Convex hull problem W(n log n)
Given a convex hull algorithm, I can use it to sort .
Sort the numbers

x --ð (x, x2)
x1 = 7 --ð (7, 49)
x2 = 5 --ð (5, 25)
x3= 17 --ð (17, 172)
x4 = 42 --ð (42, 422)
x5 = 3 --ð ( 3, 9)

Voronoi Diagrams

The concept is more than a century old, discussed in 1850 by Dirichlet and in 1908 in a paper of Voronoi. A Voronoi Diagram records everything one would want to know about proximity to a set of points (or more general objects).

Applications:

The post office problem
Where pizza parlors are located
Air rescue stations

Given the coordinates of a point quickly tell which station should respond. The closest one should respond. Each site should get a portion of the plane to cover.
One site - covers the whole world
Two sites - each sites cover half plane

Slides:

Voronoi Diagrams:
Applications from Archeology through Zoology

• Archeology and Anthropology - Identify the parts of a region under the influence of different neolithic clans, chiefdoms, ceremonial centers, or hill forts. (Singh 1976, Renfrew 1973, Hammond 1972, Cunnliff 1971)
  
• Astronomy - Identify clusters of stars and clusters or galaxies. (Icke and Van de Weygaert 1987)
  
• Biology, Ecology, Forestry - Model and analyze plant competition. (Brown 1965 "area potentially available to a tree", Mead "plant polygons" 1966, Firbank and Watkinson 1987)
  
• Cartography - Piece together satellite photographs into large "mosaic" maps. (Manacher and Zobrist 1983)
  
• Crystallography and Chemistry - Study chemical properties of metallic sodium (Wigner and Seitz "Wigner-Seitz regions" 1933) Modeling alloy structures as sphere packings (Frank and Kaspar "domain of an atom" 1958).
  
• Finite Element Analysis - Generating finite element meshes which avoid small angles (Baker 1989, Chew 1989)
  
• Geography - Analyzing patterns of urban settlements. (Boots 1975)
  
• Geology - Estimation of ore reserves in a deposit using information obtained from bore holes. (Boldyrev 1909, Davis and Harding 1920-21, 1923) Modeling crack patterns in basalt due to contraction on cooling (Stiny 1929, Smalley 1966)
  
• Geometric Modeling - Finding "good" triangulations of 3-D surfaces (Barnhill 1977)
  
• Marketing - Model market areas of US metropolitan areas (Bogue 1949), market area extending down to individual retail stores (Snyder 1962, Dacey 1965).
  
• Mathematics - Study of positive definite quadratic forms in two- and three- dimensions (Dirichlet "Dirichlet Tessellation" 1850) and m-dimensions (Voronoi "Voronoi Diagram" 1908)
  
• Metallurgy - Modeling "grain growth" in metal films (Johnson and Mehl 1939, Evans 1945, Glass 1973, Frost and Thompson 1987, Schaudt and Drysdale 1991, many others)
  
• Meteorology - Estimate regional rainfall averages, given data at discrete rain gauges. (Thiessen 1911, Horton 1917, Whitney 1929)
  
• Pattern Recognition - Find simple descriptors for shapes that extract 1-D characterizations from 2-D shapes. (Blum "Medial Axis" or "Skeleton" of a contour 1967, 1973)
  
• Physiology - Analysis of capillary distribution in cross-sections of muscle tissue to compute oxygen transport ("capillary domains:"). (Hoofd et.al. 1985, Egginton et.al. 1989)
  
• Robotics - Path planning in the presence of obstacles (O'Dunlaing, Sharir, and Yap 1986)
  
• Statistics and Data Analysis - Analyze statistical clustering (Sibson 1980), "Natural Neighbors" interpolation (Sibson 1981)
  
• Zoology - Model and analyze the territories of animals. (Tanemura and Hasegawa 1980)
  

Delaunay Triangulations - see diagram page 174--176
A triangulation of a set of points: connect any 2 points, connect another point using any edge as long as it doesn't cross an already existing edge.
Whenever you triangulate ....to be continued

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Supplemental Notes:

Leonard Euler in 1758 noted that the sum of the number of vertices and faces is always two more than the number of edges in all polyhedra.
Euler's formula: Let V, E, and F be the number of vertices, edges, and faces respectively of a polyhedron then V - E + F = 2
A formal proof of Euler's formula is given on page 119.


Given a graph with 5 regions, as in figure 1 below, a triangulation is found by:
place a vertex (point) in each region.
connecting two regions if they have a common edge as shown in figure 2.
the complete triangulation is shown in figure 3.

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Mailto: dimacs-www@dimacs.rutgers.edu
Last modified: October 3, 1996

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