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

[Laurence Finston]

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


From: Larry Siebenmann <address@hidden>
To: address@hidden, address@hidden, address@hidden
Date: Fri, 22 Apr 2005 16:01:01 -0400



Hi Laurence F. and others,

   Having been button-holed as follows on Wed, 20 Apr 2005
22:54:47 +0200:

 > I hope you don't mind if I take the liberty of asking you
 > whether there's an algebraic formula for the curve on a plane
 > of projection representing the perspective projection of a
 > circle. (I may take to button-holing mathematicians on the
 > street.)

I answered "off-the-cuff":

 > On the image plane it is a conic section, ie an ellipse, a
 > parabola, or hyperbola.

If you almost understood this answer, read on; otherwise stop
right here.

When you get a simple answer to a simple geometric question
in computer graphics,  BEWARE that it is likely to be no more
than "generically true". That's the case here. There are
other things you can sometimes get by projecting a circle in
3-space from a point onto a plane.  For example, if the
projection point is on the target plane, which happens to be
disjoint from the circle, then the projected circle is the
EMPTY SET.

However, from an answer that is generically true one can
usually (with some work)  deduce the general answer -- which
is often rather complicated and correspondingly OBSCURE.  For
example, the generic (and stable) affine classification of
complete bezier cubic loci (forgetting parameters) is just
twofold: doublepoint versus inflexion pair. But the full
classification is a bit of a mess. See the MP list for
details

Incidentally, the parabolic case above is not generic or
stable *unless* one considers movies, or one states that the
union

(elliptic case) \cup (parabolic case) \cup (hyperbolic) case)

is stable and generic.

 ----------------

I went on to say:

 > The image on the retina of the observer is an 
 > ellipse or part thereof.

The retina of the eye of an observer is (mathematically) not
a plane but rather a small sphere around the center of the
eye. So what did I mean by an ellipse on the retina??

I meant the intersection, with that sphere, of a cone having
center the eye's center, and with base an ellipse in a well
chosen plane far from the eye.

My statement has this interesting corollary: A (generic)
photograph of a small piece of an ellipse cannot be
distinguished from a photograph of a small piece of of a
hyperbola or parabola.  Crime labs beware!

 ----------------

As for algebraic formulae, I assume you are happy with
pointwise execution so I suggest you just compose two
perspectivities in R^3:

(1) the given perspsectivity projecting from the given circle
to the target plane in 3-space.

(2) the perspectivity with center the eye of the (imaginary)
television camera, that maps from the mentioned target
plane to the computer screen.

This lets you move any number of points from your original
circle to the computer screen in such a way that you "see"
on the screen the given projection of the circle.

Once you have 5 points of the screen image, you can (if you
wish) get all other points on screen by programming
2-dimensional Pascal's theorem in MP.

Cheers

Laurent S.