1. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin 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 doesn’t always produce minimum-size BSP trees, but it generates reasonably small ones.

3. UNC Chapel Hill M. C. Lin 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(n 2 log n). For any set of n non-intersecting triangles in R 3, a BSP tree of size O(n 2 ) 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. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin 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((d+1)n) nodes and can be constructed in O((d+1)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((d+1)m) time.

6. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin 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. UNC Chapel Hill M. C. Lin Techniques Discussed Plane-Sweep Incremental Construction Randomized Algorithms Divide-and-Conquer Techniques Hierarchies & Recursion Transform using Duality

12. UNC Chapel Hill M. C. Lin 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)