The distance trisector curve

Given points p and q in the plane, we are interested in separating them by two curves C1 and C2 such that every point of C1 has equal distance to p and to C2, and every point of C2 has equal distance to C1 and to q. We show by elementary geometric means that such C1 and C2 exist and are unique. Moreover, for p=(0,1) and q=(0,−1), C1 is the graph of a function , C2 is the graph of −f, and f is convex and analytic (i.e., given by a convergent power series at a neighborhood of every point). We conjecture that f is not expressible by elementary functions and, in particular, not algebraic. We provide an algorithm that, given x∈R and ε>0, computes an approximation to f(x) with error at most ε in time polynomial in . The separation of two points by two “trisector” curves considered here is a special (two-point) case of a new kind of Voronoi diagram, which we call the zone diagram and which we investigate in a companion paper.