04-E-02. Fary's Theorem在线视频

计算几何课程列表：

00. Introduction

-Before we start

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-Evaluation

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-Online Judge

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-Lecture notes

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-Discussion

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-A. History of This Course

--00-A. History of This Course

-B. What's Computational Geometry

--00-B. What's Computational Geometry

-B. What's Computational Geometry--作业

-C. How to Learn CG Better

--00-C. How to Learn CG Better

-C. How to Learn CG Better--作业

-D. Why English

--00-D. Why English

01. Convex Hull

-A. Convexity

--01-A-01. Why Convex Hull

--01-A-02. Nails In The Table

--01-A-03. Paint Blending

--01-A-04. Color Space

--01-A-05. Convex Hull

-A. Convexity--作业

-B. Extreme Points

--01-B-01. Extremity

--01-B-02. Strategy

--01-B-03. In-Triangle Test

--01-B-04. To-Left Test

--01-B-05. Determinant

-B. Extreme Points--作业

-C. Extreme Edges

--01-C-01. Definition

--01-C-02. Algorithm

--01-C-03. Demonstration

-C. Extreme Edges--作业

-D. Incremental Construction

--01-D-01. Decrease and Conquer

--01-D-02. In-Convex-Polygon Test

--01-D-03. Why Not Binary Search

--01-D-04. Support-Lines

--01-D-05. Pattern Of Turns

--01-D-06. Exterior/Interior

-D. Incremental Construction--作业

-E. Jarvis March

--01-E-01. Selectionsort

--01-E-02. Strategy

--01-E-03. Coherence

--01-E-04. To-Left Test

--01-E-05. Degeneracy

--01-E-06. Lowest-Then-Leftmost

--01-E-07. Implementation

--01-E-08. Output Sensitivity

-E. Jarvis March--作业

-F. Lower Bound

--01-F-01. Reduction

--01-F-02. CAO Chong's Methodology

--01-F-03. Transitivity

--01-F-04. Reduction: Input

--01-F-05. Reduction: Output

--01-F-06. Sorting ≤_N 2d-CH

-F. Lower Bound--作业

-G. Graham Scan: Algorithm

--01-G-01. Preprocessing

--01-G-02. Scan

--01-G-03. Simplest Cases

-G. Graham Scan: Algorithm--作业

-H. Graham Scan: Example

--01-H-01. Example (1/2)

--01-H-02. Example (2/2)

-H. Graham Scan: Example--作业

-I. Graham Scan: Correctness

--01-I-01. Left Turn

--01-I-02. Right Turn

--01-I-03. Presorting

-I. Graham Scan: Correctness--作业

-J. Graham Scan: Analysis

--01-J-01. Ω(n) Backtracks

--01-J-02. Planarity

--01-J-03. Amortization

--01-J-04. Simplification

-J. Graham Scan: Analysis--作业

-K. Divide-And-Conquer (1)

--01-K-01. Merge

--01-K-02. Common Kernel

--01-K-03. Interior

--01-K-04. Exterior

-K. Divide-And-Conquer (1)--作业

-L. Divide-And-Conquer (2)

--01-L-01. Preprocessing

--01-L-02. Common Tangents

--01-L-03. Topmost + Bottommost ?

