Mesh representations and data structures Luca Castelli Aleardi - PowerPoint PPT Presentation

Mesh representations and data structures Luca Castelli Aleardi

Shared vertex representation Half-edge DS Winged edge Triangle based DS Corner Table

easy to implement Shared vertex representation quite compact not efficient for traversal 8 5 6 9 for each face (of degree d ), store: 7 • d references to incident vertices 3 0 f for each vertex, store: 4 2 11 • 1 reference to its coordinates 10 1 faces vertex locations ( x 0 , y 0 , z 0 ) t 0 v 0 v 1 v 3 . . . Memory cost class Point{ ( x 1 , y 1 , z 1 ) t 1 v 1 v 3 v 1 v 2 double x; double y; 3 × f = 6 n t 2 v 2 v 2 v 3 . . . . . . } t 3 v 3 . . . Size (number of references) geometric information t 4 t 5 . . . . . . . . . Queries/Operations class Vertex{ Point p; List all vertices or faces } Test adjacency between u and v class Face{ Vertex[] vertices; . . . . . . . . . Find the 3 neighboring faces of f } v n − 1 combinatorial information List the neighbors of vertex v t f

Half-edge data structure: polygonal (orientable) meshes 8 5 6 9 7 3 0 e 4 2 11 10 1 class Point{ double x; f + 5 × h + n ≈ 2 n + 5 × (2 e ) + n = 32 n + n double y; } Size (number of references) geometric information opposite ( e ) vertex ( e ) class Halfedge{ Halfedge prev, next, opposite; Vertex v; e Face f; next ( e ) } class Vertex{ Halfedge e; prev ( e ) face ( e ) Point p; } class Face{ Halfedge e; } combinatorial information

Half-edge data structure: polygonal (orientable) meshes 8 5 6 9 7 3 0 e 4 2 11 10 1 class Point{ double x; 3 × h + n ≈ 3 × (2 e ) + n = 18 n + n double y; } Size (number of references) geometric information opposite ( e ) vertex ( e ) class Halfedge{ Halfedge prev, next, opposite; Vertex v; e Face f; next ( e ) } class Vertex{ Halfedge e; Point p; } class Face{ Halfedge e; } combinatorial information

Half-edge data structure: efficient traversal e = v . halfedge v e . next opposite ( e ) vertex ( e ) e next ( e ) prev ( e )

Half-edge data structure: efficient traversal e e . next e . next . opposite opposite ( e ) vertex ( e ) e next ( e ) prev ( e )

Half-edge data structure: efficient traversal e e . next . opposite e . next . opposite . next opposite ( e ) vertex ( e ) e next ( e ) prev ( e )

Half-edge data structure: polygonal manifold meshes opposite ( e ) vertex ( e ) e can we represent them? next ( e ) prev ( e ) yes

Triangle based DS: for triangle meshes (used in CGAL) class Point{ class Triangle{ for each triangle, store: float x; Triangle t1, t2, t3; • 3 references to neighboring faces float y; Vertex v1, v2, v3; float z; } • 3 references to incident vertices class Vertex{ } for each vertex, store: Triangle root; Point p; • 1 reference to an incident face } connectivity T V ( x 0 , y 0 , z 0 ) t 2 t 0 v 0 . . . . . . . . . v 1 v 3 . . . ( x 1 , y 1 , z 1 ) t 1 v 1 t 5 t 2 t 0 v 3 v 1 v 2 . . . t 2 v 2 v 2 v 3 . . . . . . . . . v 0 t 2 t 3 v 3 . . . . . . t 4 . . . . . . . . . t 5 v 1 t 5 . . . . . . . . . . . . . . . . . . t 1 v 2 t 0 t 2 v 3 . . . . . . . . . . . . . . . . . . (3 + 3) × f + n = 6 × 2 n + n = 13 n v n − 1 Size (number of references) t f

Triangle based DS: mesh traversal operators class Point{ class Triangle{ the data structure supports the following operators float x; Triangle t1, t2, t3; float y; Vertex v1, v2, v3; int cw ( int i ) { return ( i + 2 )% 3 ; } v = vertex ( △ , i ) float z; } int ccw ( int i ) { return ( i + 1 )% 3 ; } △ = face ( v ) class Vertex{ } v Triangle root; i = vertexIndex ( v, △ ) z Point p; g 0 = neighbor ( △ , i ) g 2 i g 1 } g 1 = neighbor ( △ , ccw ( i )) connectivity △ g 2 = neighbor ( △ , cw ( i )) z = vertex ( g 2 , faceIndex ( g 2 , △ )) i+1 i+2 g 0 q int degree ( int v ) { we can turn around int d = 1 ; a vertex, by com- int f = face ( v ); int g = neighbor ( f , cw ( vertexIndex ( v , f ))); bining the operators p while ( g ! = f ) { above int next = neighbor ( g , cw ( faceIndex ( f , g ))); int i = faceIndex ( g , next ); we can locate a point, by per- g = next ; forming a walk in the triangula- d + +; } tion return d ; }

Triangle based DS: mesh update operators class Point{ class Triangle{ the data structure supports the following operators float x; Triangle t1, t2, t3; float y; Vertex v1, v2, v3; removeVertex ( v ) float z; } } class Vertex{ splitFace ( f ) Triangle root; removeVertex ( v ) Point p; } edgeFlip ( e ) g 0 connectivity splitFace ( f ) v the data structure is modifiable all these operators can be performed in O (1) time g 1 g 1 e edgeFlip ( e ) g 2 g 2