Encoding normal vectors using optimized spherical coordinates

We present a method for encoding unit vectors based on spherical coordinates that out-performs existing encoding methods both in terms of accuracy and encoding/decoding time. Given a tolerance ǫ, we solve a simple, discrete optimization problem to find a set of points on the unit sphere that can trivially be indexed such that the difference in angle between the encoded vector and the original are no more than ǫ apart. To encode a unit vector, we simply compute its spherical coordinates and round the result based on the prior optimization solution. We also present a moving frame method that further reduces the amount of data to be encoded when vectors have some coherence. Our method is extremely fast in terms of encoding and decoding both of which take constant time O(1). The accuracy of our encoding is also comparable or better than previous methods for encoding unit vectors.