Zig-zagging in a Triangulation

Let T = (V, F,E) be a 2-dimensional triangular tiling of R 2 in quad-edge representation [2], consisting of a set of vertices V , a set of faces F and a set of half-edges E. We say T is locally finite iff for any circle in R 2 it holds that the set consisting of all edges (vertices, faces) in T that intersect the circle (circumference or interior) is finite. Let e ∈ E be some half-edge of this mesh starting at vertex e1 and ending at vertex e2. With inv(e) we denote the twin of e such that inv(e)1 = e2 and inv(e)2 = e1. With face(e) we denote the face f ∈ F that is, by convention, to the left of the half-edge e. With next(e) we denote the next half-edge (in the face winding order) that is also a side of face(e), i.e.: face(next(e)) = face(e). With prev(e) we denote the previous half-edge (in the face winding order) that is also a side of face(e), i.e.: face(prev(e)) = face(e). Because we are dealing with a triangular mesh, in particular, it holds: next(next(e)) = prev(e) and vice versa. The problem of point-location [1] that we consider in this note can be formulated as follows: Given a point p ∈ R 2 and an initial half-edge einit ∈ E find some half-edge egoal ∈ E, using only next(⋅), prev(⋅) and inv(⋅) operations, such that p ∈ face(egoal), i.e.: the point is on the face. We, additionally, require the algorithm to be oblivious, meaning that the next chosen half-edge only depends on the last-visited half-edge and the goal location.