242-535 ADA: 14. Intro to CG 1 • Objective o give a non-technical overview of Computational geometry, concentrating on its main application areas Algorithm Design and Analysis (ADA) 242-535, Semester 1 2014-2015 14. Introduction to Computational Geometry
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242-535 ADA: 14. Intro to CG1 Objective o give a non-technical overview of Computational geometry, concentrating on its main application areas Algorithm.
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242-535 ADA: 14. Intro to CG
1
• Objectiveo give a non-technical overview of
Computational geometry, concentrating on its main application areas
Algorithm Design and Analysis
(ADA)242-535, Semester 1 2014-2015
14. Introduction to Computational
Geometry
242-535 ADA: 14. Intro to CG
2
1. What is Computational Geometry?2. Uses in Computer Graphics3. Uses in Robotics4. Uses in GIS5. Uses in CAD/CAM6. A Textbook
Overview
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.
· 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.
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• Is point q inside simple polygon P?
Point in Polygon Testing
P n-gon
q
Naïve: O(n) per test
CG: O(log n)
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• 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(n2)
CG: O(n log n) detect, O(k+n log n) report
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• Find “smallest” (tightest fitting) pair of bounding boxes
• Motivation: o Best outer approximationo Bounding volume hierarchies
The 2-Box Cover Problem
Triangulation of Polygons
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Collision Detection
3. Uses in Robotics
· Motion planning
· Grasping
· Parts orienting
· Optimal placement
ProximityClosest coffee shop in PSU?
Voronoi diagram
Delaunay triangulation
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• 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.