I need to average out the normal vectors of polygons that neighbor other polygons. I also thought I should average out the tangents along with the normals. What I expect from the result is that if:
normal_1 and tangent_1 and bitangent_1 are vectors orthogonal to each other,
then:
if there's another group of three vectors, normal_2 and tangent_2 and bitangent_2,
that if I average normal_1 with normal_2, tangent_1 with tangent2, and bitangent_1 with bitangent_2,
that the resulting three vectors (the averaged vectors) are all are orthogonal to each other also.
I'd also like to know if it's the case when averaging any number of vectors, in other words if there are 4 groups of three orthogonal vectors, normal, tangent and bitangent, that:
the average of normal_1, normal_2, normal_3 and normal_4 is orthogonal to the average of tangent_1, tangent_2, tangent_3 and tangent_4, and that these are both orthogonal to the average of bitangent_1, bitangent_2, bitangent_3 and bitangent_4.
Consider two copies of the standard basis in $\mathbb{R}^3$. Match the vector in the first set $e_i$ to the vector in the second set $e_{i+1 \pmod 3}$ and average the pairs.
Then the angle between any two averaged vectors is $\frac{\pi}{3}$, not $\frac{\pi}{2}$.