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The Half-Edge Data Structure

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.

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The Half-Edge Data Structure

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  1. 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

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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. v2 f1 edge f2 v1 Adjacency Queries • 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

  13. 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

  14. 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 = 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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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