On Floating‐Point Normal Vectors

Abstract

In this paper we analyze normal vector representations. We derive the error of the most widely used representation, namely 3D floating-point normal vectors. Based on this analysis, we show that, in theory, the discretization error inherent to single precision floating-point normals can be achieved by 2 50.2 uniformly distributed normals, addressable by 51 bits. We review common sphere parameterizations and show that octahedron normal vectors perform best: they are fast and stable to compute, have a controllable error, and require only 1 bit more than the theoretical optimal discretization with the same error.

Encoding normal vectors using optimized spherical coordinates

“…We demonstrate the performance of our method by comparing it to several other methods including Healpix [3], octahedral subdivision (Octa) [6,7,8,9], octahedral normal vectors (ONV) [4], sextant encoding (Sextant) [5], and the sphere1 covering (Sphere1) [9]. As test data, we use normal vectors generated from the polygons of common surfaces found in Computer Graphics.…”

“…While the same procedure can be used to decode the vector, the more common implementation is to use a table lookup. The latter is quite fast, but as Meyer et al [4] point out, for high levels of accuracy the lookup table can dominate the storage costs and may not even fit in memory.…”

“…The Octahedron Normal Vector method [4] uses an octahedron to encode unit vectors and does so by flattening the octahedron into a 2D square. The authors then place a regular grid over the square and encode the vector as an index.…”

“…Meyer et al [4] showed that this representation is redundant and only 51 bits are sufficient to represent unit vectors within floating point precision. However, floating point accuracy is not always necessary, and thus good encoding/decoding techniques for 3D unit vectors that bound the maximum encoding error are needed.…”

Coarse-grained multiresolution structures for mobile exploration of gigantic surface models

“…In CATP, we improve over these local quantization approaches by expressing positions of mesh fragment vertices in the barycentric coordinate system relative to the containing tetrahedron. Hardwarefriendly normal compression is achieved through an octahedral parametrization of normals [Meyer et al 2010]. To our knowledge CATP is the first method fully supporting local quantization in a general adaptive 3D mesh structure.…”