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Let us place the billiard table in the plane
in the three-dimensional Euclidean space .
We now also let the billiard ball move in the z direction
by giving it a z-velocity 1/l if l is the length of the trajectory
between successive impacts
on the boundary of the two-dimensional cylinder table
The special choice of the z velocity has the consequence that a
ball starting at a point of the cylinder hits the cylinder again
in the point . The union of all billiard
ball trajectories which were tangent to a caustic
now form a ruled surface, a one-parameter family of straight lines.
The line of striction of the ruled surface constructed from the table
with a caustic projects along the z direction (see Figure 8a)
to the caustic. However, unlike the caustic itself, the
line of striction does not need to have self intersections.
The example in Figure 8a illustrates this.
Consider a nowhere differentiable caustic as constructed above. The
two first coordinates of the line of striction are the two coordinates
of the caustics and are therefore nowhere differentiable functions.
Because the third coordinate of the line of striction is
, which is a nowhere differentiable Weierstrass function,
all coordinate functions are nowhere differentiable.
Fri Jun 12 13:34:37 CDT 1998