SphericalLayerDistance.hlsl

/*----------------------------------------------------------------------------------------------------------------------
 * Let's imagine two spheres with common center and different radiuses. This function computes distance from a point 
 * on the sphere with smaller radius (RMin) to the surface on the sphere with bigger radius (RMax) in the direction 
 * define by Mu (cosine between direction from the point on the smaller sphere to the point on the bigger sphere and 
 * direction from the point on the smaller sphere to the sphere center)
 * 
 * Complexity :  3 mad, 1 sqrt
 *--------------------------------------------------------------------------------------------------------------------*/
float DistanceToSphericalLayer(float RMin, float RMinSq, float RMaxSq, float Mu)
{
    return RMin * Mu + sqrt(RMinSq * (Mu * Mu - 1.0) + RMaxSq);
}


/*----------------------------------------------------------------------------------------------------------------------
 * Complexity : 3 mad, 2 sqrt, 1 sub
 *--------------------------------------------------------------------------------------------------------------------*/
float DistanceBetweenSphericalLayers(float RMinSq, float RMaxSq1, float RMaxSq2, float Mu)
{
    // Reference :
    //const float D1 = RMin * Mu + sqrt(RMinSq * (Mu * Mu - 1.0) + RMaxSq1);
    //const float D2 = RMin * Mu + sqrt(RMinSq * (Mu * Mu - 1.0) + RMaxSq2);
    //return D2-D1;
    const float MuSqMinusOne = Mu * Mu - 1.0;
    const float D1Partial = sqrt(RMinSq * MuSqMinusOne + RMaxSq1);
    const float D2Partial = sqrt(RMinSq * MuSqMinusOne + RMaxSq2);
    return D2Partial - D1Partial;
}

/*----------------------------------------------------------------------------------------------------------------------
 * Complexity : 2 mad, 2 sqrt, 1 sub, 1 mul
 *--------------------------------------------------------------------------------------------------------------------*/
float DistanceBetweenSphericalLayers2(float RMinSq, float RMaxOverRMinSqMinusOne1, float RMaxOverRMinSqMinusOne2, float Mu)
{
    const float MuSq = Mu * Mu;
    const float D1Partial = sqrt(MuSq + RMaxOverRMinSqMinusOne1);
    const float D2Partial = sqrt(MuSq + RMaxOverRMinSqMinusOne2);
    return (D2Partial - D1Partial) * RMin;
}

/*----------------------------------------------------------------------------------------------------------------------
 * Complexity : 2 mad, 1 sqrt, 1 mul
 *--------------------------------------------------------------------------------------------------------------------*/
float DistanceToSphericalLayer2(float RMin, float RMaxOverRMinSqMinusOne, float Mu)
{
    return RMin * Mu + RMin * sqrt(Mu * Mu + RMaxOverRMinSqMinusOne);
}