http://www.geom.umn.edu/docs/reference/CRC-formulas/node26.html

Section 7.3 Additional Properties of Ellipses

This information is kinda condensed but it contains the illustrated answer to your
problem. You may want
to visit the section on conic basics to get warmed up.

Consider a triangle where 1 vertex is the center of the ellipse C, 1 vertex is a focus
of the ellipse F
and the 3rd vertex is point P on the circumference of the ellipse.  These are derived
from a listed
equation.

The natural motion for an ellipse is to sweep equal area with equal time. The tutorial
does not show
this, but that's Kepler's 3rd Law of Planetary Motion.

Start with line C-P to be at right angle to line C-F. Point P is where the ellipse
touches the floor.
Line C-P is the distance from the center to the floor. Rotate the ellipse by a given
amount, say 5
degrees, in time T. This changes the angle of line CP by 5 degrees over time period T.
This also moves P
to point P2. Use the polar formula to find the length of C-P2 and calculate the area
of the triangle
P-P2-C. The second movement will involve calculating the distance P2-P3 from the area
of the triangle.
The area stays the same for equal time units. This also gives you the linear movement
across the floor,
which is also cyclic.

Another way to locate P2 is to rotate the ellipse by 5 degrees over time T. This tilts
line C-P as well
as line C-F 5 degrees in relation to the floor. Use a triangle with vertex F1, vertex
F2 and vertex P,
which is perpendicular to the line F1-C-F2. The angle of coincidence equals the angle
of reflection, so
changing the angle of F1-P also changes the angle of F2-P by the same amount. Use your
regular trig to
locate the new point P2 so that the angle of coincidence remains equal to the angle of
reflectance with
the floor. The length C-P tells you how far the center is from the floor and tilting
the angle of the
line C-P to perpindicular will give you the translation amount parallel to the floor.

All of the equations used are derived from 1 master equation that describes all
conics. The circle is
just a special case of the ellipse, so if you wanted to get really cute you could have
a sphere morphing
into an ellipse while rolling across the floor.

I'm busy on a programming project but I'd sure enjoy working this out. I am now
convinced it will work
for all possible conics.

steve

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