[help-3dldf] Re: [metafont] Re: button-hole problem

[Laurence Finston]

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