MeshTools: cache angles and cross product in generateSmoothNormals().

It allocates one more array, but that speeds up the calculation to twice
as fast. Before the benchmark was around 1 ms for flat normals and 12 ms
for smooth, now it's 6 ms for smooth. There is probably more I could do
(I feel like I could save at least one more normalization), however this
is good enough for now.

src/Magnum/MeshTools/GenerateNormals.cpp

@@ -131,6 +131,29 @@ template<class T> void generateSmoothNormalsInto(const Containers::StridedArrayV
     /* Now, triangleCount should be all zeros, we don't need it anymore and the
        underlying `normals` array is ready to get filled with real output. */
 
+    /* Precalculate cross product and interior angles of each face --- the loop
+       below would otherwise calculate it for every vertex, which is at least
+       3x as much work */
+    Containers::Array<std::pair<Vector3, Math::Vector3<Rad>>> crossAngles{Math::NoInit, indices.size()/3};
+    for(std::size_t i = 0; i != crossAngles.size(); ++i) {
+        const Vector3 v0 = positions[indices[i*3 + 0]];
+        const Vector3 v1 = positions[indices[i*3 + 1]];
+        const Vector3 v2 = positions[indices[i*3 + 2]];
+
+        /* Cross product */
+        crossAngles[i].first = Math::cross(v2 - v1, v0 - v1);
+
+        /* Inner angle at each vertex of the triangle. The last one can be
+           calculated as a remainder to 180°. */
+        using namespace Math::Literals;
+        crossAngles[i].second[0] = Math::angle(
+            (v1 - v0).normalized(), (v2 - v0).normalized());
+        crossAngles[i].second[1] = Math::angle(
+            (v0 - v1).normalized(), (v2 - v1).normalized());
+        crossAngles[i].second[2] = Rad(180.0_degf)
+            - crossAngles[i].second[0] - crossAngles[i].second[1];
+    }
+
     /* For every vertex v, calculate normals from all faces it belongs to and
        average them */
     for(std::size_t v = 0; v != positions.size(); ++v) {
@@ -146,23 +169,15 @@ template<class T> void generateSmoothNormalsInto(const Containers::StridedArrayV
 
             /* Cross product is a vector in direction of the normal with length
                equal to size of the parallelogram */
-            const Vector3 cross = Math::cross(positions[v2i] - positions[v1i],
+            const std::pair<Vector3, Math::Vector3<Rad>>& crossAngle = crossAngles[triangleIds[t]];
-                                              positions[v0i] - positions[v1i]);
 
             /* Angle between two sides of the triangle that share vertex `v`.
-               The shared vertex can be one of the three, so three ways to
+               The shared vertex can be one of the three. */
-               calculate the angle */
+            Rad angle;
-            Vector3 a{Math::NoInit}, b{Math::NoInit};
+            if(v == v0i) angle = crossAngle.second[0];
-            if(v == v0i) {
+            else if(v == v1i) angle = crossAngle.second[1];
-                a = positions[v1i] - positions[v0i];
+            else if(v == v2i) angle = crossAngle.second[2];
-                b = positions[v2i] - positions[v0i];
+            else CORRADE_ASSERT_UNREACHABLE(); /* LCOV_EXCL_LINE */
-            } else if(v == v1i) {
-                a = positions[v0i] - positions[v1i];
-                b = positions[v2i] - positions[v1i];
-            } else if(v == v2i) {
-                a = positions[v0i] - positions[v2i];
-                b = positions[v1i] - positions[v2i];
-            } else CORRADE_ASSERT_UNREACHABLE(); /* LCOV_EXCL_LINE */
 
             /* The normal is cross.normalized(), we need to multiply it it by
                surface area which is cross.length()/2. Since normalization is
@@ -172,7 +187,7 @@ template<class T> void generateSmoothNormalsInto(const Containers::StridedArrayV
                that as well. Finally we need to weight by the angle, and in
                that case only the ratio is important as well, so it doesn't
                matter if degrees or radians. */
-            normals[v] += cross*Float(Math::angle(a.normalized(), b.normalized()));
+            normals[v] += crossAngle.first*Float(angle);
         }
 
         /* Normalize the accumulated direction */

src/Magnum/MeshTools/GenerateNormals.h

@@ -121,7 +121,7 @@ extern template MAGNUM_MESHTOOLS_EXPORT Containers::Array<Vector3> generateSmoot
 A variant of @ref generateSmoothNormals() that fills existing memory instead of
 allocating a new array. The @p normals array is expected to have the same size
 as @p positions. Note that even with the output array this function isn't fully
-allocation-free --- it still allocates two additional internal arrays for
+allocation-free --- it still allocates three additional internal arrays for
 adjacent face calculation.
 @see @ref generateFlatNormalsInto()
 */