Josh,

Check out the bottom of the conics web page. The equation for rotating diagonals
indirectly yields the
value you are seeking. The path that point P follows and the path length are what you
need for this
approach.

However, you can get the same information by calculating the path that the Center
point follows and it's
length. The equations for this are given directly.

If this hasn't been answered clearly by the time I get back home, I'll look it up in
Texareau's book and
post the equations.

steve

Josh English wrote:

> Thanks, Steve. Luckily I found a method of rolling the scaled sphere, which does
make an ellipse, around
> the origin. What I need now is a method for finding the length of an arc on an
ellipse. I've found
> equations, but they don'tmake much sense to me. I'm still working on it, though. The
arc length is the
> last thing I need to finish the problem.
>
> I wonder if Andrew Cocker has the solution as is simply testing me : ) I know I'll
be making a tutorial
> about this one!
>
> Josh
>
> Steve wrote:
>
> > 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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