1 files changed, 83 insertions, 16 deletions

diff --git a/intern/cycles/kernel/kernel_montecarlo.h b/intern/cycles/kernel/kernel_montecarlo.h
index 9b96bb80c32..dde93844dd3 100644
--- a/intern/cycles/kernel/kernel_montecarlo.h
+++ b/intern/cycles/kernel/kernel_montecarlo.h
@@ -187,7 +187,10 @@ ccl_device float2 regular_polygon_sample(float corners, float rotation, float u,
 ccl_device float3 ensure_valid_reflection(float3 Ng, float3 I, float3 N)
 {
 	float3 R = 2*dot(N, I)*N - I;
-	if(dot(Ng, R) >= 0.05f) {
+
+	/* Reflection rays may always be at least as shallow as the incoming ray. */
+	float threshold = min(0.9f*dot(Ng, I), 0.01f);
+	if(dot(Ng, R) >= threshold) {
 		return N;
 	}
 
@@ -195,24 +198,88 @@ ccl_device float3 ensure_valid_reflection(float3 Ng, float3 I, float3 N)
 	 * The X axis is found by normalizing the component of N that's orthogonal to Ng.
 	 * The Y axis isn't actually needed.
 	 */
-	float3 X = normalize(N - dot(N, Ng)*Ng);
-
-	/* Calculate N.z and N.x in the local coordinate system. */
-	float Iz = dot(I, Ng);
-	float Ix2 = sqr(dot(I, X)), Iz2 = sqr(Iz);
-	float Ix2Iz2 = Ix2 + Iz2;
-
-	float a = safe_sqrtf(Ix2*(Ix2Iz2 - sqr(0.05f)));
-	float b = Iz*0.05f + Ix2Iz2;
-	float c = (a + b > 0.0f)? (a + b) : (-a + b);
+	float NdotNg = dot(N, Ng);
+	float3 X = normalize(N - NdotNg*Ng);
+
+	/* Calculate N.z and N.x in the local coordinate system.
+	 *
+	 * The goal of this computation is to find a N' that is rotated towards Ng just enough
+	 * to lift R' above the threshold (here called t), therefore dot(R', Ng) = t.
+	 *
+	 * According to the standard reflection equation, this means that we want dot(2*dot(N', I)*N' - I, Ng) = t.
+	 *
+	 * Since the Z axis of our local coordinate system is Ng, dot(x, Ng) is just x.z, so we get 2*dot(N', I)*N'.z - I.z = t.
+	 *
+	 * The rotation is simple to express in the coordinate system we formed - since N lies in the X-Z-plane, we know that
+	 * N' will also lie in the X-Z-plane, so N'.y = 0 and therefore dot(N', I) = N'.x*I.x + N'.z*I.z .
+	 *
+	 * Furthermore, we want N' to be normalized, so N'.x = sqrt(1 - N'.z^2).
+	 *
+	 * With these simplifications, we get the final equation 2*(sqrt(1 - N'.z^2)*I.x + N'.z*I.z)*N'.z - I.z = t.
+	 *
+	 * The only unknown here is N'.z, so we can solve for that.
+	 *
+	 * The equation has four solutions in general:
+	 *
+	 * N'.z = +-sqrt(0.5*(+-sqrt(I.x^2*(I.x^2 + I.z^2 - t^2)) + t*I.z + I.x^2 + I.z^2)/(I.x^2 + I.z^2))
+	 * We can simplify this expression a bit by grouping terms:
+	 *
+	 * a = I.x^2 + I.z^2
+	 * b = sqrt(I.x^2 * (a - t^2))
+	 * c = I.z*t + a
+	 * N'.z = +-sqrt(0.5*(+-b + c)/a)
+	 *
+	 * Two solutions can immediately be discarded because they're negative so N' would lie in the lower hemisphere.
+	 */
+	float Ix = dot(I, X), Iz = dot(I, Ng);
+	float Ix2 = sqr(Ix), Iz2 = sqr(Iz);
+	float a = Ix2 + Iz2;
+
+	float b = safe_sqrtf(Ix2*(a - sqr(threshold)));
+	float c = Iz*threshold + a;
+
+	/* Evaluate both solutions.
+	 * In many cases one can be immediately discarded (if N'.z would be imaginary or larger than one), so check for that first.
+	 * If no option is viable (might happen in extreme cases like N being in the wrong hemisphere), give up and return Ng. */
+	float fac = 0.5f/a;
+	float N1_z2 = fac*(b+c), N2_z2 = fac*(-b+c);
+	bool valid1 = (N1_z2 > 1e-5f) && (N1_z2 <= (1.0f + 1e-5f));
+	bool valid2 = (N2_z2 > 1e-5f) && (N2_z2 <= (1.0f + 1e-5f));
+
+	float2 N_new;
+	if(valid1 && valid2) {
+		/* If both are possible, do the expensive reflection-based check. */
+		float2 N1 = make_float2(safe_sqrtf(1.0f - N1_z2), safe_sqrtf(N1_z2));
+		float2 N2 = make_float2(safe_sqrtf(1.0f - N2_z2), safe_sqrtf(N2_z2));
+
+		float R1 = 2*(N1.x*Ix + N1.y*Iz)*N1.y - Iz;
+		float R2 = 2*(N2.x*Ix + N2.y*Iz)*N2.y - Iz;
+
+		valid1 = (R1 >= 1e-5f);
+		valid2 = (R2 >= 1e-5f);
+		if(valid1 && valid2) {
+			/* If both solutions are valid, return the one with the shallower reflection since it will be closer to the input
+			 * (if the original reflection wasn't shallow, we would not be in this part of the function). */
+			N_new = (R1 < R2)? N1 : N2;
+		}
+		else {
+			/* If only one reflection is valid (= positive), pick that one. */
+			N_new = (R1 > R2)? N1 : N2;
+		}
 
-	float Nz = safe_sqrtf(0.5f * c * (1.0f / Ix2Iz2));
-	float Nx = safe_sqrtf(1.0f - sqr(Nz));
+	}
+	else if(valid1 || valid2) {
+		/* Only one solution passes the N'.z criterium, so pick that one. */
+		float Nz2 = valid1? N1_z2 : N2_z2;
+		N_new = make_float2(safe_sqrtf(1.0f - Nz2), safe_sqrtf(Nz2));
+	}
+	else {
+		return Ng;
+	}
 
-	/* Transform back into global coordinates. */
-	return Nx*X + Nz*Ng;
+	return N_new.x*X + N_new.y*Ng;
 }
 
 CCL_NAMESPACE_END
 
-#endif /* __KERNEL_MONTECARLO_CL__ */
+#endif  /* __KERNEL_MONTECARLO_CL__ */