1 mesh data structures - technion

Digital Geometry Processing, Technion - Spring 2008

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Mesh Data Structures


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Geometry processingMesh data

Data structure selection depends on• Mesh type• Algorithm requirements

Geometry Connectivity


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Mesh gallery

Single component,closed, triangular, orientable manifold

Multiple components

With boundaries

Not only triangles

Not orientable

Non manifold


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In this courseOnly orientable, triangular, manifold meshes

Single component,closed, triangular, orientable manifold

Multiple components

With boundaries

Not only triangles

Not orientable

Non manifold


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Algorithms’ requirements

Data structure should be efficient for• Rendering

• Geometry queriesWhat are the vertices of face #3?Are vertices i and j adjacent?Which faces are adjacent to vertex #7?

• Geometry operationsRemove/add a vertex/faceVertex split, edge collapse


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Storing mesh data

How “good” is a data structure?

• Time to construct - preprocessing

• Time to answer a query

• Time to update

• Space complexity

• Redundancy


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List of facesList of vertices - 3D coordinates

List of faces - triplets of vertex pointers (v1,v2,v3)

f1 f2

f3

f4v1

v2

v3

v4v5

v6

......

(x3, y3, z3)v3

(x2, y2, z2)v2

(x1, y1, z1)v1

coordinatesvertex

......

(v2, v4, v3)f2

(v1, v2, v3)f1

vertices (ccw)face


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List of faces - queries

What are the vertices of face #3?• O(1) - third triplet from face list

Are vertices i and j adjacent?• Requires a full pass over all faces

f1 f2

f3

f4v1

v2

v3

v4v5

v6

......

(x3, y3, z3)v3

(x2, y2, z2)v2

(x1, y1, z1)v1

coordinatesvertex

......

(v2, v4, v3)f2

(v1, v2, v3)f1

vertices (ccw)face


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List of faces – pros and cons

Pros:• Simple• Low memory requirements• Can represent “polygon soups”

Cons:• O(|F|) for queries like

Vertices adjacencyFaces near vertex


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Adjacency matrixList of vertices - 3D coordinatesList of faces - triplets of vertex pointers (v1,v2,v3)

Adjacency matrix A of size |V|*|V|• A(i,j) = 1 iff vi and vj are adjacent

f1 f2

f3f4v1

v2

v3

v4v5

v6

......

(x3, y3, z3)v3

(x2, y2, z2)v2

(x1, y1, z1)v1

coordinatesvertex

......

(v2, v4, v3)f2

(v1, v2, v3)f1

vertices (ccw)face

1

1

1

1

v4 v6v5v3v2v1

11v6

1v5

1111v4

111v3

11v2

11v1


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Adjacency matrix - queries

What are the vertices of face #3?• O(1) – third triplet from face list

Are vertices i and j adjacent?• O(1): A(i,j) == 1?

Which faces are adjacent to vertex j?• Requires a full pass on all faces

......

(x3, y3, z3)v3

(x2, y2, z2)v2

(x1, y1, z1)v1

coordinatesvertex

......

(v2, v4, v3)f2

(v1, v2, v3)f1

vertices (ccw)face

1

1

1

1

v4 v6v5v3v2v1

11v6

1v5

1111v4

111v3

11v2

11v1


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Adjacency matrix – pros and cons

Pros:• Information on vertices adjacency• Can represent non-manifold meshes

Cons:• No connection between a vertex and its

adjacent facesO(|F|) for faces near vertex query


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DCEL (Doubly-Connected Edge List)

Record for each face, edge and vertex:

• Geometric information• Topological information• Attribute information

aka Half-Edge Structure


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DCEL Vertex record:

CoordinatesPointer to an outgoing half-edge

Face record:Pointer to an adjacent half-edge

v

f


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DCELHalf-edge record:

Pointer to origin• origin(e)

Pointer to twin half-edge• twin(e)

Pointer to adjacent face• IncidentFace(e)• face left of e from origin to

destination

Next and previous edge of IncidentFace(e) • next(e)• prev(e)

e


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DCEL

etw

in(e)

origin(e)

IncFace(e)prev(e)

next(e)


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DCEL (cont.)

