ADA: 14. Intro to CG1 Objective o give a non-technical overview of Computational geometry, concentrating on its main application areas Algorithm Design and Analysis (ADA) , Semester Introduction to Computational Geometry

ADA: 14. Intro to CG2 1.What is Computational Geometry? 2.Uses in Computer Graphics 3.Uses in Robotics 4.Uses in GIS 5.Uses in CAD/CAM 6.A TextbookOverview

1. What is Computational Geometry? The systematic study of algorithms and data structures for geometric objects, with a focus on exact algorithms that are asymptotically fast.

ADA: 14. Intro to CG4 CG in Context Theoretical Computer Science Applied Computer Science Applied Math Geometry Computational Geometry Efficient Geometric Algorithms Design Analyze Apply

2. Uses in Computer Graphics  Intersect geometric primitives (lines, polygons, polyhedra, etc.)  Determine primitives lying in a region.  Hidden surface removal – determine the visible part of a 3D scene while discard the occluded part from a view point.  Deal with moving objects and detect collisions.

ADA: 14. Intro to CG6 Is point q inside simple polygon P? Point in Polygon Testing P n-gon q Naïve: O(n) per test CG: O(log n)

ADA: 14. Intro to CG7 Given n line segments in the plane, determine: o Does some pair intersect? (DETECT) o Compute all points of intersection (REPORT) Segment Intersection Naïve: O(n 2 ) CG: O(n log n) detect, O(k+n log n) report

ADA: 14. Intro to CG8 Find “smallest” (tightest fitting) pair of bounding boxes Motivation: o Best outer approximation o Bounding volume hierarchies The 2-Box Cover Problem

Triangulation of Polygons

ADA: 14. Intro to CG10 Collision Detection

3. Uses in Robotics  Motion planning  Grasping  Parts orienting  Optimal placement

Proximity Closest coffee shop in PSU? Voronoi diagram Delaunay triangulation

ADA: 14. Intro to CG13 A Voronoi diagram is a way of dividing space into smaller regions. A set of points (called seeds, sites, or "coffee shops") is specified beforehand and for each seed there will be a corresponding region consisting of all points closer to that seed than to any other. The regions are called Voronoi cells. Closely related to Delaunay triangulation A Voronoi Diagram

Voronoi Diagrams in Nature Dragonfly wing Honeycomb Constrained soap bubbles Giraffe pigmentation

ADA: 14. Intro to CG15 A Delaunay triangulation for a set points results in a series of triangles connecting those points. A circle drawn through the three points in a triangle will contain no other points. Delaunay Triangulation Delaunay triangulation

Path Planning Robot How can a robot find a short route to the destination that avoids all obstacles?

ADA: 14. Intro to CG17 Mobile Robotic Guard Watchman Route Problem

Determine the smallest number of cameras needed to see all of a given area. 5 cameras are enough to see everywhere (what about 4 cameras? 3?) How Many Cameras? viewable area for this camera

4. Uses in GIS Storage of geographical data (contours of countries, height of mountains, course of rivers, population, roads, electricity lines, etc.)  Large amount of data – requiring efficient algorithms.  Geographic data storage (e.g., map of roads for car positioning or computer display).  Interpolation between nearby sample data points  Overlay of multiple maps.

5. Uses in CAD/CAM  Intersection, union, and decomposition of objects.  Testing on product specifications.  Design for assembly – modeling and simulation of assembly.  Testing design for feasibility.

ADA: 14. Intro to CG21 Bounding Volume Hierarchy BV-tree: Level 0

ADA: 14. Intro to CG22 BV-tree: Level 1

ADA: 14. Intro to CG23 BV-tree: Level 2

ADA: 14. Intro to CG24 BV-tree: Level 5

ADA: 14. Intro to CG25 BV-tree: Level 8

ADA: 14. Intro to CG26 Computational Geometry in C o Joseph O’Rourke, Cambridge University Press, 2nd ed., A Textbook books/compgeom.html