In a recent blog post1 (inspired from reading Peter Shirley’s Ray Tracing in One Weekend) Karthik Karanth derives generating uniform points inside the unit sphere from scratch. This is just a quick note on how this can be improved.


Uniform points in sphere


If we’re given a black-box implementation of point on unit sphere we could adapt that to an interior point by properly scaling the result. We need a uniform parameterization of the volume: the volume of a sphere is $\frac{4}{3}\pi~r^3$, so the relative cumulative volume is $r^3$ which give a scale factor which is the cube-root of a uniform float.

In a previous post2 I sketched out generating uniform points on sphere as transformed points in disc. Adding the scaling factor to the example code gives:

void uniform_s2_filled(vec3_t* p)
{
  float d,s,r;
  vec2_t v;

  d    = uniform_disc(&v);      // point on disc
  r    = cbrtf(rng_f32());      // uniform param of radius
  s    = 2.f*sqrtf(1.f-d)*r;    // composed scales for x,y
  p->x = s*v.x;
  p->y = s*v.y;
  p->z = r*(1.f-2.f*d);
}


The uniform_disc function from the previous post is a rejection method with rejection rate of ~0.21. A drop in replacement without rejection would look like:

float uniform_disc(vec2_t* p)
{
  float d = rng_f32();
  float a = 2.f*PI*rng_f32();
  float r = sqrtf(d);
  p->x = r*cosf(a);
  p->y = r*sinf(a);

  return d;                   // r^2
}


So a disc transform based method drops the inverse trig op, one pair for forward trig ops without rejection and is trig free for rejection based.



Uniform points in spherically capped cone


Tossing out another adaption of my previous post: If we take the uniform point on spherical cap and rescale as above we get a uniform point in the volume of the intersection of the unit sphere and conic:

// 'h' is height of the cap
void uniform_intersect_s2_cone(vec3_t* p, float h)
{
  float k,s,r;
  vec2_t v;

  r = cbrtf(rng_f32());
  k = h*uniform_disc(&v);
  s = sqrtf(h*(2.f-k))*r;
  p->x = s*v.x;
  p->y = s*v.y;
  p->z = r*(1.f-k);
}

References and Footnotes

  1. Generating Random Points in a Sphere, Karthik Karanth, 2018 link 

  2. Uniform points on disc, circle, sphere and caps, 2016 local post