Computational Geometry II

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Computational Geometry II

• Brian Chen
• Rice University Computer Science

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uhh. . . Recap, Please?

• Arrangements are sets of Lines in the plane, in
  general position
• Each pair intersects in exactly one point
• Can be described by Euclidean Coordinates (y mx
  b and all that)

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Purposes and Topics

• Present a mathematical framework for discussing
  the topics in the remainder of the course
• Duality, Convex Hulls, Envelopes, Voronoi
  Diagrams, and Delauney Triangulations

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Duals, duality and dual spaces

• Lines can be uniquely designated by the equations
  which describe them
• Example y 3x 4 is uniquely described by the
  values 3 and 4.
• Points can be uniquely designated by their
  coordinates on the plane
• Example (3, 4) is obviously the only point at x
  3, y 4.

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Who cares, Brian?

• Since lines and points are uniquely determined by
  two values, that means that there exists a
  bijection between them.
• A Bijection is a mapping which correlates one
  member of one set with exactly one member of the
  other set, and also correlates one member of the
  other set with one of the first.

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A new point of view

• This bijection means that we can now transform
  problems about points in space into problems
  about lines, and vice versa.
• It also leads to some very interesting properties
  on the plane and the dual plane
• definitions dual of an arrangement, dual of a
  constellation, dual space.

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Duality Mappings

• There isnt just one mapping between points and
  lines.
• As long as the mapping is one-one and onto (e.g.
  a bijection) then the mapping is a duality
  mapping. (Examples)
• (note we ignore pairs with identical x-values,
  because this results in parallel lines)

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What about sets of points?

• Sets of points in dual space are very
  interesting
• a Segment is a set of points along a line. Since
  the segment doesnt have infinite slope, all the
  points have lines in the dual space which
  intersect. In fact they all intersect at the
  same point. (Proof)

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What about more than just 2 points?

• A line segment is composed of an infinite number
  of collinear points
• In the dual space, it looks like two wedges

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Convex Hulls

• An object which is Convex is an object where for
  any pair of points p and q inside the object, the
  line segment pq is entirely contained within the
  object.
• The Convex Hull of a set A of points is the
  smallest convex set of points which contain all
  the points in A.

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Pictures (on board)
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Why do we want to know?

• Knowing that we are operating on convex sets lets
  us write fast collision detection algorithms, and
  we can use a convex shape to approximate many
  complicated objects
• Convex Hulls also let us do fast visibility
  calculations (draw pic)

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Interesting Duality Envelopes

• The Upper Envelope is the intersection of all
  upper half planes of an arrangement of lines.
• The Lower Envelope is the intersection of all
  lower half planes of the lines.

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3d Generalization

• Points in 3d turn into planes in the dual space
• segments in 3d turn into X-prisms in the dual
  space (drawing)

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Putting ideas together

• Now we can exactly correlate constellations of
  points with arrangements of lines
• Constellations have the interesting property of
  Convex Hulls
• Lines have the interesting property of Envelopes
• coughCORRELATIONcough

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Are Hulls and Envelopes related?

• We have a notion of boxing in points
• We have a notion of boxing in lines
• Because of duals, EVERY point is a line in dual
  space, and every line in dual space is a point.

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The dual of an arrangement

• Is a set of points in cartesian space.
• Doesnt APPEAR to have anything to do with the
  set of lines. (hint Im lying)

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Tell us already!

• Upper Convex Hulls Correspond to lower Envelopes,
  and Lower Convex Hulls Correspond to upper
  Envelopes.

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Section II Voronoi Diagrams

• What are Voronoi Diagrams?
• A Voronoi Diagram Is a partition of a space
  defined for an individual constellation of points
  in the space.

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Defintion

• The space is partitioned into cells, such that a
  point on the plane is in a cell iff the
  constellation point in the cell is the closest
  constellation point.
• A point is on a Boundary if the point is
  equidistant between two or more contellation
  points.

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Pictures, Pictures, Pictures

• Simple picture with 2 points
• More complicated picture with 3 points
• Definition of general position (no three on a
  line, no four on a circle ltall points on the
  circle are equidistant from the centergt)
• General picture

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Interesting Properties

• definition Voronoi vertex, Voronoi edge, Voronoi
  boundary.
• A point p is a Voronoi vertex iff the largest
  empty circle C(p) around p contains at least
  three Voronoi sites.
• A point p is on the Voronoi boundary iff the
  largest empty circle surrounding it contains
  exactly two sites.

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Delaunay Triangulations

• Are the border connectivity graphs of Voronoi
  diagrams
• Picture

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Properties

• Delaunay Graphs are always planar.
• If T is a triangulation of points P, then T is
  Delaunay iff the circumcircle of any triangle
  contains no other point of P.
• Definition of Legal Triangulation (see next
  slide)

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General Triangulation

• aribitrarily making triangles everywhere.
• Triangulations used to describe height maps from
  point samples.
• Need triangulations which dont do stupid things

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Last Property

• Delauney Triangulations are the only legal
  triangulation
• If a Triangulation T is Delaunay iff it is legal.
• Delauney Triangulations are the best
  Triangulations

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Next Topic Collision detection

• Begin using the structures described this week
  and last week.