Visibility of "Tame" Terrain

Abstract Let f be a continuous, one-valued function defined on a finite interval I. We call f “tame” if I can be decomposed into a finite set of subintervals on each of which f is either convex or concave. If P is a point on or above (the graph of) a tame function f, we say that a point Q=(x, f(x)) of f is visible from P if the open line segment PQ lies above f. We show that the points of f visible from P subtend an angular sector at P. Moreover, these visible points comprise a finite number of arcs of f, and each gap between these arcs corresponds to a different concavity of f; thus, different subsets of the concavities can be regarded as defining different “aspects” of f as seen from points on or above f. We also show that there exist finite sets of points P from which all of f is visible, but that this need not be true if the points are required to lie on f. Unfortunately, it seems to be difficult to extend these results to a continuous, one-valued “terrain” surface defined on a finite planar region. Even if every vertical profile of such a function (i.e., every cross section by a vertical plane) is tame, and even if the region consists of a finite number of subregions of simple shapes on which the surface is convex or concave, there can exist points from which infinitely many connected pieces of the surface are visible, and there may not exist a finite set of viewpoints from which the entire surface is visible.