Geodesics in Euclidean Space with Analytic Obstacle

Abstract

Abstract.In this note we are concerned with the behavior of geodesies in Euclidean «-space with a smooth obstacle. Our principal result is that if the obstacle is locally analytic, that is, locally of the form xn = f(xx, ... , xn_x) for a real analytic function /, then a geodesic can have, in any segment of finite arc length, only a finite number of distinct switch points, points on the boundary that bound a segment not touching the boundary.This result is certainly false that for a C°° boundary. Indeed, even in E , where our result is obvious for analytic boundaries, we can construct a C°°b oundary so that the closure of the set of switch points is of positive measure.We denote by M the closure of the complement of the obstacle in Euclidean space En and by S the boundary of the obstacle. Thus M is an «-dimensional Riemannian manifold-with-boundary embedded in En and S is its boundary surface.A geodesic on a Riemannian manifold-with-boundary, M, is defined to be a locally shortest path in M. In our context the geodesies are easy to visualize; in E the geodesic is the path of a stretched string with the boundary considered as the surface of an obstacle around which the string must bend or into which the string must plunge and end. In [ABB1, ABB2] the properties of these geodesies are explored in the general setting of a Riemannian manifold-with-boundary.We describe briefly the elementary properties of the geodesies in M. A geodesic contacting the boundary in a segment is a geodesic of the boundary; a geodesic segment not touching the boundary is a straight line segment. A segment on the boundary joins a segment in the ambient space in a differentiable join. We call the endpoints on the boundary S of a segment not touching the boundary switch points. Cluster points of switch points, necessarily points at which the geodesic contacts the boundary, we call intermittent points or chatter points. As we will see, even for a C°° surface we can have sets of positive measure of such chatter points, and so it is reassuring to observe that the acceleration of a geodesic at a chatter point is 0. Indeed, acceleration is well defined and bounded everywhere except at the necessarily countable set of switch points; where it is defined, acceleration is normal and outward-pointing

Geodesics in 3-dimensional Euclidean Space with One or Two Analytic Obstacles

“…Our Ph.D. thesis verified this finiteness for shortest-length curves in any semi-algebraic subset of R 2 . Earlier in [1], F. Albrecht and I.D. Berg proved this is true for a geodesic in a closed region of n-dimensional Euclidean space with a smooth real analytic hypersurface as boundary.…”

Pursuit-evasion in a two-dimensional domain

“…Theorem 4.1 (Albrecht and Berg [2]). If M is a 2-dimensional analytic manifold with boundary embedded in E 2 , and γ is a geodesic in M , then the switch points on γ have no accumulation points.…”

“…We restrict ourselves to analytic boundary, instead of smooth (C 2 , or even C ∞ ) boundary, to avoid some potentially pathological behavior of geodesics. For example, Albrecht and Berg [2] construct a geodesic in C ∞ environment, that achieves a Cantor set of positive measure as the accumulation of switch points. This unusual geometry hampers our ability to confine the evader in a well-defined connected component.…”

Two-Dimensional Pursuit-Evasion in a Compact Domain with Piecewise Analytic Boundary

“…This strategy was adapted for pursuit in polygonal regions by Isler, Kannan and Khanna [14]. Their adaptation relies heavily on the vertices of the polygon P and gives a capture time of n • diam(P ) 2 , where n is the number of vertices of P . We give a topological version of lion's strategy that succeeds in any compact CAT(0) domain D (including polygons) with capture time bounded by diam(D) 2 .…”