--01-L-04. Stitch

--01-L-05. Zig-Zag

--01-L-06. Time Cost

--01-L-07. More Considerations

-L. Divide-And-Conquer (2)--作业

-M. Wrap-Up

--01-M. Wrap-Up

02. Geometric Intersection

-0. Introduction

--02-0. Introduction

-0. Introduction--作业

-A. Preliminary

--02-A-01. EU

--02-A-02. Min-Gap

--02-A-03. Max-Gap

--02-A-04. IEU

-A. Preliminary--作业

-B. Interval Intersection Detection

--02-B-01. Algorithm

--02-B-02. Lower Bound

-B. Interval Intersection Detection--作业

-C. Segment Intersection Reporting

--02-C-01. Brute-force

--02-C-02. Hardness

-C. Segment Intersection Reporting--作业

-D. BO Algorithm: Strategy

--02-D-01. Proximity & Separability

--02-D-02. Comparability & Ordering

--02-D-03. Data Structures

--02-D-04. Possible Cases

-D. BO Algorithm: Strategy--作业

-E. BO Algorithm: Implementation

--02-E-01. Degeneracy

--02-E-02. Event Queue

--02-E-03. Events & Operations

--02-E-04. Sweepline Status

-E. BO Algorithm: Implementation--作业

-F. BO Algorithm: Analysis

--02-F-01. Correctness

--02-F-02. Example

--02-F-03. Retesting

--02-F-04. Complexity of Event Queue

--02-F-05. Complexity of Status Structure

-F. BO Algorithm: Analysis--作业

-G. Convex Polygon Intersection Detection

--02-G-01. Problem Specification

--02-G-02. Monotone Partitioning

--02-G-03. Criterion

--02-G-04. Decrease-And-Conquer

--02-G-05. Example Cases

--02-G-06. Complexity

-G. Convex Polygon Intersection Detection--作业

-H. Edge Chasing

--02-H-01. Eliminating Sickles

--02-H-02. Example

--02-H-03. Analysis

-H. Edge Chasing--作业

-I. Plane Sweeping

--02-I. Plane Sweeping

-I. Plane Sweeping--作业

-J. Halfplane Intersection Construction

--02-J-01. The Problem

--02-J-02. Lower Bound

--02-J-03. Divide-And-Conquer

-J. Halfplane Intersection Construction--作业

03. Triangulation

-0. Methodology

--03-0. Methodology

-0. Methodology--作业

-A. Art Gallery Problem

--03-A-01. Definition

--03-A-02. Lower & Upper Bounds

--03-A-03. Hardness

--03-A-04. Approximation & Classification

-A. Art Gallery Problem--作业

-B. Art Gallery Theorem

--03-B-01. Necessity of floor(n/3)

--03-B-02. Sufficiency by Fan Decomposition

-B. Art Gallery Theorem--作业

-C. Fisk's Proof

--03-C-01. Triangulation

--03-C-02. 3-Coloring

--03-C-03. Domination

--03-C-04. Pigeon-Hole Principle

--03-C-05. Generalization

-C. Fisk's Proof--作业

-D. Orthogonal Polygons

--03-D-01. Necessity of floor(n/4)

--03-D-02. Sufficiency by Convex Quadrilateralization

--03-D-03. Generalization

-D. Orthogonal Polygons--作业

-E. Triangulation

--03-E-01. Existence

--03-E-02. Ear & Mouth

--03-E-03. Two-Ear Theorem

--03-E-04. Well-Order

--03-E-05. Ear Candidate

--03-E-06. Induction

--03-E-07. Well-Order (Again)

