Jack Snoeyink

Ray shooting in polygons using geodesic triangulations

LetP be a simple polygon withn vertices. We present a simple decomposition scheme that partitions the interior ofP intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles. This decomposition can be used to preprocessP in a very simple manner, so that any ray-shooting query can be answered in timeO(logn). The data structure requiresO(n) storage andO(n logn) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time toO(n). We also extend our general technique to the case of ray shooting amidstk polygonal obstacles with a total ofn edges, so that a query can be answered inO(√ logn) time.

Ray Shooting in Polygons Using Geodesic Triangulations

LetP be a simple polygon withn vertices. We present a simple decomposition scheme that partitions the interior ofP intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles. This decomposition can be used to preprocessP in a very simple manner, so that any ray-shooting query can be answered in timeO(logn). The data structure requiresO(n) storage andO(n logn) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time toO(n). We also extend our general technique to the case of ray shooting amidstk polygonal obstacles with a total ofn edges, so that a query can be answered inO(√ logn) time.

An efficient algorithm for finding the CSG representation of a simple polygon

We consider the problem of converting boundary representations of polyhedral objects into constructive-solid-geometry (CSG) representations. The CSG representations for a polyhedron P are based on the half-spaces supporting the faces of P. For certain kinds of polyhedra this problem is equivalent to the corresponding problem for simple polygons in the plane. We give a new proof that the interior of each simple polygon can be represented by a monotone boolean formula based on the half-planes supporting the sides of the polygon and using each such half-plane only once. Our main contribution is an efficient and practical O(n log n) algorithm for doing this boundary-to-CSG conversion for a simple polygon of n sides. We also prove that such nice formulæ do not always exist for general polyhedra in three dimensions.

Computing a face in an arrangement of line segments and related problems

This paper presents a randomized incremental algorithm for computing a single face in an arrangement of n line segments in the plane that is fairly simple to implement. The expected running time of the algorithm is

Stabbing parallel segments with a convex polygon

O(n\alpha (n)\log n)

Compliant motion in a simple polygon

. The analysis of the algorithm uses a novel approach that generalizes and extends the Clarkson–Shor analysis technique [in Discrete Comput. Geom., 4 (1989), pp. 387–421]. A few extensions of the technique, obtaining efficient randomized incremental algorithms for constructing the entire arrangement of a collection of line segments and for computing a single face in an arrangement of Jordan arcs are also presented.

Sweeping arrangements of curves

We present an algorithm that, given a set of n parallel line segments in the plane, finds a convex polygon whose boundary intersects each segment at least once, or determines that none exists. Our algorithm runs in O(n log n) steps and linear space, which is optimal. Our solution involves a reduction to a bipartite stabbing problem, using a “point-sweeping” or “chain-unwrapping” technique. We use geometric duality to solve bipartite stabbing.

A trivial knot whose spanning disks have exponential size

We consider motion planning under the compliant motion model, in which a robot directed to walk into a wall may slide along it. We examine several variants of compliant motion planning for a point robot inside a simple polygon with n sides, where the goal is a fixed vertex or edge. For the case in which the robot moves with perfect control, we build a data structure that lets us in O (log n ) time determine the range of directions in which the robot can move from a query point to the goal in a single step. This structure lets us solve a variety of other problems: we can find a similar query data structure for multi-step paths; we can solve single-step problems allowing uncertainty in control and position sensing; and we can explicitly compute the set of all points that can reach the goal in a single step, even allowing uncertainty in control. Our algorithms run in O ( n log n ) time and linear space; they use a novel method for maintaining convex hulls of simple paths that may be of independent interest.

Input-Sensitive Compliant Motion in the Plane

We consider arrangements of curves that intersect pairwise in at most k points. We show that a curve can sweep any such arrangement and maintain the k-intersection property if and only if k equals 1 or 2. We apply this result to an eclectic set of problems: finding Boolean formulae for polygons with curved edges, counting triangles and digons in arrangements of pseudocircles, and finding extension curves for arrangements. We also discuss implementing the sweep.

Counting and cutting cycles of lines and rods in space

If a closed curve in space is a trivial knot (intuitively, one can untie it without cutting) then it is the boundary of some disk with no self-intersections. In this paper we investigate the minimum number of faces of a polyhedral spanning disk of a polygonal knot with <italic>n</italic> segments. We exhibit a knot whose minimal spanning disk has exp(<italic>cn</italic>) faces, for some positive constant <italic>c</italic>.