17674541
Some methods of computational geometry applied to computer graphics
Abstract Windowing a two-dimensional picture means to determine those line segments of the picture that are visible through an axis-parallel window. A study of some algorithmic problems involved in windowing a picture is offered. Some methods from computational geometry are exploited to store the picture in a computer such that (1) those line segments inside or partially inside of a window can be determined efficiently, and (2) the set of those line segments can be maintained efficiently while the window is moved parallel to a coordinate axis and/or it is enlarged or reduced.
31366541
An optimal algorithm for constructing the weighted voronoi diagram in the plane
Let S denote a set of n points in the plane such that each point p has assigned a positive weight w(p) which expresses its capability to influence its neighbourhood. In this sense, the weighted distance of an arbitrary point x from p is given by de(x,p)/w(p) where de denotes the Euclidean distance function. The weighted Voronoi diagram for S is a subdivision of the plane such that each point p in S is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of S. 
 
An algorithm which constructs the weighted Voronoi diagram for S in O(n2) time is outlined in this paper. The method is optimal as the diagram can consist of Θ(n2) faces, edges and vertices.
33731652
Rectangular Point Location in d Dimensions with Applications
12582971
Stabbing line segments
An algorithm for the geometric problem of determining a line (called a stabbing line) which intersects each ofn given line segments in the plane is presented. As a matter of fact, the algorithm computes a description of all stabbing lines. A purely geometric fact is proved which infers that this description requiresO(n) space to be specified. Our algorithm computes it inO(n logn) time which is optimal in the worst case.Using the description of the stabbing lines, we are able to decide inO(logn) time whether or not a specified line is a stabbing line. Finally, the problem of maintaining the description of all stabbing lines while inserting and deleting line segments is addressed.
30538700
Computing a Ham-Sandwich Cut in Two Dimensions
Let B be a set of n"b black points and W a set of n"w, white points in the Euclidean plane. A line h is said to bisect B (or W) if, at most, half of the points of B (or W) lie on any one side of h. A line that bisects both B and W is called a ham-sandwich cut of B and W. We give an algorithm that computes a ham-sandwich cut of B and W in 0((n"h+n"w) log (min {n"b, n"w}+ 1)) time. The algorithm is considerably simpler than the previous most efficient one which takes 0((n"b + n"w) log (n"b + n"w)) time.
2265636
Polygonal Intersection Searching
17780302
Triangulating topological spaces
Given a subspace<inline-equation><f><blkbd>X⊆R</blkbd><sup>d</sup></f></inline-equation> and a finite set <inline-equation><f>S⊆<blkbd>R</blkbd><sup>d</sup></f></inline-equation>, we introduce the Delaunay simplicial complex, <inline-equation><f><sc>D</sc><inf><blkbd>X</blkbd></inf></f></inline-equation>, restricted by <inline-equation><f><blkbd>X</blkbd></f></inline-equation>. Its simplices are spanned by subsets<inline-equation><f>T⊆S</f></inline-equation> for which the common intersection of Voronoi cells meets <inline-equation><f><blkbd>X</blkbd></f></inline-equation> in a non-empty set. By the nerve theorem,<inline-equation><f><lim align="c"><op>⋃</op></lim><sc>D</sc><inf><blkbd>X</blkbd></inf></f></inline-equation> and <inline-equation><f><blkbd>X</blkbd></f></inline-equation> are homotopy equivalent if all such sets are contractible. This paper shows that <inline-equation><f><lim align="c"><op>⋃</op></lim><sc>D</sc><inf><blkbd>X</blkbd></inf></f></inline-equation> and <inline-equation><f><blkbd>X</blkbd></f></inline-equation> <?Pub Caret1>are homeomorphic if the sets can be further subdivided in a certain way so they form a regular CW complex.
3265202
Emerging Challenges in Computational Topology
Here we present the results of the NSF-funded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology.
46103497
Loops in Reeb Graphs of 
2-Manifolds
Abstract
 Given a Morse function f over a 2-manifold with or without boundary,
 the Reeb graph is obtained by contracting the connected components
 of the level sets to points.
 We prove tight upper and lower bounds on the
 number of loops in the Reeb graph
 that depend on the genus, the number of boundary components,
 and whether or not the 2-manifold is orientable.
 We also give an algorithm that constructs the Reeb graph in time
 O(n log n), where n is the number of edges in the
 triangulation used to represent the 2-manifold and the Morse function.


9524011
On the Definition and the Construction of Pockets in Macromolecules
The shape of a protein is important for its functions. This includes the location and size of identifiable regions in its complement space. We formally define pockets as regions in the complement with limited accessibility from the outside. Pockets can be efficiently constructed by an algorithm based on alpha complexes. The algorithm is implemented and applied to proteins with known three-dimensional conformations.
444430
Loops in reeb graphs of 2-manifolds
Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(nlogn), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.