课程简介
众所周知，几何学的历史至少可追述至古希腊时代，但不同人对“计算几何”的理解却不尽相同。本课程讨论的计算几何，源自于古典离散/组合几何学与现代计算机科学的结合。M. I. Shamos在1978年完成的博士论文，标志着这个学科分支的诞生。从那时起，“计算几何”往往特指针对离散与组合几何结构的算法研究。简而言之，她也可认为是算法设计与分析的几何版。 本课程的教学目标有三： 其一、对计算几何理论的总体认识，在日后的研究工作中，这种认识为你提供几何的视角。其次、对几何问题求解范式及策略的全面领会，包括递增式构造、平面扫描、分而治之、分层化、近似以及随机化等。最后、对基本几何结构及其算法的透彻掌握，包括凸包、多边形细分、Voronoi图、Delaunay三角剖分，以及几何求交、点定位、范围查找、截窗查询等。
课程大纲
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
期末考试
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
期末考试