Computational Geometry The study of algorithms for combinatorial, topological, and metric problems concerning sets of points, typically in Euclidean space. Representative areas of research include geometric search, convexity, proximity, intersection, and linear programming. Online Computing Dictionary CS691G Computational Geometry - UMass Amherst

Discrete Geometry Covering Tiling Packing Tiling: from http://www.uwgb.edu/dutchs/symmetry/archtil.htm Covering: from http://www.ics.uci.edu/~eppstein/junkyard/all.html Packing: from http://www.sciencemag.org/cgi/content/full/301/5637/1186a Kepler conjectured in 1611 that the density of a packing of congruent spheres in three dimensions is never greater than / 18. This maximum density (approximately 74.05%) is achieved with face-centered cubic packing, which is just the grocer's method of arranging oranges on a fruit-stand (each orange touches four in the layer beneath it). The Kepler conjecture received the attention of some of the greatest mathematicians, including Newton and Gauss. It even made it to the Pantheon of open problems of 1900, Hilbert's famous list of 23 unsolved problems. Proven in the 90s using computers. But not entirely verified. Most mathematicians dislike and mistrust computer-assisted proofs, because such proofs seem to violate their Platonic a priorism and "contaminate" the field with the contingencies of experimental science. But the reason mathematicians have been able to manage, so far, with the tiny computer between their shoulders, is that they only proved (relatively) trivial results. What makes the proofs of the four-color and the Kepler conjectures even more significant than their already immense face-values (as proofs of long-standing, extremely difficult problems) is that they are harbingers and iconic examples of future mathematics, which will all be computer-assisted and eventually computer-generated. Packing CS691G Computational Geometry - UMass Amherst

Video Games CS691G Computational Geometry - UMass Amherst

What we saw… Walking through large model Collisions Dynamic simulation (Compare with automated movie generation) CS691G Computational Geometry - UMass Amherst

What to look for… Algorithms Complexity Data structures Geometric primitives CS691G Computational Geometry - UMass Amherst

Proximity Queries CS691G Computational Geometry - UMass Amherst

Dynamic Simulation CS691G Computational Geometry - UMass Amherst

Dynamic Simulation CS691G Computational Geometry - UMass Amherst

Multi-Player Games CS691G Computational Geometry - UMass Amherst

Multi-Player Games Some players might be computer generated (animations) Distributed state representation CS691G Computational Geometry - UMass Amherst

Motion Planning CS691G Computational Geometry - UMass Amherst

Kinetic Data Structures CS691G Computational Geometry - UMass Amherst

The Post Office Problem Which is the closest post office to every house? (Don Knuth) Given n sites in the plane Subdivision of plane based on proximity Georgy Voronoi 1868-1908 CS691G Computational Geometry - UMass Amherst

Voronoi Diagram CS691G Computational Geometry - UMass Amherst

Descartes in 1644: Gravitational Influence of stars René Descartes 1596-1650 CS691G Computational Geometry - UMass Amherst

Distribution of McDonalds in SF CS691G Computational Geometry - UMass Amherst

Soap Bubble in a Frame CS691G Computational Geometry - UMass Amherst

Honeycomb CS691G Computational Geometry - UMass Amherst

Dragonfly’s Wing CS691G Computational Geometry - UMass Amherst

Graphic by D'Arcy Thompson CS691G Computational Geometry - UMass Amherst

Installation by Scott Snibbe, 1998 CS691G Computational Geometry - UMass Amherst

Uses for Voronoi Diagram Anthropology and Archeology -- Identify the parts of a region under the influence of different Neolithic clans, chiefdoms, ceremonial centers, or hill forts. Astronomy -- Identify clusters of stars and clusters of galaxies (Here we saw what may be the earliest picture of a Voronoi diagram, drawn by Descartes in 1644, where the regions described the regions of gravitational influence of the sun and other stars.) Biology, Ecology, Forestry -- Model and analyze plant competition ("Area potentially available to a tree", "Plant polygons") Cartography -- Piece together satellite photographs into large "mosaic" maps Crystallography and Chemistry -- Study chemical properties of metallic sodium ("Wigner-Seitz regions"); Modelling alloy structures as sphere packings ("Domain of an atom") Finite Element Analysis -- Generating finite element meshes which avoid small angles Geography -- Analyzing patterns of urban settlements Geology -- Estimation of ore reserves in a deposit using information obtained from bore holes; modelling crack patterns in basalt due to contraction on cooling Geometric Modeling -- Finding "good" triangulations of 3D surfaces Marketing -- Model market of US metropolitan areas; market area extending down to individual retail stores Mathematics -- Study of positive definite quadratic forms ("Dirichlet tessellation", "Voronoi diagram") Metallurgy -- Modelling "grain growth" in metal films Meteorology -- Estimate regional rainfall averages, given data at discrete rain gauges ("Thiessen polygons") Pattern Recognition -- Find simple descriptors for shapes that extract 1D characterizations from 2D shapes ("Medial axis" or "skeleton" of a contour) Physiology -- Analysis of capillary distribution in cross-sections of muscle tissue to compute oxygen transport ("Capillary domains") Robotics -- Path planning in the presence of obstacles Statistics and Data Analysis -- Analyze statistical clustering ("Natural neighbors" interpolation) Zoology -- Model and analyze the territories of animals CS691G Computational Geometry - UMass Amherst

Voronoi Graph Voronoi region Vor(p) (p in set S) Voronoi Graph VOR(S) the set of points on the plane that are closer to p than to any othe rpoint in S Voronoi Graph VOR(S) dual to voronoi region graph two points are adjacent if their voronoi regions have common contiguous boundary (segment) CS691G Computational Geometry - UMass Amherst

Voronoi Graph Voronoi Graph in the rectilinear plane Rectilinear distance: p = (x, y); p’=(x’,y’) Voronoi region of b ab b a bc c ac CS691G Computational Geometry - UMass Amherst

Now we will find such X-Y-path e1, e2,…,ek in VG(S) THEOREM: For any set S of points in the plane, MST is subgraph of the Voronoi Graph VG(S) PROOF Let an edge XY between two points X and Y does not belong to the Voronoi graph VG(S). We will show that there is an X-Y- path in VG(S) which contains edges e1, e2,…,ek each shorter than XY, this will imply that XY not belong to MST. Indeed, for each edge eI (I=1,…,k) there is an MST path pI connecting ends of eI consisting of MST edges each no longer than eI. The path obtained by concatenating paths p1,…,pk connects X to Y and contains MST edges each shorter than XY. Thus XY does not belong to MST. Now we will find such X-Y-path e1, e2,…,ek in VG(S) CS691G Computational Geometry - UMass Amherst

Delaunay Triangulation (1934) Boris Nikolaevich Delone (1890 - 1980) Dual of Voronoi (graph theoretic, topological, combinatorial) CS691G Computational Geometry - UMass Amherst

Delaunay Triangulation Properties maximizes minimum angle in each triangle minimizes maximum radius of circumcircle and enclosing circle minimizes sum of inscribed radii many more… CS691G Computational Geometry - UMass Amherst

Finite Element Analysis CS691G Computational Geometry - UMass Amherst