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.
8396933
Hierarchical morse complexes for piecewise linear 2-manifolds
We present algorithms for constructing a hierarchy of increasingly coa rse Morse complexes that decompose a piecewise linear 2-manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse complexes by cancelling pairs of critical points in order of increasing persistence.
44420378
Triangulating Topological Spaces
Given a subspace ${\Bbb X} \subseteq {\Bbb R}^d$ and a finite set S⊆ℝd, we introduce the Delaunay complex, ${\cal D}_{\Bbb X}$, restricted by ${\Bbb X}$. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets ${\Bbb X}$ in a non-empty set. By the nerve theorem, $\bigcup {\cal D}_{\Bbb X}$ and ${\Bbb X}$ are homotopy equivalent if all such sets are contractible. This paper proves a sufficient condition for $\bigcup {\cal D}_{\Bbb X}$ and ${\Bbb X}$ be homeomorphic.
9599628
The union of balls and its dual shape
Efficient algorithms are described for computing topological,combinatorial, and metric properties of the union of finitely many ballsin <inline-equation><f><blkbd>R</blkbd><sup>d</sup></f></inline-equation>. These algorithms are based on a simplicial complexdual to a certain decomposition of the union of balls, and on shortinclusion-exclusion formulas derived from this complex. The algorithmsare most relevant in <inline-equation><f><blkbd>R</blkbd><sup>3</sup></f></inline-equation> <?Pub Caret>where unions of finitely many balls arecommonly used as models of molecules.
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.
2265636
Polygonal Intersection Searching
33731652
Rectangular Point Location in d Dimensions with Applications
53237055
Three-dimensional alpha shapes
Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that purpose, this article introduces the formal notion of the family of α-shapes of a finite point set in R 3 . Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter α e R controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in time 0(n 2 ) , worst case. A robust implementation of the algorithm is discussed, and several applications in the area of scientific computing are mentioned.