Here are some questions I have:

Q1- why are we adding 1 to Rz?
Q2- why is the magnitude multiplied by 2?
Q3- why are we adding 0.5?

Q1: On the last page of the PPT presentation that Alvaro links, you should see "eye-space sphere normal n = (rx, ry, rz) + (0, 0, 1)" with the notation for (0,0,1) "direction to eye in local space." Thus, n (eye space) = (rx, ry, rz+1) by vector addition.

Q2 and Q3:

First, the variable m is just a convenient factor to calculate before u & v are calc'd as it applies to both equations - i.e., calculate it once rather than twice.

Actually, it's easier to understand if you work backwards from the tex coord equations:

u = Rx/m + 1/2

v = Ry/m + 1/2

Texture coordinates run from 0 to 1 over the width and height of the texture (sphere map). The sphere map is an unusual beast which maps sphere normalized coordinates to colors. Without getting into the actual generation of the sphere map (that's left as an exercise for the student), the first parameter to calculate is the normalized sphere coordinates. The sphere normal vector "points" to a position on the sphere which is the desired color.

CXnormalized = Rx/|n|

CYnormalized = Ry/|n|

Due to the way sphere map textures are generated, Rx / |n| maps to the range -1 to +1. Dividing by 2 changes that range to -0.5 to 0.5 from map bottom-left to top-right. Tex coords need to be in the range 0 to 1, which is done by adding 0.5: e.g., (-0.5 + 0.5, 0.5 + 0.5) == (0, 1).

So:

u = Rx / (2 * |n| ) + 1/2

v = Ry / (2 * |n|) + 1/2

No need to calculate 2 * |n| twice, so define m = 2 * sqrt[ rx2 + ry2 + (rz+1)2 ]

EDIT: had |R| rather than |n| in several places.