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

[Laurence Finston]

------ Forwarded message -------


From: Peter Vanroose <address@hidden>
Reply-to: address@hidden: address@hidden
Date: Mon, 25 Apr 2005 17:47:21 +0200 (CEST)

> So is a perspective projection of a conic section always a conic
> section?

Yes.
Otherwise stated, "being a conic section" is a projective invariant.

This means: projecting a circle, ellipse, hyperbola, or parabola,
no matter how, always gives either a circle or an ellipse or a
hyperbola or a parabola.

(Well, I could have dropped "circle" since cicles are ellipses.)

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.

Take again my example with the vertical circle on the horizontal
projection plane.

Let's have it more concrete: the horizontal projection plane is the
plane Z=0, the circle lies in the plane X=0, with centre (0,0,2) and
radius 2 (so it touches the projection plane in (0,0,0)), and the eye
point is (-3,0,5).

The projection is an ellipse [guaranteed !]
with main axis Y=Z=0 [since it's a symmetry axis of the projection],
with top points (0,0,0) and (12,0,0) [= projection of (0,0,4)],
so the centre is (6,0,0),
but the projection of the circle's centre (0,0,2) is the point (2,0,0):
still on the main axis, but not in the middle.

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".


--      Peter.