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04-F-03. Vertex & Face

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04-F-03. Vertex & Face课程教案、知识点、字幕

好 接下来我们来看顶点

所有的顶点也会构成一张顶点的表

或者一个几何

在这个几何中的每一项

分别对于一个顶点

而每一个顶点

都应该同样要记录类似的

几方面的信息

identifier我们不说了

x和y是特别的

因为在这里我们补充说明一下

既是每一个vertex

在这里都是已经嵌入到

平面上去了

所以它都应该具有

一个准确的 具体的坐标

x和y就是这样

请注意一般的graph

它中间的顶点

是没有这个信息的

所以我们更倾向于

将指定了这样的坐标

并且嵌入到平面之后的

这样图中的顶点称作为vertex

而不是泛泛的称之为node

接下来呢

我们可以看到作为一个顶点的记录

除了刚才的两项以外

接下来唯一还要记录的

就是一个incidence关系

这个incidence关系

指的是什么呢

依然是一个关联关系了

那么我们知道

任何一个顶点

与之都会有若干条边相关联

而我们这里的incidence呢

指的是其中特定的那样一条

这一条其实是随机的

当然在这里头

我们故意的称之为

the first outgoing incident half-edge

第一条向外发出的关联的半边

在后面我们会看到这个

first incident edge

实际上只是我们为了

围绕这个顶点x所做遍历

找到第一个起始点

好 接下来我们再看第三张表

也就是面表中的每一个元素

它对应的其实就是一张一张的face

我们可以看到

这里记录的信息更少

相对于顶点

把x y坐标也去掉了

原因很简单

因为这个face

没有必要去记录一个它的准确的位置

那么我们这里唯一要解释的就是

这里同样有一个incidence

这个incidence指的是什么呢

从这个图可以看出来

是指的是一条

incident edge

或者叫incident half-edge

没错 我们刚才讲过每一张面

都是由若干条half-edges围成的

我们将来确实像刚才所提到的那样

有必要围绕着它做一个遍历

而遍历的起点

其实就是由这个

first incident half-edge所给出来的

而数据结构中的这一项

也恰好就是起到这样的一个作用

好了 现在我们已经将这个数据结构中

所有的组成部分逐一的进行了剖解

但是我相信

你和我刚开始学这个的时候一样

你依然没有形成一个总体的认识

计算几何课程列表:

00. Introduction

-Before we start

--html

-Evaluation

--html

-Online Judge

--html

-Lecture notes

--html

-Discussion

--html

-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

04-F-03. Vertex & Face笔记与讨论

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