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Algorithm Design and Analysis (ADA). 242-535 , Semester 1 2013-2014. Objective give a non-technical overview of Computational geometry, concentrating on its main application areas. 14. Introduction to Computational Geometry. Overview. What is Computational Geometry ?

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## Algorithm Design and Analysis (ADA)

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**Algorithm Design and Analysis (ADA)**242-535, Semester 1 2013-2014 • Objective • give a non-technical overview of Computational geometry, concentrating on its main application areas 14. Introduction to Computational Geometry**Overview**• What is Computational Geometry? • Uses in Computer Graphics • Uses in Robotics • Uses in GIS • Uses in CAD/CAM • A Textbook**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.**CG in Context**Geometry Theoretical Computer Science Applied Math Computational Geometry Efficient GeometricAlgorithms Design Analyze Applied Computer Science 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.**Point in Polygon Testing**• Is point q inside simple polygon P? Naïve: O(n) per test CG: O(log n) q P n-gon**Segment Intersection**• Given n line segments in the plane, determine: • Does some pair intersect? (DETECT) • Compute all points of intersection (REPORT) Naïve: O(n2) CG: O(n log n) detect, O(k+n log n) report**The 2-Box Cover Problem**• Find “smallest” (tightest fitting) pair of bounding boxes • Motivation: • Best outer approximation • Bounding volume hierarchies**3. Uses in Robotics**• Motion planning • Grasping • Parts orienting • Optimal placement**Proximity**Closest coffee shop in PSU? Delaunay triangulation Voronoi diagram**A Voronoi Diagram**• 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**Voronoi Diagrams in Nature**Honeycomb Dragonfly wing Giraffe pigmentation Constrained soap bubbles**Delaunay Triangulation**• 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**Path Planning**How can a robot find a short route to the destination that avoids all obstacles? Robot**Mobile Robotic Guard**Watchman Route Problem**How Many Cameras?**Determine the smallest number of cameras needed to see all of a given area. viewable area for this camera 5 cameras are enough to see everywhere(what about 4 cameras? 3?)**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 datapoints • Overlay of multiple maps.**5. Uses in CAD/CAM**• Intersection, union, and decomposition of objects. • Testing on product specifications. • Testing design for feasibility. • Design for assembly – modeling and simulation of assembly.**Bounding Volume Hierarchy**BV-tree: Level 0**5. A Textbook**• Computational Geometry in C • Joseph O’Rourke,Cambridge University Press, 2nd ed.,1998 http://cs.smith.edu/~orourke/books/compgeom.html

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