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[help-3dldf] Re: [metafont] Re: button-hole problem
From: |
Laurence Finston |
Subject: |
[help-3dldf] Re: [metafont] Re: button-hole problem |
Date: |
Mon, 25 Apr 2005 21:17:34 +0200 |
User-agent: |
IMHO/0.98.3+G (Webmail for Roxen) |
Peter Vanroose wrote:
> > So is a perspective projection of a conic section always a conic
> > section?
>
> Yes.
> Otherwise stated, "being a conic section" is a projective invariant.
Thank you. This is very handy and makes me think that it would definitely be
worthwhile to pursue this tack. Am I right in assuming that "projectively
invariant" implies that this characteristic of conic sections is invariant for
any projection whatsoever, such as projection onto a sphere or a cylinder? It
seems to me that it ought to be possible to construct a transformation for
which this condition would not hold. If this is true, does it imply that not
all transformations are projections? If this is so, are projections a subset
of transformations? Or am I comparing apples and oranges? Please excuse me
if these are naive questions.
> On the other hand, the *centre* of a circle or an ellipse is *not* a
> projective invariant (it's an affine invariant), so the centre is not
> necessarily projected onto the centre of the projection.
[...]
> Projections don't preserve distances nor distance fractions, so they in
> general don't preserve "middle of a line segment", so in particular
> they don't preserve "centre of an ellipse".
I don't consider these to be problems. I may not need the centers of the
projections, and previous remarks in this discussion have indicated that it's
possible to find the centers, foci, etc., if needed.
However, I'm still a bit at sea about the problem of finding equations
(implicit and/or parametric) for the projections and finding the intersections
of conic sections. I realize it's up to me to learn this material, I'm just
finding it extremely difficult. It seems like a very long road.
Thank you very much for your help.
Laurence