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Tsinghua University

Computational Geometry

Tsinghua University via XuetangX

Overview

Geometry can be traced back to ancient Greece, but Computational Geometry evolved less than 40 years as a branch of computer science. The Computational Geometry taught in this course is derived from classical discrete/combinatorial geometry and modern computer science.

Computational Geometry first appeared on the horizon when M. I. Shamos presented his Ph.D. dissertation in 1978. Since then, this phrase has been used to refer to algorithmic study on discrete and combinatorial geometric structures and can also be regarded as the geometric version of Algorithm Design and Analysis. Computational Geometry is now considered the basis of robotics, computer aided design and manufacturing (CAM and CID), and geographic information systems (GIS).

Syllabus

  • 00. Introduction
    • Before we start
    • Evaluation
    • Online Judge
    • Lecture notes
    • Discussion
    • A. History of This Course
    • B. What's Computational Geometry
    • C. How to Learn CG Better
    • D. Why English
  • 01. Convex Hull
    • A. Convexity
    • B. Extreme Points
    • C. Extreme Edges
    • D. Incremental Construction
    • E. Jarvis March
    • F. Lower Bound
    • G. Graham Scan: Algorithm
    • H. Graham Scan: Example
    • I. Graham Scan: Correctness
    • J. Graham Scan: Analysis
    • K. Divide-And-Conquer (1)
    • L. Divide-And-Conquer (2)
    • M. Wrap-Up
  • 02. Geometric Intersection
    • 0. Introduction
    • A. Preliminary
    • B. Interval Intersection Detection
    • C. Segment Intersection Reporting
    • D. BO Algorithm: Strategy
    • E. BO Algorithm: Implementation
    • F. BO Algorithm: Analysis
    • G. Convex Polygon Intersection Detection
    • H. Edge Chasing
    • I. Plane Sweeping
    • J. Halfplane Intersection Construction
  • 03. Triangulation
    • 0. Methodology
    • A. Art Gallery Problem
    • B. Art Gallery Theorem
    • C. Fisk's Proof
    • D. Orthogonal Polygons
    • E. Triangulation
    • F. Triangulating Monotone Polygons
    • G. Monotone Decomposition
    • I. Tetrahedralization
  • 04. Voronoi Diagram
    • A. Introduction
    • B. Terminologies
    • C. Properties
    • D. Complexity
    • E. Representation
    • F. DCEL
    • G. Hardness
    • H. Sorted Sets
    • I. VD_sorted
    • J. Naive Construction
    • K. Incremental Construction
    • L. Divide-And-Conquer
    • M. Plane-Sweep
  • 05. Delaunay Triangulation
    • A. Point Set Triangulation
    • B. Delaunay Triangulation
    • C. Properties
    • D. Proximity Graph
    • E. Euclidean Minimum Spanning Tree
    • F. Euclidean Traveling Salesman Problem
    • G. Minimum Weighted Triangulation
    • H. Construction
    • I. RIC With Example
    • J. Randomized Incremental Construction
    • K. RIC Analysis
  • 06. Point Location
    • 0. Online/Offline Algorithms
    • A. Introduction
    • B. Slab Method
    • C. Persistence
    • D. Path Copying
    • E. Node Copying
    • F. Limited Node Copying
    • G. Kirkpatrick Structure
    • H. Trapezoidal Map
    • I. Constructing Trapezoidal Map
    • J. Performance Of Trapezoidal Map
  • 07. Geometric Range Search
    • A. Range Query
    • B. BBST
    • C. kd-Tree: Structure
    • D. kd-Tree: Algorithm
    • E. kd-Tree: Performance
    • F. Range Tree: Structure
    • G. Range Tree: Query
    • H. Range Tree: Performance
    • I. Range Tree: Optimization
  • 08. Windowing Query
    • A. Orthogonal Windowing Query
    • B. Stabbing Query
    • C. Interval Tree: Construction
    • D. Interval Tree: Query
    • E. Stabbing With A Segment
    • F. Grounded Range Query
    • G. 1D-GRQ Using Heap
    • H. Priority Search Tree
    • I. 2D-GRQ Using PST
    • J. Segment Tree
    • K. Vertical Segment Stabbing Query
  • Final Exam

    Taught by

    Junhui Deng

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