Lihong Ma

Voronoi Diagram for services neighboring a highway

We are given a transportation line where displacements happen at a bigger speed than in the rest of the plane. A shortest time path is a path between two points which takes less than or equal time to any other. We consider the time to follow a shortest time path to be the time distance between the two points. In this paper, we give a simple algorithm for computing the Time Voronoi Diagram, that is, the Voronoi Diagram of a set of points using the time distance.

Smallest Color-Spanning Objects

Motivated by questions in location planning, we show for a set of colored point sites in the plane how to compute the smallest-- by perimeter or area--axis-parallel rectangle and the narrowest strip enclosing at least one site of each color.

Convex Distance Functions In 3-Space Are Different

The bisector systems of convex distance functions in 3-space are investigated and it is shown that there is a substantial difference to the Euclidean metric which cannot be observed in 2-space. This disproves the general belief that Voronoi diagrams in convex distance functions are, in any dimension, analogous to Euclidean Voronoi diagrams. The fact is that more spheres than one can pass through four points in general position. In the L4-metric, there exist quadrupels of points that lie on the surface of three L4-spheres. Moreover, for each n ≥ 0 one can construct a smooth and symmetric convex distance function d and four points that are contained in the surface of exactly 2n+1+ d-spheres, and this number does not decrease if the four points are disturbed independently within 3-dimensional neighborhoods. This result implies that there is no general upper bound to the complexity of the Voronoi diagram of four sites based on a convex distance function in 3-space.

On bisectors for different distance functions

Abstract Let γ C and γ D be two convex distance functions in the plane with convex unit balls C and D . Given two points, p and q , we investigate the bisector, B ( p , q ), of p and q , where distance from p is measured by γ C and distance from q by γ D . We provide the following results. B ( p , q ) may consist of many connected components whose precise number can be derived from the intersection of the unit balls, C and D . The bisector can contain bounded or unbounded two-dimensional areas. Even more surprising, pieces of the bisector may appear inside the region of all points closer to p than to q . If C and D are convex polygons over m and n vertices, respectively, the bisector B ( p , q ) can consist of at most min( m , n ) connected components which contain at most 2( m + n ) vertices altogether. The former bound is tight, the latter is tight up to an additive constant. We also present an optimal O( m + n ) time algorithm for computing the bisector.

Convex distance functions in 3-space are different

We investigate the bisector systems of convex distance functions in 3-space and show that there is a substantial difference to the Euclidean metric which cannot be observed in 2-space. Namely, more than one sphere can pass through four points in general position. We show that in the L4-metric there exist quadrupels of points that lie on the surface of three L4-spheres, and that this number does not decrease if the four points are disturbed independently within 3-dimensional neighborhoods. Moreover, for each n ≥ 2 we construct a smooth and symmetric convex distance function d and four points that are contained in the surface of exactly n d-spheres. This result implies that there is no general upper bound to the complexity of the Voronoi diagram of four sites based on a convex distance function in 3-space.

Java applets for the dynamic visualization of Voronoi diagrams

This paper is dedicated to Thomas Ottmann on the occasion of his 60th birthday. We discuss the design of several Java applets that visualize how the Voronoi diagram of n points continuously changes as individual points are moved across the plane, or as the underlying distance function is changed. Moreover, we report on some experiences made in using these applets in teaching and research. The applets can be found and tried out at http://wwwpi6.fernuni-hagen.de/GeomLab/.

The optimal way for looking around a corner

Let two walls form a wedge of angle less than 180/spl deg/. At one of the walls, a robot is located, facing the corner where the walls meet. The robots task is to eye the other wall. To this end, it can freely move around in the area outside the wedge. Suppose the robot does not know the angle of the wedge. How should it move to minimize path length? It is shown that there is a strategy which guarantees that, for any possible value of the angle, the length of the path the robot walks before it can look around the corner is bounded by the length of the shortest path to do so, times the constant c /spl ap/ 1.21218. It is proved that the strategy is optimal in that no smaller competitive factor than c can be achieved. A simple formula is given for the robot to find the optimal path.