Solid angles of spherical triangle and square

SolidAngle.hlsl

float3 SolidAngle(float3 p0, float3 p1, float3 p2)
{
    // edges
    float cos01 = dot(p0, p1);
    float cos12 = dot(p1, p2);
    float cos20 = dot(p2, p0);

    float sin01Inv = 1 / sqrt(1 - cos01 * cos01);
    float sin12Inv = 1 / sqrt(1 - cos12 * cos12);
    float sin20Inv = 1 / sqrt(1 - cos20 * cos20);

    float cos0 = (cos12 - cos01 * cos12) * (sin01Inv * sin12Inv);
    float cos1 = (cos20 - cos12 * cos01) * (sin12Inv * sin01Inv);
    float cos2 = (cos12 - cos12 * cos20) * (sin12Inv * sin20Inv);
    return acos(cos0) + acos(cos1) + acos(cos2) - UNITY_PI;
}

// p0, p1, p2, p3 are points in the unit sphere such that
// the path p0->p1->p2->p3 defines a boundary of a spherical square.
float4 SolidAngle(float3 p0, float3 p1, float3 p2, float3 p3)
{
    // edges
    float cos01 = dot(p0, p1);
    float cos12 = dot(p1, p2);
    float cos23 = dot(p2, p3);
    float cos30 = dot(p3, p0);

    // diagonals
    float cos20 = dot(p2, p0);
    float cos31 = dot(p3, p1);

    float sin01Inv = 1 / sqrt(1 - cos01 * cos01);
    float sin12Inv = 1 / sqrt(1 - cos12 * cos12);
    float sin23Inv = 1 / sqrt(1 - cos23 * cos23);
    float sin30Inv = 1 / sqrt(1 - cos30 * cos30);

    // p0,p1,p3
    float cos0 = (cos31 - cos01 * cos30) * (sin01Inv * sin30Inv);
    // p1,p2,p0
    float cos1 = (cos20 - cos12 * cos01) * (sin12Inv * sin01Inv);
    // p2, p3, p1
    float cos2 = (cos31 - cos12 * cos23) * (sin12Inv * sin23Inv);
    // p3, p0, p2
    float cos3 = (cos20 - cos23 * cos30) * (sin23Inv * sin30Inv);
    return acos(cos0) + acos(cos1) + acos(cos2) + acos(cos3) - UNITY_TWO_PI;
}