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 ƒ: R → R, 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 log 1+|x|/ε.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 <i>Voronoi diagram with neutral zone</i> and which we investigate in a companion paper.