--03-E-08. Properties

-E. Triangulation--作业

-F. Triangulating Monotone Polygons

--03-F-01. Monotone Polygon

--03-F-02. Monotonicity Testing

--03-F-03. Strategy

--03-F-04. Stack-Chain Consistency

--03-F-05. Same Side + Reflex

--03-F-06. Same Side + Convex

--03-F-07. Opposite Side

--03-F-08. Example

--03-F-09. Analysis

-F. Triangulating Monotone Polygons--作业

-G. Monotone Decomposition

--03-G-01. Cusps

--03-G-02. Helper

--03-G-03. Helper Candidate

--03-G-04. Sweep-Line Status

--03-G-05. Possible Cases

--03-G-06. Example

--03-G-07. Analysis

-G. Monotone Decomposition--作业

-I. Tetrahedralization

--03-I-01. Polyhedron Decomposition

--03-I-02. Schonhardt's Polyhedron

--03-I-03. Seidel's Polygon

-I. Tetrahedralization--作业

04. Voronoi Diagram

-A. Introduction

--04-A-01. A First Glance

--04-A-02. Dining Halls on Campus

--04-A-03. More Analogies & Applications

--04-A-04. Voronoi

-A. Introduction--作业

-B. Terminologies

--04-B-01. Site & Cell

--04-B-02. Intersecting Halfspaces

--04-B-03. Voronoi Diagram

--04-B-04. Planar Voronoi Diagram

-B. Terminologies--作业

-C. Properties

--04-C-01. Non-Empty Cells

--04-C-02. Empty Disks

--04-C-03. Nearest = Concyclic

--04-C-04. Number of Nearest Sites = Degree

--04-C-05. Split & Merge

-C. Properties--作业

-D. Complexity

--04-D-01. Linearity

--04-D-02. Proof

-D. Complexity--作业

-E. Representation

--04-E-01. Subdivision

--04-E-02. Fary's Theorem

--04-E-03. Representing VD

-E. Representation--作业

-F. DCEL

--04-F-01. Twin Edges

--04-F-02. Half-Edge

--04-F-03. Vertex & Face

--04-F-04. Traversal

--04-F-05. True Or False

--04-F-06. Application

-F. DCEL--作业

-G. Hardness

--04-G-01. 1D Voronoi Diagram

--04-G-02. 2D Voronoi Diagram

--04-G-03. Voronoi Diagram In General Position

-G. Hardness--作业

-H. Sorted Sets

--04-H-01. Convex Hull Made Easier

--04-H-02. Convex Hull As A Combinatorial Structure

--04-H-03. Voronoi Diagram As A Geometric Structure

-H. Sorted Sets--作业

-I. VD_sorted

--04-I-01. ε-Closeness

--04-I-02. Lifting

--04-I-03. Projection

--04-I-04. Case A

--04-I-05. Case B

--04-I-06. Sorting Not Made Easier

-I. VD_sorted--作业

-J. Naive Construction

--J. Naive Construction

-J. Naive Construction--作业

-K. Incremental Construction

--04-K-01. Royal Garden

--04-K-02. Disjoint Union

--04-K-03. Complexity

-K. Incremental Construction--作业

-L. Divide-And-Conquer

--04-L-01. Strategy

--04-L-02. Solving Overlaps

--04-L-03. Contour

--04-L-04. Bisectors

--04-L-05. Y-Monotonicity

--04-L-06. Common Tangents

--04-L-07. Contour Length

--04-L-08. Clip & Stitch

--04-L-09. Intersecting with Cells

--04-L-10. Convexity

--04-L-11. Avoiding Rescans

-L. Divide-And-Conquer--作业

-M. Plane-Sweep

--04-M-01. A First Glance

--04-M-02. Backtracking

--04-M-03. Fortune's Trick

--04-M-04. Frozen Region

--04-M-05. Beach Line

--04-M-06. Lower Envelope

--04-M-07. Break Points

--04-M-08. Events

--04-M-09. Circle Event: What, When & Where

--04-M-10. Circle Event: Why

--04-M-11. Circle Event: How

--04-M-12. Site Event: What

--04-M-13. Site Event: How

-M. Plane-Sweep--作业

05. Delaunay Triangulation

-A. Point Set Triangulation

--05-A-01. Definition

--05-A-02. Edge Flipping

--05-A-03. Upper Bound

--05-A-04. Lower Bound

-A. Point Set Triangulation--作业

-B. Delaunay Triangulation

--05-B-01. Dual Graph

--05-B-02. Triangulation

--05-B-03. Hardness

--05-B-04. History

-B. Delaunay Triangulation--作业

-C. Properties

--05-C-01. Empty Circumcircle

--05-C-02. Empty Circle

--05-C-03. Nearest Neighbor

--05-C-04. Complexity

-C. Properties--作业

-D. Proximity Graph

--05-D-01. Gabriel Graph

--05-D-02. Relative Neighborhood Graph

--05-D-03. Lower Bounds

-D. Proximity Graph--作业

-E. Euclidean Minimum Spanning Tree

--05-E-01. Definition

--05-E-02. Construction

--05-E-03. Subgraph of RNG

--05-E-04. Example

-E. Euclidean Minimum Spanning Tree--作业

-F. Euclidean Traveling Salesman Problem

