Mesh Representation, part II based on: Data Structures for Graphics, chapter 12 in Fundamentals of Computer Graphics, 3 rd ed. (Shirley & Marschner) Slides

Mesh Representation, part II

based on: Data Structures for Graphics, chapter 12 inFundamentals of Computer Graphics, 3rd ed.

(Shirley & Marschner)Slides by Marc van Kreveld

1

3D objects

facets

edges vertices

2

3D objects

• We typically represent the boundary; the interior is implied to be the bounded subspace

• We will assume linear boundaries here, so we have facets, edges, and vertices (and the interior)

3

Triangle meshes

• Many freeform shapes consist of (many) triangles

4

Triangle meshes

• How are triangles, edges, and vertices allowed to connect?

• How do we represent (store) triangle meshes?• How efficient are such schemes?

Separate triangle meshIndexed triangle meshTriangle neighbor structureWinged-edge structure

5

Winged-edge structure

• Stores connectivity at edges instead of vertices• For one edge, say, e1:– Two vertices are important: v4 and v6

– Two triangles are important: T5 and T6

– Four edges are important: e2, e14, e5, and e8

T6T5

T4T3

T2

T1

T7

v7

v6

v5

v4

v3

v2

v1

v8

e7

e6

e5

e4

e3

e2

e1

e8

e12

e11

e10e9

e13

e14

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• Give e1 a direction, then– v4 is the tail and v6 is the head

– T5 is to the left and T6 is to the right

– e2 is previous on the left side, e14 is next on the left side, e5 is previous on the right side, and e8 is next on the right side

T6T5

T4T3

T2

T1

T7

v7

v6

v5

v4

v3

v2

v1

v8

e7

e6

e5

e4

e3

e2

e1

e8

e12

e11

e10e9

e13

e14

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• Give e1 a direction, then– v4 is the tail and v6 is the head

– T5 is to the left and T6 is to the right

– e2 is previous on the left side, e14 is next on the left side, e5 is previous on the right side, and e8 is next on the right side

T6T5

v6

v4

e5

e2

e1

e8e14 lnext

left

head

tail

right

lprev

rnext

rprev

8


lnext

left

head

tail

right

lprev

rnext

rprev

Edge { Edge lprev, lnext, rprev, rnext; Vertex head, tail; Triangle left, right}

Vertex { double x, y, z; Edge e; // any incident}

Triangle { Edge e; // any incident}

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• Also works for meshes that do not only use triangles– There is still one head and one tail– There is still one left face and one right face– There are still previous and next edges in the left face and

in the right face: e13, e7, e6, and e8

e7

e6e4

e3 e1

e8

e12

e11

e10

e13

Note: The triangle neighbor structure does not generalize

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• It also works for planar subdivisions like country maps

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Edge { Edge lprev, lnext, rprev, rnext; Vertex head, tail; Face left, right}

Vertex { double x, y; Edge e; // any incident}

Face { Edge e; // any incident}

e7

e6e4

e3 e1

e8

e12

e11

e10

e13

12

Winged-edge structure, storage

• A triangular mesh with nv vertices has 3nv edges and 2nv triangles

• A vertex needs 3(4) units of storage (with z coord.)• An edge needs 8 units of storage• A triangle needs 1 unit of storage

3(4) + 38 + 21 = 29(30) nv units of storage

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• However, the arbitrary orientation of edges makes traversal of the boundary of a face awkward

e7

e6e4

e3 e1

e8

e12

e11

e10

e13e2

e5

F5

With consistent orientations, we could report the vertices of a face iteratively:

while ( e estart ) { e = e.lnext; report e.tail.coordinates;}

But consistent orientations usually do not exist

x

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while ( e estart ) { if (forward) { report e.tail.coordinates; enew = e.lnext; if (enew.head == e.head) forward = false; } else { report e.head.coordinates; enew = e.rprev; if (enew.tail == e.tail) forward = true; } e = enew;}

eface

enew

eface

enew

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Half-edge structure

• A.k.a. doubly-connected edge list, DCEL• Allows the purely forward traversal of the boundary

of a face• Every edge is represented as two half-edges

(split lengthwise!)

