Below are the results of some atan2-tests I just did.

Some constants:

  pi/2 = 1.570796326794897
  pi   = 3.141592653589793
3*pi/2 = 4.712388980384690
2*pi   = 6.283185307179586

tau = 2*pi

Z = 1e-100


Here the functions are given valid arguments:

The results for atan2() are all in the closed interval [-pi, +pi]

atan2(-0, -1) = -3.141592653589793
atan2(-Z, -1) = -3.141592653589793

atan2(-1, -Z) = -1.570796326794897
atan2(-1, -0) = -1.570796326794897
atan2(-1, +0) = -1.570796326794897
atan2(-1, +Z) = -1.570796326794897

atan2(-Z, +1) = -0
atan2(-0, +1) = -0

atan2(+0, +1) = +0
atan2(+Z, +1) = +0

atan2(+1, +Z) = +1.570796326794897
atan2(+1, +0) = +1.570796326794897
atan2(+1, -0) = +1.570796326794897
atan2(+1, -Z) = +1.570796326794897

atan2(+Z, -1) = +3.141592653589793
atan2(+0, -1) = +3.141592653589793


Atan2_Fn =
    function(a, b) {
        mod(atan2(a, b) + tau, tau)
    }

The results for Atan2_Fn() are all in the half-open interval [+0, +tau)

Atan2_Fn(-Z, +1) = +0
Atan2_Fn(-0, +1) = +0
Atan2_Fn(+0, +1) = +0
Atan2_Fn(+Z, +1) = +0

Atan2_Fn(+1, +Z) = +1.570796326794897
Atan2_Fn(+1, +0) = +1.570796326794897
Atan2_Fn(+1, -0) = +1.570796326794897
Atan2_Fn(+1, -Z) = +1.570796326794897

Atan2_Fn(+Z, -1) = +3.141592653589793
Atan2_Fn(+0, -1) = +3.141592653589793
Atan2_Fn(-0, -1) = +3.141592653589793
Atan2_Fn(-Z, -1) = +3.141592653589793

Atan2_Fn(-1, -Z) = +4.712388980384690
Atan2_Fn(-1, -0) = +4.712388980384690
Atan2_Fn(-1, +0) = +4.712388980384690
Atan2_Fn(-1, +Z) = +4.712388980384690


n_atan2 =
    function(a, b) {
        atan2(a, b) + select(a, tau, 0)
    }

The results for n_atan2() should all be in the half-open interval [+0, +tau)

n_atan2(-Z, +1) = +6.283185307179586  <--- NB
n_atan2(-0, +1) = +0
n_atan2(+0, +1) = +0
n_atan2(+Z, +1) = +0

n_atan2(+1, +Z) = +1.570796326794897
n_atan2(+1, +0) = +1.570796326794897
n_atan2(+1, -0) = +1.570796326794897
n_atan2(+1, -Z) = +1.570796326794897

n_atan2(+Z, -1) = +3.141592653589793
n_atan2(+0, -1) = +3.141592653589793
n_atan2(-0, -1) = -3.141592653589793  <--- NB !!!
n_atan2(-Z, -1) = +3.141592653589793

n_atan2(-1, -Z) = +4.712388980384690
n_atan2(-1, -0) = +4.712388980384690
n_atan2(-1, +0) = +4.712388980384690
n_atan2(-1, +Z) = +4.712388980384690


f_normalized_atan2 =
    function(a, b) {
        select(
            a,
            atan2(a, b) + tau,
            select(b, pi, 0),
            atan2(a, b)
        )
    }

The results for f_normalized_atan2() are all in the half-open interval [+0,
+tau)

f_normalized_atan2(-Z, -1) = +3.141592653589793
f_normalized_atan2(-0, -1) = +3.141592653589793
f_normalized_atan2(+0, -1) = +3.141592653589793
f_normalized_atan2(+Z, -1) = +3.141592653589793

f_normalized_atan2(-Z, +1) = +6.283185307179586  <--- NB
f_normalized_atan2(-0, +1) = +0
f_normalized_atan2(+0, +1) = +0
f_normalized_atan2(+Z, +1) = +0

f_normalized_atan2(+1, -Z) = +1.570796326794897
f_normalized_atan2(+1, -0) = +1.570796326794897
f_normalized_atan2(+1, +0) = +1.570796326794897
f_normalized_atan2(+1, +Z) = +1.570796326794897

f_normalized_atan2(-1, -Z) = +4.712388980384690
f_normalized_atan2(-1, -0) = +4.712388980384690
f_normalized_atan2(-1, +0) = +4.712388980384690
f_normalized_atan2(-1, +Z) = +4.712388980384690


Here the functions are given invalid arguments:

Atan2 =
    function(a, b) {
        atan2(a, b)
    }

Atan2(-0, -0) = -3.141592653589793
Atan2(-0, +0) = -0
Atan2(+0, +0) = +0
Atan2(+0, -0) = +3.141592653589793


Atan2_Fn =
    function(a, b) {
        mod(atan2(a, b) + tau, tau)
    }

Atan2_Fn(+0, +0) = +0
Atan2_Fn(+0, -0) = +3.141592653589793
Atan2_Fn(-0, -0) = +3.141592653589793
Atan2_Fn(-0, +0) = +0


n_atan2 =
    function(a, b) {
        atan2(a, b) + select(a, tau, 0)
    }

n_atan2(+0, +0) = +0
n_atan2(+0, -0) = +pi
n_atan2(-0, -0) = -pi  <--- NB !!!
n_atan2(-0, +0) = +0


f_normalized_atan2 =
    function(a, b) {
        select(
            a,
            atan2(a, b) + tau,
            select(b, pi, 0),
            atan2(a, b)
        )
    }

f_normalized_atan2(+0, +0) = +0
f_normalized_atan2(-0, +0) = +0
f_normalized_atan2(-0, -0) = +0
f_normalized_atan2(+0, -0) = +0


When looking at the results from both the valid and invalid arguments, it seems
to me that POV-Ray's built in atan2() gives the most consistent results,
followed by Atan2_Fn().

--
Tor Olav
http://subcube.com
https://github.com/t-o-k

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