Re: [Algorithms] Sphere to rectangle

[Fabian G. <f.g...@49...>]

Benjamin Rouveyrol wrote:
> Hi,
> 
> The way I solved this (but a bit math heavy):
> If R is the light Radius and D the distance between the camera and the 
> light:
> NewR = D * tan(asin(R/D))
> 
> Then you compute Up and right vectors based on the camera and light 
> positions (and the camera Up).
> You end up with 4 corners: LightCenter +|- Up * NewR  +|- Right * NewR
> 
> 
> You can project them and this should give you your bounding rectangle. 
> If someone has a simpler solution, I'd be most interested ^_^

Eric Lengyel explains one way to solve this on the last page of his "The 
Mechanics of Robust Stencil Shadows" article:

http://www.gamasutra.com/view/feature/2942/the_mechanics_of_robust_stencil_.php?page=6

I derived an alternative method for some hobby project around 2004 that 
is somewhat more straightforward. I don't have the original derivation 
anymore, but I remember that it was fairly straightforward trig:

// Calculates bounding rectangle in normalized device coordinates for
// the view-space sphere with center "center" and radius "r". "zoom"
// contains the first two diagonal entries of your projection matrix
// (which is assumed to be perspective). You should do a rough rejection
// test of the sphere against the frustum first.

static void CalcSphereBounds(const Vector& center, float r, const float 
zoom[2], float minb[2], float maxb[2])
{
   // by default, assume that full screen covered
   minb[0] = minb[1] = -1.0f;
   maxb[0] = maxb[1] =  1.0f;

   // once for x, once for y
   for(int i=0;i<2;i++)
   {
     float x = center[i];
     float z = center.z;
     float ds = x*x + z*z;
     float l = ds - r * r;

     if(l > 0.0f)
     {
       float s,c;
       l = sqrt(l);

       s = x * l - z * r; // ds*sin(alpha)
       c = x * r + z * l; // ds*cos(alpha)
       if(z*ds > -r*s) // left/top intersection has positive z
         minb[i] = max(-1.0f, s*zoom[i]/c);

       s = z * r + x * l; // ds*sin(beta)
       c = z * l - x * r; // ds*cos(beta)
       if(z*ds > r*s) // right/bottom intersection has positive z
         maxb[i] = min(1.0f, s*zoom[i]/c);
     }
   }
}

I'll try to re-derive this so as not to leave a bunch of unexplained 
formulas standing in the room, but it's a relatively short solution and 
it's worked fine for me, so I guess it's of interest.

Kind regards,
-Fabian