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# The Half-Edge Data Structure PowerPoint PPT Presentation

The Half-Edge Data Structure. Computational Geometry, WS 2006/07 Lecture 9, Part I Prof. Dr. Thomas Ottmann Khaireel A. Mohamed. Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg. Overview.

The Half-Edge Data Structure

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## The Half-Edge Data Structure

Computational Geometry, WS 2006/07

Lecture 9, Part I

Prof. Dr. Thomas Ottmann

Khaireel A. Mohamed

Algorithmen & Datenstrukturen, Institut für Informatik

Fakultät für Angewandte Wissenschaften

Albert-Ludwigs-Universität Freiburg

### Overview

• Planar subdivision representation

• Boundary representation structure

• Baumgart’s winged-edge data structure

• Doubly-connected-edge-list (DCEL)

• Overlaying planar subdivisions

• Analyses

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### Representing a Polygon Mesh

• We require a convenient and efficient way to represent a planar subdivision.

• Components in the planar subdivision:

• A list of vertices

• A list of edges

• A list of faces storing pointers for its vertices

• Must preserve adjacency relationships between components.

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

Point anywhere on the polygon mesh and ask:

• Which faces use this vertex?

• Which edges use this vertex?

• Which faces border this edge?

• Which edges border this face?

• Which faces are adjacent to this face?

Planar subdivision

Euler’s formular: v – e + f = 2

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### Boundary Representation Structures

• To represent such queries efficiently, we use the boundary representation (B-rep) structure.

• B-rep explicitly model the edges, vertices, and faces of the planar subdivision PLUS additional adjacency information stored inside.

• Two most common examples of B-rep:

• Baumgart’s winged-edge data structure

• Doubly-connect-edge-list (DCEL)

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

e_v1[4]

v1

f1

e

f2

v2

e_v2[4]

### Baumgart’s Winged-Edge DS

• The Edge DS is augmented with pointers to:

• the two vertices it touches (v1, v2),

• the two faces it borders (f1, f2), and

• pointers to four of the edges which emanate from each end point (e_v1[4], v2[4]).

• We can determine which faces or vertices border a given edge in constant time.

• Other types of queries can require more expensive processing.

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### The Doubly-Connected-Edge-List (DCEL)

• DCEL is a directed half-edgeB-rep data structure.

• Allows all adjacency queries in constant time (per piece of information gathered). That is, for example;

• When querying all edges adjacent to a vertex, the operation will be linear in the number of edges adjacent to the vertex, but constant time per edge.

• The DCEL is excellent in representing manifold surfaces:

• Every edge is bordered by exactly two faces.

• Cross junctions and internal polygons are not allowed.

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

f

e_next

HE_edge

v_orig

e_twin

e_prev

### DCEL Component – Half-edge

• The half-edges in the DCEL that border a face form a circular linked-list around its perimeter (anti-clockwise); i.e. each half-edge in the loop stores a pointer to the face it borders (incident).

• Each half-edge is directed and can be described in C as follows:

struct HE_edge {

HE_vert *v_orig;

HE_edge *e_twin;

HE_face *f;

HE_edge *e_next;

HE_edge *e_prev;

};

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

edge

HE_vert

p=(x,y)

### DCEL Component - Vertex

• Vertices in the DCEL stores:

• their actual point location, and

• a pointer to exactly ONE of the HE_edge, which uses the vertex as its origin.

• There may be several HE_edge whose origins start at the same vertex. We need only one, and it does not matter which one.

struct HE_vert {

Gdiplus::PointF p;

HE_edge *edge;

};

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### DCEL Component – Face I

• The “bare-bones” version of the face component needs only to store a single pointer to one of the half-edges it borders.

• In the implementation by de Berg et al. (2000), edge is the pointer to the circular loop of the OuterComponent (or the outer-most boundary) of the incident face.

