Instructor: Scot Drysdale | June 27, 1996 |

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)

Note that this works because of the function chosen. For example this will not work on a sin curve.

Homework review

Ceiling function (round up) ...

Floor function (round down) ...

- 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:

**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)

- 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:

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

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.

Mailto: dimacs-www@dimacs.rutgers.edu

Last modified: October 3, 1996