Reflection coefficient representation for convex planar sets

We combine certain results from two disparate areas, kinematics and geophysics, to obtain a convenient representation for the class of convex compact planar sets, in terms of a sequence of complex valued reflection coefficients. This gives a one-to-one relation between any convex compact planar set S and any set of parameters comprising: a) the coordinates of a reference point in /spl Sscr/, b) the circumference of the set, and c) a complex reflection coefficient sequence, {k/sub 1/, k/sub 2/,...}, such that 1) k/sub 1/=0, 2) |k/sub n/|/spl les/1, /spl forall/n, 3) if |k/sub N/|=1 for some N then k/sub n/=0, /spl forall/n>N. For a finite duration reflection coefficient sequence, where k/sub n/=0, /spl forall/n>N, if 0<|k/sub N/|<1 then the boundary of S is an infinitely differentiable convex curve, and if |k/sub N/|=1, then the boundary is an N-sided convex polygon.