Conic sections look like a neat approach for elementary arithmetic, but I'm
not sure they're particularly helpful for other functions. Maybe good
enough for small neighborhoods, but handling the edge cases sounds like a
mess....
But again, I think the hard part here is getting Maxima to systematically
handle a new kind of quasi-number correctly in every place where it makes
sense. There is both a general problem -- what framework do you use -- and
a specific one -- how do you add relevant code to all the right places (and
ideally minimal additional code)?
Conceptually, some sort of object system seems like the right framework.
One way I have proposed to move forward in the past is to use bigfloats as
guidance. Every place bigfloats are treated specially, new kinds of
quasi-numbers would probably have to be treated specially. In addition,
many quasi-numbers don't have all the properties numbers do (e.g. strict
total order), etc., which Maxima assumes all numbers do, so there will be
additional cases.
-s
On Tue, Feb 12, 2013 at 4:06 PM, Henry Baker <hbaker1 at pipeline.com> wrote:
> How about the following idea for representing complex intervals: _conic
> sections_.
>
> The problem I'm trying to solve is the following one:
>
> How do you provide a set of atomic intervals that is expressive enough to
> handle both real intervals -- e.g., [3,5] -- as well as circular regions in
> the complex plane -- e.g., {x | (x-C)<r^2}.
>
> Possible answer: conic sections, including degenerate conic sections.
>
> The interval [3,5] is a degenerate ellipse with foci at (3,0) and (5,0),
> where the string is completely taut.
>
> The interval [3,INF) is a degenerate parabola with the focus at (3,0) and
> stretching to the right; (MINF,3] is the degenerate parabola with the focus
> at (3,0) stretching to the left.
>
> The interval [3,3] is a degenerate ellipse with foci at (3,0) and (3,0),
> or alternatively, a circle with center (3,0) and radius=0.
>
> One tradition extension of intervals to the complex plane uses _circles_
> (Henrici's "circular arithmetic"), so we can represent these.
>
> So far, we have used only conics with eccentricities <=1. I don't know if
> there are any useful "intervals" which are hyperbolic.
>
> I don't know enough about elliptic functions to know if elliptical regions
> in the complex plane are particularly useful.
>
>