16

Half-edge structure

• A consistent orientation around every face now works!• Every half-edge is

incident only tothe face to its left(by convention)

then every facehas its half-edges oriented counterclockwise

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Half-edge structureHEdge { HEdge next, pair; Vertex head; Face f;}

Vertex { double x, y; HEdge h; // any incident half-edge // pointing to this vertex}

Face { HEdge h; // any incident half-edge in its boundary}

next

f

head

pair

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Half-edge structure

• A half-edge h can find itstail as h.pair.head

• A half-edge h can findthe other face incidentto the edge it is part ofas h.pair.f

• A half-edge cannot find its prev easily (prev is the opposite of next); it is a design choice to include an extra prev in HEdge or not

next

f

head

pair

19

Half-edge structure, storage

• A triangular mesh with nv vertices has 3nv edges and 2nv triangles

• A vertex needs 3(4) units of storage (with z coord.)• A half-edge needs 4 units of storage• A triangle needs 1 unit of storage


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• A half-edge needs 5 units of storage when prev is also stored


21


• For country maps instead of triangular meshes, a map with nv vertices has nv edges and << nv faces

3 + 24 = 11 nv units of storage

22

Half-chain structure

• For country maps instead of triangular meshes, a map with nv vertices has nv edges and << nv faces

• In GIS, the long sequences of degree-2 vertices (chains) defining the shape of a border are stored differently, with the vertices in an array, so next and prev are not needed

• We can define half-chains• The whole half-chain has the same left face• Each half-chain has a next half-chain, a prev half-chain,

and a pair half-chain23

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• Chains are made by splitting the subdivision at all vertices of degree at least 3 or exactly 1; onlythese vertices are objects in the half-chainstructure

• All vertices in between two suchvertices have degree 2

• A chain always starts and ends at a vertex of degree 1 or at least 3, and has zero or more degree 2 vertices

• Half-chains are the two “sides”of a chain, splitting length-wise


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next

pair

head

f

Sequences of degree-2 vertices inside a chain are stored in an array, referenced by both half-chains that form a pair

prev

Half-chain structure, storage

• Assume nv vertices of which nw are of degree 1 or at least 3 (so only nw vertex objects)

• If nw << nv, then we have nv edges, < 3nw chains, and < 2nw faces

• The total storage requirements are just 2nv + 29nw units (much better than the 11nv units for the half-edge structure!)

• For example, a part of Europe with 20 three-country points and 100,000 vertices (for the shape of borders) requires less than 200,600 units

26

Half-edge structure, disconnected

27

Subdivisions have loose components like island

Half-edge structure, disconnected

28

Internal holes can also exist

Half-edge structure

• Planar subdivisions may have “islands/holes” in faces: faces from which pieces are excluded

• In this case the graph of vertices and edges is not a connected graph

f

29

Half-edge structure

• Faces with holes are common in 3D CAD/CAM models too

30

Half-edge structure for subdivisions with holes

HEdge { HEdge next, pair; Vertex head; Face f;}

Vertex { double x, y, x; HEdge h; // any incident half-edge // pointing to this vertex}

Face { HEdge h; // any incident half-edge in its boundary HEdge inner[k]; // for each hole, any incident half-edge; // allows up to k holes in the face}

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f

f.h

f.inner[0]

f.inner[1]

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• Every bounded face has exactly one counterclockwise cycle of half-edges bounding it from the outside, and zero or more clockwise cycles of half-edges bounding that face from the inside (holes)

• The structure also works for any planar straight-line graph (PSLG), with possibly:– dangling edges or other dangling structures– loose edges

33

Planar versus non-planar

• None of the structures allow edge-crossings, as faces would not be well-defined, and points can lie in multiple faces at once

