( ang );

u = RANDOM( 1 );
v = RANDOM( 1-ht ) + ht;

omega = 2*Q_PI*u;
alpha = acos( v );

orient = orient.Rot_z( alpha );// rotates around the z axis
orient = orient.Rot_y( omega );// rotates around the y axis

return orient;
}

/*******************************************************
Given t angles in radians generate a random vector
around the y axis in between the 2 angles specified
*******************************************************/
vec3 Gen_Rand_Point_in_band_on_Unit_Sphere( float ang1, float ang2 ){
float u,v;
float alpha, omega;
double sc, ht1, ht2;
vec3 orient( 0, 1, 0 );

if( (ang1>ang2) ) return vec3( 0, 0, 1 );
if( (ang1<0) || (ang1>Q_PI) ) return vec3( 0, 0, 1 );
if( (ang2<0) || (ang2>Q_PI) ) return vec3( 0, 0, 1 );

ht1 = cos( ang1 );
ht2 = cos( ang2 );
sc = fabs( ht1 - ht2 );

u = RANDOM( 1 );
v = RANDOM( sc ) + ht2;

omega = 2*Q_PI*u;
alpha = acos( v );

orient = orient.Rot_z( alpha );// rotates around the z axis
orient = orient.Rot_y( omega );// rotates around the y axis

return orient;
}

Since this was a quick solution I suspect I may have messed up in one of the following ways:
- I assumed v as a value between -1 and 1 in the original algorithm refers to a height range. All I do in the other functions is restrict the height range based on the height of the angle cos(ang). I suspect doing this destroys the uniformed distribution. I am I little lacking in attention to really study up and follow through on this but I suspect it so be warned.
- Maybe my use of the cos(ang) as a height restriction does not accurately reflect the angle desired. It appears to be correct in testing but I am just eyeballing it.

By: nobody ( Nobody/Anonymous )
RE: Random distribution of points / vectors  
2004-02-15 09:53

Here is also a the link I am using to the Wolfram site:
http://mathworld.wolfram.com/SpherePointPicking.html