Overview of Last Lecture

1
Overview of Last Lecture

• About Final Course Project
• presentation, demo, write-up
• More geometric data structures
• Binary Space Partitions (BSP)
• hidden surface removal, set operation, visibility
  
• Quadtrees
• mesh generation, image analysis, GIS, graphics
• Comprehensive Reviews
• Current Future Trends of the Field

2
Binary Space Partitions

• Recursively splitting the plane (space) with a
  line (hyperplane), which may cut the plane
  (space) as well as the objects themselves.
• The size of a BSP tree is the total number of the
  object fragments generated by the partitioning
• Auto-partition a BSP that uses only the cutting
  lines (planes) to be the set of extensions of the
  input segments. NOTE Auto-partition doesnt
  always produce minimum-size BSP trees, but it
  generates reasonably small ones.

3
Data Structure Analysis

• Let S be a set of n segments in a plane. A BSP
  of size O(n log n) can be computed in expected
  O(n2 log n).
• For any set of n non-intersecting triangles in
  R3, a BSP tree of size O(n2) exists. Moreover,
  there are configurations for which the lower
  bound size of any BSP is quadratic. Despite
  this fact, in general, BSP trees perform
  reasonably well.

4
Quadtrees

• Quadtree a rooted tree in which every internal
  node has 4 children. Every node corresponds to a
  square.
• Construction split the current square into 4
  quadrants, partition the point set accordingly,
  and recursively construct quadtrees for each
  quadrant with its associated point set. It stops
  when the point set contains less than 2 points.
• The point set is not necessarily split well. It
  is possible that all points lie in the same
  quadrant. Thus, a quadtree can be quite
  unbalanced. It is not possible to express the
  size and depth of a quadtree as a number of
  points it stores. But, other quantification is
  possible.

5
Data Structure Analysis

• The depth of a quadtree for a set P of points in
  the plane is at most log(s/c) 3/2, where c is
  the smallest distance between any two points in P
  and s is the side length of the initial square
  that contains P.
• A quadtree of depth d storing a set of n points
  has O((d1)n) nodes and can be constructed in
  O((d1)n) time.
• Let T be a quadtree with m nodes. Then, the
  balanced version of T has O(m) nodes and it can
  be constructed in O((d1)m) time.

6
Mesh Generation

• Simulation of heat transfer and interaction
  between different media require FEM. Such method
  requires dividing the region into small elements.
  The accuracy and speed of FEM depends on the
  mesh.
• Non-uniform mesh generation (fine near the edges
  of components and coarse far away from the edges)
  can be generated using quadtrees. (Examples in
  p.290-291)

7
Topics Covered

• Line-Segment Intersection
• 3D Morphing, thematic map overlay
• Polygon Triangulation
• guarding an art gallery, morphing
• Linear Programming
• manufacturing/molding, collision detection,
  polygon simplification
• Robustness Degneracies
• causes and solutions

8
Topics Covered

• Geometric Data Structures/Search
• range/window search using k-d trees, range trees,
  interval trees, priority search trees, segment
  trees, BSP, quadtrees, etc.
• crystal structure determination, database query,
  image queries, windowing/zoom
• Point Location
• GIS, polygonization of parametric surfaces, path
  planning

9
Topics Covered

• Voronoi Diagram
• post office problem, D. triangulation, CH
• Arrangements Duality
• computing discrepency, visibility graph
• Delaunay Triangulations
• height interpolation, constraint triangulation,
  meshing, etc.
• Convex Hulls
• optimal bounding volumes, V. region

10
Topics Covered

• Robot Motion Planning
• Minkowski sum, potential field methods,
  approximate cell decomposition, visibility
  graphs, etc.
• distance computation, character animation, drug
  design, image-guided surgery, radiosity
  computation, etc.
• Others
• subdivision surfaces, cloth simulation

11
Techniques Discussed

• Plane-Sweep
• Incremental Construction
• Randomized Algorithms
• Divide-and-Conquer Techniques
• Hierarchies Recursion
• Transform using Duality

12
Current Future Trends

• Toward simpler and efficient geometric data
  structures and algorithms
• Design consideration for the problem nature of
  applications
• More dynamic data structures (KDS)