TY - CHAP
AB - We give analytic inclusion-exclusion formulas for the area and perimeter derivatives of a union of finitely many disks in the plane.
AU - Cheng, Ho-Lun
AU - Herbert Edelsbrunner
ID - 3991
T2 - Computer Science in Perspective: Essays Dedicated to Thomas Ottmann
TI - Area and perimeter derivatives of a union of disks
VL - 2598
ER -
TY - JOUR
AB - Computing the volume occupied by individual atoms in macromolecular structures has been the subject of research for several decades. This interest has grown in the recent years, because weighted volumes are widely used in implicit solvent models. Applications of the latter in molecular mechanics simulations require that the derivatives of these weighted volumes be known. In this article, we give a formula for the volume derivative of a molecule modeled as a space-filling diagram made up of balls in motion. The formula is given in terms of the weights, radii, and distances between the centers as well as the sizes of the facets of the power diagram restricted to the space-filling diagram. Special attention is given to the detection and treatment of singularities as well as discontinuities of the derivative.
AU - Herbert Edelsbrunner
AU - Koehl, Patrice
ID - 3992
IS - 5
JF - PNAS
TI - The weighted-volume derivative of a space-filling diagram
VL - 100
ER -
TY - JOUR
AB - We present algorithms for constructing a hierarchy of increasingly coarse Morse-Smale complexes that decompose a piecewise linear 2-manifold. While these complexes are defined only in the smooth category, we extend the construction to the piecewise linearcategory by ensuring structural integrity and simulating differentiability. We then simplify Morse-Smale complexes by canceling pairs of critical points in order of increasing persistence.
AU - Herbert Edelsbrunner
AU - Harer, John
AU - Zomorodian, Afra
ID - 3993
IS - 1
JF - Discrete & Computational Geometry
TI - Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds
VL - 30
ER -
TY - JOUR
AB - The body defined by a finite collection of disks is a subset of the plane bounded by a tangent continuous curve, which we call the skin. We give analytic formulas for the area, the perimeter, the area derivative, and the perimeter derivative of the body. Given the filtrations of the Delaunay triangulation and the Voronoi diagram of the disks, all formulas can be evaluated in time proportional to the number of disks.
AU - Cheng, Ho-Lun
AU - Herbert Edelsbrunner
ID - 3994
IS - 2
JF - Computational Geometry: Theory and Applications
TI - Area, perimeter and derivatives of a skin curve
VL - 26
ER -
TY - CONF
AB - We combine topological and geometric methods to construct a multi-resolution data structure for functions over two-dimensional domains. Starting with the Morse-Smale complex, we construct a topological hierarchy by progressively canceling critical points in pairs. Concurrently, we create a geometric hierarchy by adapting the geometry to the changes in topology. The data structure supports mesh traversal operations similarly to traditional multi-resolution representations.
AU - Bremer, Peer-Timo
AU - Herbert Edelsbrunner
AU - Hamann, Bernd
AU - Pascucci, Valerio
ID - 3997
TI - A multi-resolution data structure for two-dimensional Morse-Smale functions
ER -