<CORE>/ShaderLibrary/AreaLighting.hlsl full source
#ifndef UNITY_AREA_LIGHTING_INCLUDED
#define UNITY_AREA_LIGHTING_INCLUDED
#define APPROXIMATE_POLY_LIGHT_AS_SPHERE_LIGHT
#define APPROXIMATE_SPHERE_LIGHT_NUMERICALLY
// 'sinSqSigma' is the sine^2 of the half-angle subtended by the sphere (aperture) as seen from the shaded point.
// 'cosOmega' is the cosine of the angle between the normal and the direction to the center of the light.
// This function performs horizon clipping.
real DiffuseSphereLightIrradiance(real sinSqSigma, real cosOmega)
{
#ifdef APPROXIMATE_SPHERE_LIGHT_NUMERICALLY
real x = sinSqSigma;
real y = cosOmega;
// Use a numerical fit found in Mathematica. Mean absolute error: 0.00476944.
// You can use the following Mathematica code to reproduce our results:
// t = Flatten[Table[{x, y, f[x, y]}, {x, 0, 0.999999, 0.001}, {y, -0.999999, 0.999999, 0.002}], 1]
// m = NonlinearModelFit[t, x * (y + e) * (0.5 + (y - e) * (a + b * x + c * x^2 + d * x^3)), {a, b, c, d, e}, {x, y}]
return saturate(x * (0.9245867471551246 + y) * (0.5 + (-0.9245867471551246 + y) * (0.5359050373687144 + x * (-1.0054221851257754 + x * (1.8199061187417047 - x * 1.3172081704209504)))));
#else
#if 0 // Ref: Area Light Sources for Real-Time Graphics, page 4 (1996).
real sinSqOmega = saturate(1 - cosOmega * cosOmega);
real cosSqSigma = saturate(1 - sinSqSigma);
real sinSqGamma = saturate(cosSqSigma / sinSqOmega);
real cosSqGamma = saturate(1 - sinSqGamma);
real sinSigma = sqrt(sinSqSigma);
real sinGamma = sqrt(sinSqGamma);
real cosGamma = sqrt(cosSqGamma);
real sigma = asin(sinSigma);
real omega = acos(cosOmega);
real gamma = asin(sinGamma);
if (omega >= HALF_PI + sigma)
{
// Full horizon occlusion (case #4).
return 0;
}
real e = sinSqSigma * cosOmega;
UNITY_BRANCH
if (omega < HALF_PI - sigma)
{
// No horizon occlusion (case #1).
return e;
}
else
{
real g = (-2 * sqrt(sinSqOmega * cosSqSigma) + sinGamma) * cosGamma + (HALF_PI - gamma);
real h = cosOmega * (cosGamma * sqrt(saturate(sinSqSigma - cosSqGamma)) + sinSqSigma * asin(saturate(cosGamma / sinSigma)));
if (omega < HALF_PI)
{
// Partial horizon occlusion (case #2).
return saturate(e + INV_PI * (g - h));
}
else
{
// Partial horizon occlusion (case #3).
return saturate(INV_PI * (g + h));
}
}
#else // Ref: Moving Frostbite to Physically Based Rendering, page 47 (2015, optimized).
real cosSqOmega = cosOmega * cosOmega; // y^2
UNITY_BRANCH
if (cosSqOmega > sinSqSigma) // (y^2)>x
{
return saturate(sinSqSigma * cosOmega); // Clip[x*y,{0,1}]
}
else
{
real cotSqSigma = rcp(sinSqSigma) - 1; // 1/x-1
real tanSqSigma = rcp(cotSqSigma); // x/(1-x)
real sinSqOmega = 1 - cosSqOmega; // 1-y^2
real w = sinSqOmega * tanSqSigma; // (1-y^2)*(x/(1-x))
real x = -cosOmega * rsqrt(w); // -y*Sqrt[(1/x-1)/(1-y^2)]
real y = sqrt(sinSqOmega * tanSqSigma - cosSqOmega); // Sqrt[(1-y^2)*(x/(1-x))-y^2]
real z = y * cotSqSigma; // Sqrt[(1-y^2)*(x/(1-x))-y^2]*(1/x-1)
real a = cosOmega * acos(x) - z; // y*ArcCos[-y*Sqrt[(1/x-1)/(1-y^2)]]-Sqrt[(1-y^2)*(x/(1-x))-y^2]*(1/x-1)
real b = atan(y); // ArcTan[Sqrt[(1-y^2)*(x/(1-x))-y^2]]
return saturate(INV_PI * (a * sinSqSigma + b));
}
#endif
#endif
}
// The output is *not* normalized by the factor of 1/TWO_PI (this is done by the PolygonFormFactor function).
