Strips: Triangle and Quad

Jyun-Ming ChenReference: 1, 2

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From Blue Book

GL_TRIANGLE_STRIP  For odd n, vertices n, n+1, and n+2 define triangle n. For even n, vertices n+1, n, and n+2 define triangle n.

GL_QUAD_STRIP  Vertices 2n-1, 2n, 2n+2, and 2n+1 define

quadrilateral n.

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Using Strips

Reduce number of glVertex calls Each glVertex call send a data

through pipeline: matrix multiplication, …

4 6v instead of 12v

3: 8v instead of 12v

In general, n: (n+2) vn: (2n+2) v

In general, n: (n+2) vn: (2n+2) v

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Syntax

Sequence of vertices: follow the arrows separating the triangles

a c

b d

e

f

abcdef[abc,bcd,cde,def] [abc,cbd,cde,edf]

alternate winding; interpret as ….

GL_TRIANGLE_STRIP 

For odd n, vertices n, n+1, and n+2 define triangle n.

For even n, vertices n+1, n, and n+2 define triangle n.

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Swap

Sometimes, additional vertices need to be added (known as swap)

a

b d

e

f

c

[abc,cbd,cdc,cde,cef]

abcdcef[abc,bcd,cdc,dce,cef]

alternate winding

A penalty: need one

more vertex for the swap

Swap is a penalty; but breaking into two strips is more costly

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Key Difference (swap)

Common edges Head-tail connected: no swap Strut exists: swap!

a c

b d

e

f

a

b d

e

f

c

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Fan: a strip with many swaps

b

c

d

e

a

bacadae

(bac, aca, cad, ada, dae)

The same geometry can be given by a triangle fan:

aedcb

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Exercise

How can this strip be made?

a

b d

c

fg

h i

j

e

Algorithmically, how does one

construct a strip given a set of

connected triangles?

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Other Topics of Triangle Strips

Winding: determined by the first triangle Related: glFrontFace, glCullFace

Shading: Smooth: specify normal vector preceding

each vertex Flat: only send (face) normal before the

face-defining vertex

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Example

a c

b d

e

fabcdef (4, 6v)All triangles are CW-winded

abcdef (4, 6v)All triangles are CCW-winded

a c

b

e

d f

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Example (Swap)

abcxcdef (5, 8v)All triangles are CCW-winded

a c

b

e

d fx

a c

b d

e

f

aabcdef (4, 7v)All triangles are CCW-winded

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Example (flat shading)

a c

b d

e

f Begin v(a) v(b) n(T1), v(c) n(T2), v(d) n(T3), v(e) n(T4), v(f)End

T1

T2 T3

T4

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Remark: Flat Shading

According to spec, the color/normal of flat shaded polygon depends on the last primitive-defining vertexSo the code should work fine if glShadeModel(GL_FLAT) is specifiedHowever, if the shade model is changed to GL_SMOOTH, the normal vectors will be assigned to either (0,0,1) or (-1,0,0) depending on the order of traversal. Be careful!

This code can be problematic!

Stripification

New word for “strip generation”

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Greedy Stripping (SGI)

Each triangle associated with an adjacency number (degree in dual graph)Start from the triangle with lowest degree, collect along the path with uncollected & fewer degree triangle

1

1

3

32 1

3

2

2

Tend to minimize leaving isolated triangles

Details (OpenMesh)Assign each face with the following integer property: degree (face valence); collected

For triangular mesh, four possible values for degree: 0 (isolated), 1, 2, 3

No need to sort; just start from any triangle with degree 1. When exhausted, start with degree 2, then degree 3.

Output degree 0 triangle as GL_TRIANGLES Collect the triangle.idx into an STL vector

Degree update When a triangle is collected, decrement the degree

of its neighbors

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1

1

3

32 1

3

2

2

1

1

2

32 1

3

2

2

1

0

2

22 1

3

1

2

1

0

1

22 1

3

1

1

1

0

1

22 1

2

0

1

1

0

1

21 0

2

0

0

1

0

1

21 0

1

0

0 Degree Update

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Supplement

Strip collection is related to finding Hamliton paths in the dual graphWhile a single Hamilton path seems impossible, longer strips are preferred (for better rendering speed)

Wikipedia: In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected graph which visits each vertex exactly once.

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Stripification (Kommann)

Starting triangle: one with least number of adjacency that are not part of any strip Idea: process isolated triangles first

Step Evaluate the weight all neighboring

triangles; choose the one with minimum weight to continue the strip

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Kommann (cont)

Weight evaluation Face Connectivity: 0,1,2 (# of

triangles not visited) – include poorly connected triangles first

Node connectivity: use connectivity of nodes of the current triangle to decide which side to add: +1 for highest connected node, -1 for all other nodes

Swap required: +1(yes), -1(no)

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Weight assignment

References: 1, 2

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Strip for a Cube

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Quad strip

SyntaxWinding (consistent)Shading smooth shading: averaging vertex

colors Flat shading: the color of the last

defining vertex

GL_QUAD_STRIP 

Vertices 2n-1, 2n, 2n+2, and 2n+1

define quadrilateral n.

Compare GL_QUADS 

Vertices 4n-3, 4n-2, 4n-1, and 4n

define quadrilateral n.

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Example (quadstrip)0

1

2

3

4

5

Quads are formed as (0,1,3,2) and (2,3,5,4)Using quadstrip: we give 0,1,2,3,4,5

Using quads: we need to give 0,1,3,2, 2,3,5,4

1. Add the vertices of the (i=1) to the strip. From the “stand alone” vertex

2. Increment i, adding i

3. Consider the existence of i If no, add the “other v” of i If yes, find the common vertex between i & i-1

If the common vertex is the last (tail) vertex in the strip, add the “other v” of i

If not, add the common vertex to the strip (swap), then add the “other v” of i

4. Proceed to step 2, until all triangles have been added

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Algorithm: Strip construction

Assuming all triangles are CCW-oriented

i=1

Stand-alone

Step 1

Stand alone vertex Find the common vertices between

D1 & D2 Take the one that’s left alone

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Step 3 Cases

No Di+1 Add “other v”

Di+1 (with swap)

Di+1 (with no swap)

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i-1i

other v

i-1i

other v

i+1

i-1i

other v

i+1

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Example

a

b d

c

fg

h i

j

e

Add acb

Add acbcd

Add acbcdce

Add acbcdcef

Add acbcdcefg

Add acbcdcefgh

Add acbcdcefghgi

Add acbcdcefghgij

12

3

45

67

8

If the common vertex between i & i-1 is not end of strip, swap is needed

If the common vertex between i & i-1 is not end of strip, swap is needed

Input: triangles to be put in a strip (1-8)Output: the vertices forming the strip (the 13 vertices)

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Remark

The underlying data structure should be able to answer these queries efficiently Stand-alone vertex of triangle “other-v” of a triangle Common vertex of two triangles For Kommann stripification

Number of unvisited neighbors of a triangle Number of triangles connecting to a vertex

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A quicker way to find the intersection of two sets

Stripifier by OpenMesh

OpenMesh/Tools/UtilsThe strips generated are not as good

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Stripifier

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