## response from Prof. B.

Yes, it will be good to get to some things beyond polyhedra, and yes,
it is hard to deal with curved objects, especially spheres, that are so
homogeneous. You do want to get information by turning things around, so
there has to be some variation of features in order to see changes.

But no, I don't think you are going to get anything surprising when
you look at ellipsoids or hyperellipsoids. Here is an analytic reason why
the projection of an ellipsoid is always an ellipse (including the case of
a circle); the higher dimensional concepts provide nothing new in this
case.

An ellipsoid is given by a formula involving x, y, and z up to the
second degree, so we could write an ellipsoid as a level set of a function
f(x,y,z) = Ax^2 + By^2 + Cz^2 + 2Dxy + 2Eyz + 2Fzx. If we project down a
vector (a,b,c) to its perpendicular plane, then the edge of the shadow
will be the image of the points on the ellipsoid where the tangent plane
contains a line parallel to (a,b,c), i.e. where the normal to that tangent
plane is perpendicular to (a,b,c). But the normal to the ellipsoid is
given by 2(Ax + Dy + Fz, Dx + By + Ez, Fx + Ey + Cz) so the condition for
this to be perpendicular to (a,b,c) is the linear condition a(Ax + Dy +
Fz) + b( Dx + By + Ez) + c( Fx + Ey + Cz) = 0. This is the equation for a
plane, and the intersection of a plane and an ellipsoid is an ellipse.
Thus the boundary of the shadow is the projection of an ellipse into the
plane, and this gives an ellipse.

Actually the formula in quadratic terms applies to other figures as
well, such as hyperboloids of one and two sheets or parabolic cylinders or
hyperbolic paraboloids. The analytic argument shows that the boundary of
the shadow of any of these objects will be a conic section. If we know
ahead of time that the object we started with is an ellipsoid, then all of
its shadows have to be bounded, contained in some very large disc. But
the only bounded conic sectios are ellipses, so the shadows of ellipsoids
will be ellipses.

For the same reason, a three-dimensional shadow of a hyperellipsoid (with
a formula generalizing the above in an analogous way) will be an ordinary
ellipsoid.

Is that a convincing argument? Can you find one that is more visual?

Good exercises by the way, and important for topics like machine vision.
How would you answer your questions, in ordinary language and using some
mathematical formalism?