real3 ComputeEdgeFactor(real3 V1, real3 V2)
{
real subtendedAngle;
real V1oV2 = dot(V1, V2);
real3 V1xV2 = cross(V1, V2); // Plane normal (tangent to the unit sphere)
real sqLen = saturate(1 - V1oV2 * V1oV2); // length(V1xV2) = abs(sin(angle))
real rcpLen = rsqrt(max(FLT_EPS, sqLen)); // Make sure it is finite
#if 0
real y = rcpLen * acos(V1oV2);
#else
// Let y[x_] = ArcCos[x] / Sqrt[1 - x^2].
// Range reduction: since ArcCos[-x] == Pi - ArcCos[x], we only need to consider x on [0, 1].
real x = abs(V1oV2);
// Limit[y[x], x -> 1] == 1,
// Limit[y[x], x -> 0] == Pi/2.
// The approximation is exact at the endpoints of [0, 1].
// Max. abs. error on [0, 1] is 1.33e-6 at x = 0.0036.
// Max. rel. error on [0, 1] is 8.66e-7 at x = 0.0037.
real y = HALF_PI + x * (-0.99991 + x * (0.783393 + x * (-0.649178 + x * (0.510589 + x * (-0.326137 + x * (0.137528 + x * -0.0270813))))));
if (V1oV2 < 0)
{
y = rcpLen * PI - y;
}
#endif
return V1xV2 * y;
}
// Input: 3-5 vertices in the coordinate frame centered at the shaded point.
// Output: signed vector irradiance.
// No horizon clipping is performed.
real3 PolygonFormFactor(real4x3 L, real3 L4, uint n)
{
// The length cannot be zero since we have already checked
// that the light has a non-zero effective area,
// and thus its plane cannot pass through the origin.
L[0] = normalize(L[0]);
L[1] = normalize(L[1]);
L[2] = normalize(L[2]);
switch (n)
{
case 3:
L[3] = L[0];
break;
case 4:
L[3] = normalize(L[3]);
L4 = L[0];
break;
case 5:
L[3] = normalize(L[3]);
L4 = normalize(L4);
break;
}
// If the magnitudes of a pair of edge factors are
// nearly the same, catastrophic cancellation may occur:
// https://en.wikipedia.org/wiki/Catastrophic_cancellation
// For the same reason, the value of the cross product of two
// nearly collinear vectors is prone to large errors.
// Therefore, the algorithm is inherently numerically unstable
// for area lights that shrink to a line (or a point) after
// projection onto the unit sphere.
real3 F = ComputeEdgeFactor(L[0], L[1]);
F += ComputeEdgeFactor(L[1], L[2]);
F += ComputeEdgeFactor(L[2], L[3]);
if (n >= 4)
F += ComputeEdgeFactor(L[3], L4);
if (n == 5)
F += ComputeEdgeFactor(L4, L[0]);
return INV_TWO_PI * F; // The output may be projected onto the tangent plane (F.z) to yield signed irradiance.
}
// See "Real-Time Area Lighting: a Journey from Research to Production", slide 102.
// Turns out, despite the authors claiming that this function "calculates an approximation of
// the clipped sphere form factor", that is simply not true.
// First of all, above horizon, the function should then just return 'F.z', which it does not.
// Secondly, if we use the correct function called DiffuseSphereLightIrradiance(), it results
// in severe light leaking if the light is placed vertically behind the camera.
// So this function is clearly a hack designed to work around these problems.
real PolygonIrradianceFromVectorFormFactor(float3 F)
{
#if 1
float l = length(F);
return max(0, (l * l + F.z) / (l + 1));
#else
real sff = saturate(dot(F, F));
real sinSqAperture = sqrt(sff);
real cosElevationAngle = F.z * rsqrt(sff);
return DiffuseSphereLightIrradiance(sinSqAperture, cosElevationAngle);
#endif
}
// Expects non-normalized vertex positions.
