Lecture # 4
|Instructor: Scot Drysdale
||June 27, 1996
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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)
Graph the points, you get a parabola - all points are on the convex hull.
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.
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).
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
Applications from Archeology through Zoology
- 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)
- Identify clusters of stars and clusters or galaxies. (Icke and Van de Weygaert
- Biology, Ecology, Forestry
- Model and analyze plant competition. (Brown 1965 "area potentially available to
a tree", Mead "plant polygons" 1966, Firbank and Watkinson 1987)
- Piece together satellite photographs into large "mosaic" maps. (Manacher and Zobrist
- 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)
- Analyzing patterns of urban settlements. (Boots 1975)
- 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)
- Model market areas of US metropolitan areas (Bogue 1949), market area extending
down to individual retail stores (Snyder 1962, Dacey 1965).
Study of positive definite quadratic forms in two- and three- dimensions (Dirichlet
"Dirichlet Tessellation" 1850) and m-dimensions (Voronoi "Voronoi Diagram" 1908)
- Modeling "grain growth" in metal films (Johnson and Mehl 1939, Evans 1945, Glass
1973, Frost and Thompson 1987, Schaudt and Drysdale 1991, many
- 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)
- Analysis of capillary distribution in cross-sections of muscle tissue to compute
oxygen transport ("capillary domains:"). (Hoofd et.al.
1985, Egginton et.al.
- 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
- 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
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.
Last modified: October 3, 1996