• For the unbounded face, this pointer is NULL.

struct HE_face_barebone {

HE_edge *edge;

};

HE_face_barebone

edge

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

innerComps[0]

HE_face

outerComp

### DCEL Component – Face II

• All holes contained inside an incident face are considered as InnerComponents. A list of pointers to half-edges of unique holes is maintained in HE_face as follows.

• In the case that there are no holes in an incident face, innerComps is set to NULL.

struct HE_face {

HE_edge *outerComp;

HE_edge **innerComps;

};

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

v2

f1

edge

f2

v1

• Given a half-edge edge, we can perform queries in constant time.

• Example:

HE_vert *v1 = edgev_orig;

HE_vert *v2 = edgee_twinv_orig;

HE_vert *f1 = edgef;

HE_vert *f2 = edge e_twinf;

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### DCEL Example

Example

vertex v1 = { (1, 2), h_edge(12) }

face f1 = {h_edge(15), [h_edge(67)] }

h_edge(54) = { v5, h_edge(45), f1, h_edge(43), h_edge(15) }

In terms of the structure definitions:

HE_vert v1;

v1p = new Point(1,2);

v1egde = e_12;

HE_face f1;

f1outerComp = e_15;

f1innerComp[0] = e_67;

HE_edge e_54;

e_54v_orig = v5;

e_54e_twin = e_45;

e_54f = f1;

e_54e_next = e_43;

e_54e_prev = e_15;

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

• Iterating over the half-edges adjacent to a given face.

• Iterating over the half-edges that are adjacent to a given vertex.

HE_edge *edge = faceouterComp;

do {

// Do something with edge.

edge = edgenext;

} while (edge != faceouterComp);

HE_edge *edge = vertexedge;

do {

// Do something with edge, edgee_twin, etc.

edge = edgee_twinnext;

} while (edge != vertexedge);

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### Face Records

Determining the boundary-type:

• Is a complete edge-loop (boundary-cycle) an outer-boundary, or the boundary of a hole in the face?

• Select the face f that we are interested in.

• Identify the lowest of the left-most vertexv of any edge-loop.

• Consider the two half-edges passing through v, and compute their angle .

• If  is smaller than 180°, then the edge-loop is an outer-boundary.

f

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

c6

c3

c2

c7

c5

c1

c8

c4

### Top Level DCEL Representation

Construct a graph G to representy boundary-cycles.

• For every boundary-cycle, there is a node in G (+ imaginary bound).

• An arc joins two cycles iff one is a boundary of a hole and the other has a half-edge immediately to the left of the left-most vertex of that hole.

Holes

c3

c1

c8

c6

c2

c7

c5

Outside

c4

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### Splitting an Edge

• Given an edge e and a point p on e, we can split e into two sub-edges e1 and e2 in constant time.

e

p

ev_orig

ee_twin

e2

p

e2e_twin

e1

ev_orig

e1e_twin

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### Splitting and Re-directing Edges

• Given an edge e and a vertex v of degree deg(v) on e, we can split and redirect the sub-edges of the DCEL at v in time O(1 + deg(v)).

e

Insertion of new edges into the flow:

» Iterate edges at v.

» Exercise.

v

ev_orig

e2

v

e1

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### Overlaying Two Planar Subdivisions

Plane sweep!

• Event-points (maintained in balanced binary search tree):

• Vertices of S1 and S2

• All intersections between edges in S1 and S2

• Status-structure (per event):

• Neighbouring edges sorted in increasing x-order.

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

2

1

3

3

1

7

4

5

8

8

L

7

2

1

P

U(P)

C(P)

7

3

1

3

C(P)

2

### Handling Intersections

• Additional handling of ‘intersection’ event points:

• Split and re-direct edges.

• Check new nearest-neighbours for intersections.

• Recall (from Lecture 3):

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

### Analysis

For a total of n vertices in both S1 and S2:

• Sorting of n vertices: O(n log n) time

• Runtime per ‘intersection’-vertex: O(1 + deg(v))

• Time to retrieve neighbouring edges per ‘interection’-vertex: O(log n)

• Total ‘intersection’-vertices: k

• Total runtime: O(n log n + k log n)

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann