[Proj] Re: Global Gauss-Kruger and libproj4---the final story

[Glynn Clements]

Gerald I. Evenden wrote:

> > > The only context that I use the term "interruption" in discussing
> > > cartographic projections are in cases like Goode's world maps or
> > > "orange peal" charts often using the sinusoidal projection.  Please
> > > define what *you* mean by interruptions.
> >
> > An interruption is any location where two points of infinitesimal
> > separation on the globe are mapped to two points having finite separation
> > on the plane. This happens somewhere on all projections. All along an
> > interruption, the projection formulæ are no longer a function; they are
> > multivalue.
> 
> Let's see.  tan(x) has two values at x=90 degrees: +inf and -inf.

Depending upon your definition of tan(), it may be undefined,
single-valued or two-valued at odd multiples of pi.

The set of real numbers doesn't include infinity, so if tan() is
defined with the real line as its range, it is undefined at such
points. If it's defined with its range as the projective (one-point)
compactification of the real line (the real line plus a single point
at infinity), it's single valued. If its range is the affine
(two-point) compactification of the real line (with separate positive
and negative infinities), then its multi-valued.

>From a programming perspective, whether you have a single infinity or
signed infinities depends upon the CPU and its configuration. The
80287 had a flag bit to select the behaviour, while the 80387 onwards
only supports signed (affine) infinities.

[One consequence of using affine infinity is that the comparison
operation only tests equivalence, not (extensional) equality:
-0 == +0 but 1/-0 =/= 1/+0.]

> So it is no longer a function, eh?
> 
> I can't find any such distinction for the definition of a "function."  Please 
> give reference.

By the set-theoretical definition, a function must be single-valued;
that's the only factor which separates functions from relations.

-- 
Glynn Clements <glynn at gclements.plus.com>