The Half-Edge Data Structure Computational Geometry, WS 2006/07 Lecture 9, Part I Prof. Dr. Thomas Ottmann Khaireel A. Mohamed Algorithmen & Datenstrukturen,

The Half-Edge Data Structure

Computational Geometry, WS 2006/07Lecture 9, Part I

Prof. Dr. Thomas OttmannKhaireel A. Mohamed

Algorithmen & Datenstrukturen, Institut für InformatikFakultät für Angewandte WissenschaftenAlbert-Ludwigs-Universität Freiburg

Computational Geometry, WS 2006/07Prof. Dr. Thomas Ottmann 2

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


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.


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 subdivisionEuler’s formular: v – e + f = 2


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)


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.

e

v1

v2

f2

f1

e_v1[4]

e_v2[4]


The Doubly-Connected-Edge-List (DCEL)

• DCEL is a directed half-edge B-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.


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;}; HE_edge

f e_next

e_prev

v_orig e_twin


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;};

HE_vert

edge

p=(x,y)


• 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.

DCEL Component – Face I

struct HE_face_barebone { HE_edge *edge;};

HE_face_barebone

edge


• 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.

DCEL Component – Face II

struct HE_face { HE_edge *outerComp; HE_edge **innerComps;};

HE_face outerComp

innerComps[0]


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;

edgef1 v2

v1 f2


DCEL Example

Examplevertex 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) }

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;

In terms of the structure definitions:


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);


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 vertex v 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


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.

c6

c7c3

c5c1c2

c4c8 c1

c6

c3

c2c5

c8

c4

c7

Outside

Holes


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.

ep

p

ee_twinev_orig

ev_orig e1

e2e2e_twin

e1e_twin


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

e1

e2

ev_orig

v

vInsertion of new edges into the flow:» Iterate edges at v.» Exercise.


Overlaying Two Planar SubdivisionsPlane 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.


Handling Intersections

• Additional handling of ‘intersection’ event points:– Split and re-direct edges.

– Check new nearest-neighbours for intersections.

• Recall (from Lecture 3): 2

13

7

7 3

12 8

U(P)

C(P)

C(P)

3

8

4

5

7

1

3

2

1

• PL


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)

Documents

The Half-Edge Data Structure Computational Geometry, WS 2006/07 Lecture 9, Part I Prof. Dr. Thomas Ottmann Khaireel A. Mohamed Algorithmen & Datenstrukturen,