What is Computational Geometry

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What is Computational Geometry

• Ref Godfried T. Toussaint

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The Problem

• Given a line (yaxb) and a point p (xp,yp),
  determine whether the point is above/below the
  line
• Solution 1
• Set xz xp
• Compute yz
• Compare yz and yp

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Alternate Solutions
Originality of this formula

• Signed area
• Negative if CW
• Positive if CCW
• How do you compare these two algorithms in terms
  of
• Robustness?
• Efficiency?

Consider special cases
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Possible Definitions of CGeom

• The science concerned with computing geometric
  properties of sets of geometric objects in space
• The design and analysis of algorithms for solving
  geometric problems
• The study of the inherent computational
  complexity of geometric problems under varying
  models of computation
• Runs faster, requires less memory space and more
  robust wrt numerical errors

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Modern Computational Geometry

• Ph.D. thesis of M. Shamos (Yale, 1978)
• Has been around for more than 2600 years starting
  with the Greeks
• Difference
• The size of input to an algorithm (n)
• (Today) Hundreds, thousands, or millions
• The dimension of a problem (d)
• (Greek) Euclidean two and three space
• (Today) not limited to the exploration of low
  dimensional problems non-Euclidean, higher
  dimension

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The Complexity of Algorithm

• Ex Convex Hull
• (naïve algorithm)
• C(n) the total number of primitive operation
  required

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Big O, Big Omega, and C(n)

• Big O
• A simplified expression with only the dominant
  terms, dropping the coefficient
• The contribution that Shamos made was the
  emphasis on including with each algorithm a
  complexity analysis in terms of Big O notation
  and the introduction of lower bounds on the
  complexity of geometric problems
• Big Omega W
• Lower bound on the time complexity of the problem
• A statement about the problem (not the algorithm)
• Optimal when complexity function of the
  algorithm matches the complexity function of the
  problem

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The Domain of CG

• Geometric Probing
• Art Gallery Theorems and Algorithms
• Computer Graphics
• Geometric Modeling
• Computer Vision
• Robotics

• Dynamic CG
• Parallel CG
• Isothetic CG
• Numerical CG
• Geodesic CG
• See also
• Eppsteins Geometry in Action

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Minimax Facility Location Problem

• Determine location X where a facility should be
  located so as to minimize the distance from X to
  its furthest customer
• Find the smallest circle that encloses a given
  set of n points

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Geometric Probing

• Implement a QC procedure to determine circularity
  of rings
• For smooth convex shape, the verification of D in
  sufficient directions should work
• Apply 3 probes, 60 degrees apart
• Does this work?
• Challenger anecdote

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Geometric Probing (cont)

• Reuleaux triangle
• (construction)
• Known as constant diameter shape
• Smooth rolling, yet highly non-circular
• Demonstrates that any number of such probes is
  insufficient to determine circularity

Verify the distance between any parallel lines is
D
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Theory of Geometric Probing

• The determination of the number of probes that
  are necessary and sufficient to determine an
  object completely
• The design of efficient algorithms for actually
  carrying out the probing strategies
• Computer vision, pattern recognition, QC, robotics

• References
• Steven Skiena Problems in Geometric Probing.
  Algorithmica 4(4) 599-605 (1989)
• Skiena Interactive Reconstruction via Geometric
  Probing (1992)

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Art Gallery Problem

• 1973 Victor Klee poses the problem
• Given n arbitrarily shaped gallery, what is the
  minimum number of cameras required to guard the
  interior of an n-wall gallery?
• 1975, Vasek Chvatals theorem n/3 are always
  sufficient and sometimes necessary.

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Observation
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Computer Graphics

• Hidden line problem hidden surface removal
  visibility problem

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Isothetic Computational Geometry

• Also rectilinear computational geometry
• Often greatly simplifies the algorithm
• Image processing, VLSI design

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Geodesic

• Euclidean distance between two points no longer a
  useful measure
• Geodesic distance the length of the shortest
  path between the two points that avoids obstacles
• Geodesic convex hull (also relative convex hull)
• Given a set S of n points inside a simple polygon
  P of n vertices, the relative convex hull of X
  (relative to P) is the minimum perimeter circuit
  that lies in p and encloses S.

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Homeworks

• Verify that the signed area of triangle is
  correct.
• Verify that for Reuleaux triangles, the distance
  between any pair parallel lines is constant.