Josh, let me locate some stuff on the web for you. The tangent to any point of any
conic cross section
is pretty standard stuff. In Jean Texareau's "How To Make A Telescope" he describes
the whole thing in
geometrical terms, so you might feel right at home. The math is differential calculus
but it is
accompanied with very clear illustrations of the various geometries of the conics and
all their
properties. You can simply examine the drawings and pick the equations you need. No
need to learn
calculus - you already know how to solve an equation. Spherics, hyperbolics,
parabolics and elliptics
are all covered. Also, the book is intended to be a primer for the amateur telescope
maker so the
presentation is easy to understand.

I'm vacationing at the lake house or I'd just grab my copy and give you the answer.
I'll be home in
about a month if the web search doesn't turn up the info.

I really believe that conics holds the key to a generalized solution to your very
interesting problem!
Once the equations are set up, you should be able to roll any conic cross section
across the floor by
altering a couple of variables. We do something very similar to this when testing and
figuring a
mirror. We measure the intersection of the conic and the light path which yields the
slope, then
calculate which conic we have. The slope is defined as being the intersection point of
a plane that is
perpendicular to the surface of the conic at that point. The ratio of the changing
slopes gives us a
scaling effect, is the ellipsoid size 1 or size 2? You can probably use this to keep
the ellipse in
constant, perfect contact with the ground plane because you'll know exactly what size
the ellipse is
and what it's slope is at every point.

The math also includes calculating the difference between a conic and a spheric, but I
don't think you
can achieve true conics simply by scaling a sphere.

I'll get back to ya on this.

steve

Josh English wrote:

> this seems more complicated to me, but I never studied optics, I just tutor high
school algebra and
> geometry. I suppose if you could find the proper intersection of plane and cone and
find a way to
> rotate it appropriately, it would work. I'm still stuck on finding a tangent to the
ellipse at any
> given point on the edge of the ellipse.
> I also haven't heard from Andrew in this thread, so I wonder if he has a solution or
not
>
> Josh English
>
> Steve wrote:
>
> > I don't have the solutions, but I have an impression that the trig approach is a
bit contrived.
> >
> > Perhaps you should be looking for a more generalized method.
> >
> > The sphere is is a conic cross section (angle 90) formed by revolution. It's
perfectly described
> > by the math for conics. Size is derived by locating the cross section along the
vertical axis.
> > Using this seems fairly simple, plus it would also solve the problem of rolling
ellipsoids along
> > the floor. Instead of scaling, use the location of the cross section to derive the
size.
> >
> > A generalized solution of this type would further yield parabolic and hyperbolic
movements.
> >
> > Just an idea from my old optics days.
> >
> > steve

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