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# Nowhere differentiable lines of striction of ruled surfaces

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.

Oliver Knill
Fri Jun 12 13:34:37 CDT 1998