Scan-converting the Sphere: Circle In-Out Test

FAST 2D/3D Texture Coordinate Evaluation
Kenny Hoff 7/15/97

In order to be as efficient as possible, we will use the normals calculated previously in screen space. For the 2D textures coordinates, we will simply assume that we have a flat 2D texture mapped onto the circular silhouette of sphere, so we can simply use the normalized x and y coordinates of the normal as the texture coords. For the 3D case, all three components will be used.

We are given the following:

Normal's x and y coordinates in the range [-1,1]
Normal's z coordinate in the range [0,1]

We would like to make them fit in the range:

Nx, Ny, and Nz in [0,1]

So for the x and y components we need to scale the range by 0.5 and translate by 0.5; the z components remain the same.

However, because of the fixed-point formats required on both ends, we have some additional difficulties:

Normal components are in 16-bit 4.12 fixed-point format. In other words, 4 bits of whole number and 12 bits of fraction. This is accomplished by a scale of 4096 or 2¹².
The texture coords must be in 16-bit 0.16 fixed-point format - all fractional and normalized within [0,1]. The scaling factor is 65536 or 2¹⁶.

Our range of values is as follows now:

Nx and Ny is in [-4096,4096] and must be in [0,2¹⁶]
Nz is in [0,4096] and must be in [0,2¹⁶]

The conversion proceeds as follows:

  TextureX = (NormalX * 2⁴ * 0.5) + (2¹⁶ * 0.5)
           = (NormalX * 2⁴ * 2^-1) + 2¹⁵
           = (NormalX * 2³) + 2¹⁵
           = (NormalX << 3) + 2¹⁵

  TextureY = (NormalY << 3) + 2¹⁵

  TextureZ = NormalZ * 2⁴
           = NormalZ << 4

Where NormalX and NormalY representing the range [-1,1] and NormalZ representing the range [0,1] in 4.12 fixed-point format:

  4.12 NormalX and NormalY = NormalX * 4096
                           = NormalX * 2¹²

              4.12 NormalZ = NormalZ * 2¹²

So the normal components are converted in texture coords by a simple scaling (accomplished through a left shift) and translation (except in the case of z). The bit-shifts are fairly straightforward, but it is important to note two options for the translation (addition of 2¹⁵). Given the 16-bit normal x and y components after the scaling phase, the addition of 2¹⁵ can be accomplished by one of the following methods:

Brute-force 16-bit addition of a 16-bit integer 2¹⁵ to the 16-bit scaled result.
Use only an 8-bit addition by adding 2⁷ (128) to the most significant byte.
Simply flip the most significant bit using an XOR operation.