--05-F-01. Definition

--05-F-02. NP-Hardness

--05-F-03. Approximation

-G. Minimum Weighted Triangulation

--05-G-01. Definition

--05-G-02. Counter-Example

--05-G-03. Hardness

-G. Minimum Weighted Triangulation--作业

-H. Construction

--05-H-01. Subtended Arc

--05-H-02. Angle Vector

--05-H-03. Maximizing The Minimum Angle

--05-H-04. Evolution By Edge Flipping

--05-H-05. Strategies

-H. Construction--作业

-I. RIC With Example

--05-I-01. Idea

--05-I-02. Point Location

--05-I-03. In-Circle Test

--05-I-04. Edge Flipping

--05-I-05. Frontier

--05-I-06. Convergence

-I. RIC With Example--作业

-J. Randomized Incremental Construction

--05-J-01. Recursive Implementation

--05-J-02. Iterative Implementation

--05-J-03. In-Circle Test

--05-J-04. Point Location

-J. Randomized Incremental Construction--作业

-K. RIC Analysis

--05-K-01. Time Cost

--05-K-02. Backward Analysis

--05-K-03. Preconditions

--05-K-04. Types Of Edge Change

--05-K-05. Number Of Edge Changes

--05-K-06. Average Degree

--05-K-07. Number Of Rebucketings

--05-K-08. Probability For Rebucketing

--05-K-09. Expectation

--05-K-10. Further Consideration

06. Point Location

-0. Online/Offline Algorithms

--06-0. Online/Offline Algorithms

-0. Online/Offline Algorithms--作业

-A. Introduction

--06-A-01. Where Am I

--06-A-02. Point Location

--06-A-03. Assumptions For Clarity

--06-A-04. Input Size

--06-A-05. Performance Measurements

--06-A-06. A Global View

-A. Introduction--作业

-B. Slab Method

--06-B-01. Slab Decomposition

--06-B-02. Ordering Trapezoids

--06-B-03. Tree of Trees

--06-B-04. Example

--06-B-05. Query Time

--06-B-06. Preprocessing Time

--06-B-07. Storage Cost

--06-B-08. Worst Case

-B. Slab Method--作业

-C. Persistence

--06-C-01. Ephemeral Structure

--06-C-02. Persistent Structure

--06-C-03. Persistent Slabs

-C. Persistence--作业

-D. Path Copying

--06-D-01. Strategy

--06-D-02. X-Search

--06-D-03. Storage Optimization

-D. Path Copying--作业

-E. Node Copying

--06-E-01. O(1) Rotation

--06-E-02. Strategy

--06-E-03. Why Red-Black

--06-E-04. Linear Space

--06-E-05. Time Penalty

-E. Node Copying--作业

-F. Limited Node Copying

--06-F-01. Idea

--06-F-02. Split

--06-F-03. Complexity

--06-F-04. Recoloring

-G. Kirkpatrick Structure

--06-G-01. Optimal And Simpler

--06-G-02. Triangulation

--06-G-03. Example

--06-G-04. Hierarchy

--06-G-05. Independent Subset

--06-G-06. The More The Better

--06-G-07. The Fewer The Better

--06-G-08. Degree

--06-G-09. Existence Of Independent Subset

--06-G-10. Construction Of Independent Subset

--06-G-11. DAG

-G. Kirkpatrick Structure--作业

-H. Trapezoidal Map

--06-H-01. Ray Shooting

--06-H-02. Decomposition

--06-H-03. Properties & Complexity

--06-H-04. Search Structure: Example

--06-H-05. Search Structure: Nodes

--06-H-06. Search Structure: Performance

-H. Trapezoidal Map--作业

-I. Constructing Trapezoidal Map

--06-I-01. Initialization

--06-I-02. Iteration

--06-I-03. Challenges

--06-I-04. Case 1: Two Endpoints

--06-I-05. Case 2: One Endpoint

--06-I-06. Case 3: No Endpoints

--06-I-07. Example

-J. Performance Of Trapezoidal Map

--06-J-01. Randomization

--06-J-02. Expectation

--06-J-03. Number Of Ray Trimmed

--06-J-04. Number Of Trapezoidals Created (1)

--06-J-05. Number Of Trapezoidals Created (2)

--06-J-06. Time For Point Location

--06-J-07. Size Of Search Structure

--06-J-08. Fixed Query Point + Randomly Created Maps

--06-J-09. Each Single Step

--06-J-10. Probability Of Enclosing Trapezoid Changed

--06-J-11. Query Time

07. Geometric Range Search

-A. Range Query

--07-A-01. 1-Dimensional Range Query

--07-A-02. Brute-force

--07-A-03. Binary Search

--07-A-04. Output Sensitivity

--07-A-05. Planar Range Query

-A. Range Query--作业

-B. BBST

--07-B-01. Structure

--07-B-02. Lowest Common Ancestor

--07-B-03. Query Algorithm

--07-B-04. Complexity (1)

--07-B-05. Complexity (2)

