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Nowhere differentiable lines of striction of ruled surfaces

Let us place the billiard table in the plane tex2html_wrap_inline1342 in the three-dimensional Euclidean space tex2html_wrap_inline1344 . 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 tex2html_wrap_inline1354 on the boundary of the two-dimensional cylinder table tex2html_wrap_inline1356 . The special choice of the z velocity has the consequence that a ball starting at a point tex2html_wrap_inline1360 of the cylinder hits the cylinder again in the point tex2html_wrap_inline1362 . The union of all billiard ball trajectories which were tangent to a caustic tex2html_wrap_inline880 now form a ruled surface, a one-parameter family of straight lines.

tex2html_wrap1388 tex2html_wrap1390
tex2html_wrap1392 tex2html_wrap1394
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.

tex2html_wrap1396 tex2html_wrap1398
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 tex2html_wrap_inline1238 , which is a nowhere differentiable Weierstrass function, all coordinate functions are nowhere differentiable.


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