// Output: F is the signed vector irradiance.
real PolygonIrradiance(real4x3 L, out real3 F)
{
#ifdef APPROXIMATE_POLY_LIGHT_AS_SPHERE_LIGHT
F = PolygonFormFactor(L, real3(0,0,1), 4); // Before horizon clipping.
return PolygonIrradianceFromVectorFormFactor(F); // Accounts for the horizon.
#else
// 1. ClipQuadToHorizon
// detect clipping config
uint config = 0;
if (L[0].z > 0) config += 1;
if (L[1].z > 0) config += 2;
if (L[2].z > 0) config += 4;
if (L[3].z > 0) config += 8;
// The fifth vertex for cases when clipping cuts off one corner.
// Due to a compiler bug, copying L into a vector array with 5 rows
// messes something up, so we need to stick with the matrix + the L4 vertex.
real3 L4 = L[3];
// This switch is surprisingly fast. Tried replacing it with a lookup array of vertices.
// Even though that replaced the switch with just some indexing and no branches, it became
// way, way slower - mem fetch stalls?
// clip
uint n = 0;
switch (config)
{
case 0: // clip all
break;
case 1: // V1 clip V2 V3 V4
n = 3;
L[1] = -L[1].z * L[0] + L[0].z * L[1];
L[2] = -L[3].z * L[0] + L[0].z * L[3];
break;
case 2: // V2 clip V1 V3 V4
n = 3;
L[0] = -L[0].z * L[1] + L[1].z * L[0];
L[2] = -L[2].z * L[1] + L[1].z * L[2];
break;
case 3: // V1 V2 clip V3 V4
n = 4;
L[2] = -L[2].z * L[1] + L[1].z * L[2];
L[3] = -L[3].z * L[0] + L[0].z * L[3];
break;
case 4: // V3 clip V1 V2 V4
n = 3;
L[0] = -L[3].z * L[2] + L[2].z * L[3];
L[1] = -L[1].z * L[2] + L[2].z * L[1];
break;
case 5: // V1 V3 clip V2 V4: impossible
break;
case 6: // V2 V3 clip V1 V4
n = 4;
L[0] = -L[0].z * L[1] + L[1].z * L[0];
L[3] = -L[3].z * L[2] + L[2].z * L[3];
break;
case 7: // V1 V2 V3 clip V4
n = 5;
L4 = -L[3].z * L[0] + L[0].z * L[3];
L[3] = -L[3].z * L[2] + L[2].z * L[3];
break;
case 8: // V4 clip V1 V2 V3
n = 3;
L[0] = -L[0].z * L[3] + L[3].z * L[0];
L[1] = -L[2].z * L[3] + L[3].z * L[2];
L[2] = L[3];
break;
case 9: // V1 V4 clip V2 V3
n = 4;
L[1] = -L[1].z * L[0] + L[0].z * L[1];
L[2] = -L[2].z * L[3] + L[3].z * L[2];
break;
case 10: // V2 V4 clip V1 V3: impossible
break;
case 11: // V1 V2 V4 clip V3
n = 5;
L[3] = -L[2].z * L[3] + L[3].z * L[2];
L[2] = -L[2].z * L[1] + L[1].z * L[2];
break;
case 12: // V3 V4 clip V1 V2
n = 4;
L[1] = -L[1].z * L[2] + L[2].z * L[1];
L[0] = -L[0].z * L[3] + L[3].z * L[0];
break;
case 13: // V1 V3 V4 clip V2
n = 5;
L[3] = L[2];
L[2] = -L[1].z * L[2] + L[2].z * L[1];
L[1] = -L[1].z * L[0] + L[0].z * L[1];
break;
case 14: // V2 V3 V4 clip V1
n = 5;
L4 = -L[0].z * L[3] + L[3].z * L[0];
L[0] = -L[0].z * L[1] + L[1].z * L[0];
break;
case 15: // V1 V2 V3 V4
n = 4;
break;
}
if (n == 0) return 0;
// 2. Integrate
F = PolygonFormFactor(L, L4, n); // After the horizon clipping.
// 3. Compute irradiance
return max(0, F.z);
#endif
}
// This function assumes that inputs are well-behaved, e.i.