-B. BBST--作业

-C. kd-Tree: Structure

--07-C-01. 2d-Tree

--07-C-02. Example

--07-C-03. Construction

--07-C-04. Example

--07-C-05. Canonical Subsets

-C. kd-Tree: Structure--作业

-D. kd-Tree: Algorithm

--07-D-01. Query

--07-D-02. Example

--07-D-03. Optimization

-D. kd-Tree: Algorithm--作业

-E. kd-Tree: Performance

--07-E-01. Preprocessing Time + Storage

--07-E-02. Query Time

--07-E-03. Beyond 2D

-E. kd-Tree: Performance--作业

-F. Range Tree: Structure

--07-F-01. x-Query + y-Query

--07-F-02. Worst Case

--07-F-03. x-Query * y-Queries

-F. Range Tree: Structure--作业

-G. Range Tree: Query

--07-G-01. Painters' Strategy

--07-G-02. X-Tree

--07-G-03. Y-Trees

--07-G-04. Algorithm

-G. Range Tree: Query--作业

-H. Range Tree: Performance

--07-H-01. Storage

--07-H-02. Preprocessing Time

--07-H-03. Query Time

--07-H-04. Beyond 2D

-H. Range Tree: Performance--作业

-I. Range Tree: Optimization

--07-I-01. Y-Lists

--07-I-02. Coherence

--07-I-03. Idea

--07-I-04. Fractional Cascading

--07-I-05. Complexity

08. Windowing Query

-A. Orthogonal Windowing Query

--08-A-01. Definition

--08-A-02. Classification

-A. Orthogonal Windowing Query--作业

-B. Stabbing Query

--08-B-01. 1D Windowing Query

--08-B-02. Stabbing Query

-C. Interval Tree: Construction

--08-C-01. Median

--08-C-02. Partitioning

--08-C-03. Balance

--08-C-04. Associative Lists

--08-C-05. Complexity

-C. Interval Tree: Construction--作业

-D. Interval Tree: Query

--08-D-01. Algorithm (1)

--08-D-02. Algorithm (2)

--08-D-03. Complexity

-D. Interval Tree: Query--作业

-E. Stabbing With A Segment

--08-E-01. Definition

--08-E-02. Interval Tree

--08-E-03. Query Algorithm (1)

--08-E-04. Query Algorithm (2)

--08-E-05. Overview

--08-E-06. Complexity

-F. Grounded Range Query

--08-F-01. O(n) Space

--08-F-02. 2D-GRQ

--08-F-03. 1D-GRQ Using Range Tree

--08-F-04. 1D-GRQ By Linear Scan

-G. 1D-GRQ Using Heap

--08-G-01. Heap

--08-G-02. Query

--08-G-03. Example

--08-G-04. Complexity

-G. 1D-GRQ Using Heap--作业

-H. Priority Search Tree

--08-H-01. PST = Heap + BBST

--08-H-02. Order Property

--08-H-03. Sibling Partitioning

--08-H-04. Construction

-H. Priority Search Tree--作业

-I. 2D-GRQ Using PST

--08-I-01. Algorithm (1/2)

--08-I-02. Algorithm (2/2)

--08-I-03. Example (1/3)

--08-I-04. Example (2/3)

--08-I-05. Example (3/3)

--08-I-06. Query Time (1/3)

--08-I-07. Query Time (2/3)

--08-I-08. Query Time (3/3)

-I. 2D-GRQ Using PST--作业

-J. Segment Tree

--08-J-01. General Windowing Query

--08-J-02. Elementary Interval

--08-J-03. Discretization

--08-J-04. Worst Case

--08-J-05. BBST

--08-J-06. Solving Stabbing Query

--08-J-07. Worst Case

--08-J-08. Common Ancestor

--08-J-09. Canonical Subsets

--08-J-10. O(nlogn) Space

--08-J-11. Constructing A Segment Tree

--08-J-12. Inserting A Segment (1)

--08-J-13. Inserting A Segment (2)

--08-J-14. Inserting A Segment (3)

--08-J-15. Query Algorithm

--08-J-16. Query Time

-K. Vertical Segment Stabbing Query

--08-K-01. Review

--08-K-02. X-Segment Tree

--08-K-03. Associative Structure

--08-K-04. Vertical Segment Stabbing Query

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