Re: Arctan2 Alan,
> If you look at the graph, you can see
>that we could consider arctan to map out a sort of helix, continually
>descending as we go counter-clockwise around the origin (as seen from
>the "top")  Rather than forcing that particular (somewhat arbitrary)
>discontinuity, we could theoretically continue the helix downward and
>map the points in quadrant IV to something less than -pi, and maintain
>continuity between quadrants III and IV.  It diverges from the usual
>range produced by the arctan function, but has the benefits of being
>mathematically meaningful, continuous, and monotonic.
>  

"Mathematically meaningful" depends on your definition of a function.

(i)  If you consider arctan(y/x) as a single valued function (by taking
the principal value), then extending the range is wrong.

(ii) Continuing on the helix, you lose inclusion isotonicity. Consider

X1 = ([-2,-1],[-2,-1]), X2 =  ([1,2],[-2,-1]), and X3 = ([-2,2],[-2,-1]).

We have X1 \\subset X3, X2 \\subset X3. Extending the range on the helix
implies

either  arctan2(X1) \\not\\subset arctan2(X3)  or  arctan2(X2) \\not\\subset
arctan2(X3).

(iii) Continuing on the helix, you also lose analyticity of the
underlying complex function. See the discussion of the argument
functions in the CoStLy library
 (http://iamlasun8.mathematik.uni-karlsruhe.de/~ae16/CoStLy.html). There
are applications, where analyticity is reqired and where the use of
non-analytic inclusion functions is prohibited.

Regards,

Markus