// that the line does not pass through the origin and
// that the light is (at least partially) above the surface.
float I_diffuse_line(float3 C, float3 A, float hl)
{
// Solve C.z + h * A.z = 0.
float h = -C.z * rcp(A.z); // May be Inf, but never NaN
// Clip the line segment against the z-plane if necessary.
float h2 = (A.z >= 0) ? max( hl, h)
: min( hl, h); // P2 = C + h2 * A
float h1 = (A.z >= 0) ? max(-hl, h)
: min(-hl, h); // P1 = C + h1 * A
// Normalize the tangent.
float as = dot(A, A); // |A|^2
float ar = rsqrt(as); // 1/|A|
float a = as * ar; // |A|
float3 T = A * ar; // A/|A|
// Orthogonal 2D coordinates:
// P(n, t) = n * N + t * T.
float tc = dot(T, C); // C = n * N + tc * T
float3 P0 = C - tc * T; // P(n, 0) = n * N
float ns = dot(P0, P0); // |P0|^2
float nr = rsqrt(ns); // 1/|P0|
float n = ns * nr; // |P0|
float Nz = P0.z * nr; // N.z = P0.z/|P0|
// P(n, t) - C = P0 + t * T - P0 - tc * T
// = (t - tc) * T = h * A = (h * a) * T.
float t2 = tc + h2 * a; // P2.t
float t1 = tc + h1 * a; // P1.t
float s2 = ns + t2 * t2; // |P2|^2
float s1 = ns + t1 * t1; // |P1|^2
float mr = rsqrt(s1 * s2); // 1/(|P1|*|P2|)
float r2 = s1 * (mr * mr); // 1/|P2|^2
float r1 = s2 * (mr * mr); // 1/|P1|^2
// I = (i1 + i2 + i3) / Pi.
// i1 = N.z * (P2.t / |P2|^2 - P1.t / |P1|^2).
// i2 = -T.z * (P2.n / |P2|^2 - P1.n / |P1|^2).
// i3 = N.z * ArcCos[Dot[P1, P2] / (|P1| * |P2|)] / |P0|.
float i12 = (Nz * t2 - (T.z * n)) * r2
- (Nz * t1 - (T.z * n)) * r1;
// Guard against numerical errors.
float dt = min(1, (ns + t1 * t2) * mr);
float i3 = acos(dt) * (Nz * nr); // angle * cos(θ) / r^2
// Guard against numerical errors.
return INV_PI * max(0, i12 + i3);
}
// Computes 1 / length(mul(transpose(inverse(invM)), normalize(ortho))).
float ComputeLineWidthFactor(float3x3 invM, float3 ortho, float orthoSq)
{
// transpose(inverse(invM)) = 1 / determinant(invM) * cofactor(invM).
// Take into account the sparsity of the matrix:
// {{a,0,b},
// {0,c,0},
// {d,0,1}}
float a = invM[0][0];
float b = invM[0][2];
float c = invM[1][1];
float d = invM[2][0];
float det = c * (a - b * d);
float3 X = float3(c * (ortho.x - d * ortho.z),
(ortho.y * (a - b * d)),
c * (-b * ortho.x + a * ortho.z)); // mul(cof, ortho)
// 1 / length(1/s * X) = abs(s) / length(X).
return abs(det) * rsqrt(dot(X, X) * orthoSq) * orthoSq; // rsqrt(x^2) * x^2 = x
}
float I_ltc_line(float3x3 invM, float3 center, float3 axis, float halfLength)
{
float3 ortho = cross(center, axis);
float orthoSq = dot(ortho, ortho);
// Check whether the line passes through the origin.
bool quit = (orthoSq == 0);
// Check whether the light is entirely below the surface.
// We must test twice, since a linear transformation
// may bring the light above the surface (a side-effect).
quit = quit || (center.z + halfLength * abs(axis.z) <= 0);
// Transform into the diffuse configuration.
// This is a sparse matrix multiplication.
// Pay attention to the multiplication order
// (in case your matrices are transposed).
float3 C = mul(invM, center);
float3 A = mul(invM, axis);
// Check whether the light is entirely below the surface.
// We must test twice, since a linear transformation
// may bring the light below the surface (as expected).
quit = quit || (C.z + halfLength * abs(A.z) <= 0);
float result = 0;
if (!quit)
{
float w = ComputeLineWidthFactor(invM, ortho, orthoSq);
result = I_diffuse_line(C, A, halfLength) * w;
}
return result;
}
#endif // UNITY_AREA_LIGHTING_INCLUDED