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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
- Adjacency relationships and queries
- 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

Possible Adjacency Queries

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

Adjacency Queries- Given a half-edge edge, we can perform queries in constant time.
- Example:

HE_vert *v1 = edgev_orig;

HE_vert *v2 = edgee_twinv_orig;

HE_vert *f1 = edgef;

HE_vert *f2 = edge e_twinf;

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;

v1p = new Point(1,2);

v1egde = e_12;

HE_face f1;

f1outerComp = e_15;

f1innerComp[0] = e_67;

HE_edge e_54;

e_54v_orig = v5;

e_54e_twin = e_45;

e_54f = f1;

e_54e_next = e_43;

e_54e_prev = e_15;

Computational Geometry, WS 2006/07

Prof. Dr. Thomas Ottmann

Iterated Adjacency Queries

- 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 = faceouterComp;

do {

// Do something with edge.

edge = edgenext;

} while (edge != faceouterComp);

HE_edge *edge = vertexedge;

do {

// Do something with edge, edgee_twin, etc.

edge = edgee_twinnext;

} while (edge != vertexedge);

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 RepresentationConstruct 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

ev_orig

ee_twin

e2

p

e2e_twin

e1

ev_orig

e1e_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

ev_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

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