Operations supported:• Walk around boundary of given face• Visit all edges incident to vertex v

Queries:• Most queries are O(1)


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DCEL - example

...

e1,1

e5,1

e2,1

outgoing edge

......

(x3, y3, z3)v3

(x2, y2, z2)v2

(x1, y1, z1)v1

coordinatevertex

......

e5,1f2

e1,1f1

adjacent edgeface

...e4,1

e2,1

prev

...e5,1

e1,1

next

...e3,1

e3,2

twin

...f2

f1

Incident Face

......v3e3,2

v2e3,1

originHalf-edge

f1 f2v1

v2

v3

e1,1e4,1

e3,2

e5,1

e3,1e2,1


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DCEL – a larger example

e7,2

e9,1

e7,1

e1,1

e5,1

e2,1

IncidentEdge

(x6,y6,z6)v6

(x5,y5,z5)v5

(x4,y4,z4)v4

(x3,y3,z3)v3

(x2,y2,z2)v2

(x1,y1,z1)v1

coordinatevertex

e8,1f4

e4,2f3

e5,1f2

e1,1f1

edgeface

f1 f2

f3

f4v1

v2

v3

v4

v5

v6

e1,1

e6,1

e4,1 e7,2 e9,1

e8,1

e3,2

e5,1

e7,1

e3,1e2,1

e4,2


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DCEL – a larger example

f1 f2

f3

f4v1

v2

v3

v4

v5

v6

e1,1

e6,1

e4,1 e7,2 e9,1

e8,1

e3,2

e5,1

e7,1

e3,1e2,1

e4,2

e6,1

e5,1

e4,1

e2,1

prev

e7,1

e3,2

e5,1

e1,1

next

e4,1

e4,2

e3,1

e3,2

twin

f3

f2

f2

f1

IncidentFace

v3e4,2

v4e4,1

v3e3,2

v2e3,1

originHalf-edge


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DCEL – pros and cons

Pros:• All queries in O(1) time• All operations are O(1) (usually)

Cons:• Represents only manifold meshes


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Corner c contains:Triangle – c.tVertex – c.vNext corner in c.t (ccw) - c.nPrevious corner - c.p(== c.n.n)Corner opposite c - c.o• Edge E opposite c not incident

on c.v• Triangle T adjacent to c.t across E• c.o.v vertex of T that is not incident on E

Right corner - c.r - corner opposite c.n (== c.n.o).Left corner - c.l (== c.p.o == c.n.n.o)

Corner table

cc.l=c.p.oc.r=c.n.o

c.o

c.pc.n

c.v

c.n.v

c.o.v

c.p.v

c.l.vc.r.v

c.t

Edge E

Corner : Coupling of vertex with one of its incident triangles.


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Corner table - example

c6NULLNULLc1c3f1v2c2

NULLNULLc6c3c2f1v1c1

c1c7NULLc6c5f2v3c4

NULLc6NULLc2c1f1v3c3

NULLc1c7c4c6f2v2c5

c7NULLc1c5c4f2v4c6

c.lc.rc.oc.pc.nc.tc.vcorner

f1 f2

f3

f4v1

v2

v3

v4

v5

v6

c1

c2c5

c3 c4

c6 c10

c12

c11

c9

c8

c7


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Corner table – pros and cons

Pros:• All queries in O(1) time• All operations are O(1) (usually)

Cons:• Only triangular, manifold meshes• High redundancy (but not too high …)


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Additional details

Corner table

For each vertex – a list of all its corners

Corners number j*3, j*3+1 and j*3+2 match face number j

Corner number = EdgeID of opposite edge


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Examples of queries

What are the vertices of face #3?• Check c.v of corners 9, 10, 11

Are vertices i and j adjacent?• Scan all corners of vertex i, check if c.p.v or c.n.v are j

Which faces are adjacent to vertex j?• Check c.t of all corners of vertex j

1 mesh data structures - technion

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