Re: Proposal for the geometric engine, RFC

[Dylan Paul Thurston <dpt@math.berkeley.edu>]

On Wed, Mar 15, 2000 at 11:13:38AM -0500, Nils Barth wrote:
> Thus spake Dylan Paul Thurston:
> > On Mon, Mar 13, 2000 at 08:56:29PM +0800, Hilaire Fernandes wrote:
> > > 
> > > The geomtric engine will go into deep change in the next month, I
> > > enclose there what i'm going to change, If you have suggestion you
> > > want to appear, this is the good time to tell me.
> > 
> > Another thought...
> > 
> > How about including inversion in a circle in the "Transformations"
> > menu?  One thing to think about is that the inverse of a circle can be
> > either a circle or a line; there are numerical stability issues to
> > think about.
> 
> ...and to be correct, if the inverse of a circle is a line, it should
> include the corresponding point at infinity ;-)

Yes, of course, though there are problems: the inverse of the point at
the center of the circle is the whole line at infinity.  For inversive
geometry, it's better to complete the plane with a single point; but
you can't really have both.

Come to think of it, the numerical stability problem I mentioned
alread comes up in the current version, since you allow specifying an
arc by three points on it.  There seem to be some numerical stability
problems when the three points are nearly collinear, and it seems that
the circle disappears altogether if the points are exactly collinear.
Maybe you should pick a more stable representation for circles; I'll
think about possible representations.

Hmm... What happens if you try to draw a circle through 3 points, one
of which is at infinity?  I think the right thing to do is to ignore
the direction of the point at infinity and just draw a straight line
between the other two points.  But what if two points are at infinity?
I think you'd have to treat it the same way as if you drew a circle
through three points, two of which were the same.  Dr Geo currently
doesn't draw the circle, which is reasonable in this case.

Interesting issues!

Best,
	Dylan Thurston