34

3D meshes• So far, we can represent 2D subdivisions

and boundaries of 3D solids, but not 3D subdivisions

35

3D tetrahedral meshes

• The 3D version of a triangle is a tetrahedron• The 3D version of a triangular mesh is a tetrahedral

mesh (and triangulation vs. tetrahedralization)

36


• A 3D tetrahedral mesh has the following features:– vertices (0-dimensional)– edges (1-dimensional)– triangles (2-dimensional)– tetrahedra (3-dimensional)

37


• In a proper 3D tetrahedral mesh:– Every vertex is endpoint of some edge– For every edge, both endpoints are vertices of the mesh– Every edge is a side of some triangle– For every triangle, its three sides are edges of the mesh– Every triangle is a facet of some tetrahedron– For every tetrahedron, its four facets are triangles of the

mesh– If edges, triangles, and tetrahedra are considered to be

open, then no two features of the mesh intersect

38


• Every polygon can be converted into a triangular mesh without needing extra vertices, but polyhedra exist that cannot be converted into a tetrahedral mesh without extra edges and vertices

Schönhardt polyhedron39


• Representation of 3D tetrahedral meshes:– Indexed tetrahedron mesh: classes for vertex and

tetrahedron; a tetrahedron can access its four vertices

– Tetrahedron neighbor structure: classes for vertex and tetrahedron; a tetrahedron can access its four vertices and its (up to) four neighbor tetrahedra

– Simplices structure (described next; for any dimension): classes for vertex, edge, triangle, and tetrahedron

40

Simplices structure

• A k-simplex is the convex hull defined by k+1 points that do not lie in any single (k-1)-dim. linear variety

• Equivalent: the k+1 points p0, ..., pk are such that the vectors p0p1, p0p2, ..., p0pk are linearly independent

– 0-simplex: point / vertex– 1-simplex: line segment / edge– 2-simplex: triangle– 3-simplex: tetrahedron– 4-simplex: convex hull of 5 non-co-hyperplanar

points in 4-space41

Simplices structure

• A vertex has access to one incident edge• An edge has access to its vertices and one

incident triangle• A triangle has access to

its three edges and both incident tetrahedra

• A tetrahedron has access to its four triangles

vertex

edge

triangle

tetrahedron

some

some

both all 4

all 3

both

42

Simplices structure

• Can also be used for 4D and even higher-D• 4D can be space-time for changing objects• Higher-D can be the space of location and orientation

of a rigid 3D object (6D)

43

Simplices structure

• Always: k-simplices have access to (k-1)-simplices and (k+1)-simplices

0-dim

1-dim

(d-1)-dim

d-dim

some

some

both all d+1

all 3

both

some all d

44

Quadrilateral meshes

45

Quadrilateral meshes

46

Not every subdivision with triangular faces is a triangular mesh

47

Meshes have multi-resolution possibilities

48

Representation of 3D models: summary

• Boundary or solid model representation• Triangle mesh, quadrilateral mesh, general faces

(good for maps)• Meshes that store incidence/adjacency of features

allow efficient traversal on the mesh faster (local) operations

• Triangle strips may be useful for transmitting meshes• Higher-dimensional mesh representations exist as

well

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Questions

1. Write code to report all vertices adjacent to a given vertex v in a half-edge structure for a triangular mesh. Is your solution clockwise or counter-clockwise? Now do it the other way around

2. Do the same for general subdivisions, clockwise and counter-clockwise

3. Write code to report all vertices in the boundary of a face that may have holes and is given in a half-edge structure

4. How do you access the four 3-simplices adjacent to a given 3-simplex in a simplices structure in 3D?

5. How many units of storage are needed in the different structures for representing tetrahedrilizations?

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Documents

Mesh Representation, part II based on: Data Structures for Graphics, chapter 12 in Fundamentals of Computer Graphics, 3 rd ed. (Shirley & Marschner) Slides