1 files changed, 97 insertions, 20 deletions

diff --git a/intern/cycles/kernel/kernel_montecarlo.h b/intern/cycles/kernel/kernel_montecarlo.h
index ba25c0e24e4..ce37bd0b15e 100644
--- a/intern/cycles/kernel/kernel_montecarlo.h
+++ b/intern/cycles/kernel/kernel_montecarlo.h
@@ -195,31 +195,108 @@ 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;
-  float NI = dot(N, I);
-  float NgR, threshold;
-
-  /* Check if the incident ray is coming from behind normal N. */
-  if (NI > 0) {
-    /* Normal reflection */
-    R = (2 * NI) * N - I;
-    NgR = dot(Ng, R);
-
-    /* Reflection rays may always be at least as shallow as the incoming ray. */
-    threshold = min(0.9f * dot(Ng, I), 0.01f);
-    if (NgR >= threshold) {
-      return N;
+  float3 R = 2 * dot(N, I) * N - I;
+
+  /* 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;
+  }
+
+  /* Form coordinate system with Ng as the Z axis and N inside the X-Z-plane.
+   * The X axis is found by normalizing the component of N that's orthogonal to Ng.
+   * The Y axis isn't actually needed.
+   */
+  float NdotNg = dot(N, Ng);
+  float3 X = normalize(N - NdotNg * Ng);
+
+  /* Keep math expressions. */
+  /* clang-format off */
+  /* 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.
+   */
+  /* clang-format on */
+
+  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;
+    }
+  }
+  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 {
-    /* Bad incident */
-    R = -I;
-    NgR = dot(Ng, R);
-    threshold = 0.01f;
+    return Ng;
   }
 
-  R = R + Ng * (threshold - NgR);            /* Lift the reflection above the threshold. */
-  return normalize(I * len(R) + R * len(I)); /* Find a bisector. */
+  return N_new.x * X + N_new.y * Ng;
 }
 
 CCL_NAMESPACE_END