Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

Abstract

We investigate 3D visibility problems for scenes that consist of n non-intersecting spheres. The viewing point υ moves on a flightpath that is part of a “circle at infinity” given by a plane P and a range of angles {α(t)|t ∈ [0∶1]} ⊂ [0∶2π], At “time” t, the lines of sight are parallel to the ray r(t) in the plane P, which starts in the origin of P and represents the angle α(t) (orthographic views of the scene). We describe algorithms that compute the visibility graph at the start of the flight, all time parameters t at which the topology of the scene changes, and the corresponding topology changes. We present an algorithm with running time O((n+k+p)log n), where n is the number of spheres in the scene; p is the number of transparent topology changes (the number of different scene topologies visible along the flightpath, assuming that all spheres are transparent); and k denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.