320491 AdvGrafx Lecture Notes Chapter 1€¦ · 320491: Advanced Graphics - Chapter 1 87 Visualization and Computer Graphics Lab Jacobs University Direct edge data structure For triangular

320491: Advanced Graphics - Chapter 1 68

Visualization and Computer Graphics LabJacobs University

ModelingModeling operations often require the access of• neighboring vertices• neighboring edges• neighboring faces

Accessing neighbors is not supported by a polygon soup.



Winged-edge data struture

Stores local information for each edge:• pointers to both vertices belonging to the edge• pointers to both faces adjacent to the edge• pointers to preceding and subsequent edges for both

adjacent faces










+ all important local information is provided+ no computationally intense searches are necessary

- 8 pointers per edge may be confusing- additional memory for storage required



Split-edge data structure

Stores local information for each edgeby splitting it into two parts.

Each split-edge stores• the respective vertex• the adjacent face• the subsequent split-edge• the corresponding opposite split-edge







The subsequent edge can (consistently!) be stored in• clockwise or• counter-clockwisedirection.

Some implementions use pointers in both directions.






Split-edge data structureFace loop (pointers to face are omitted):



Split-edge data structureVertex loop (pointers to face are omitted):



Split-edge data structureBoundaries (pointers to face are omitted):



Half-edge data structureSimilar to split-edge data structure, but edges arehalved instead of split.

(pointers to faces can be added)



Half-edge data structure



Half-edge data structureFace loop:



Half-edge data structureVertex loop:



Half-edge data structureBoundary:



Half-edge data structureImplementation:



Half-edge data structureFace loop:



Direct edge data structure

For triangular meshes, one can use the followingindexing scheme:• Let f be the index of a face, then its half-edges get

assigned the indices 3f, 3f+1, and 3f+2.• For each half-edge with index h, the corresponding

face has index h/3 using integer division.• For each half-edge with index h, the subsequent half-

edge has index 3f+[(h+1) mod 3].



Direct edge data structure

Direct meshes reduce memory consumption, as • only the respective vertex and the opposite half-edge

need to be stored,• while the adjacent face and the subsequent half-edge

are given implicitly by the indexing scheme.



1.3 Subdivision Surfaces



1.3.1 Subdivision Concept



References

Subdivision Methods for Geometric Design:A Constructive Approach. Joe Warren & Henrik Reimer.Morgan Kaufmann Publishers Inc., San Francisco, CA.1st edition, 2002.



Curvature-continuous surface representation

Piecewise polynomial surface representation



Uniform B-splines

Repeated integration of box function:



Convolution

Definition:

Translation:

Dilation:



Convolution of box function

320491 AdvGrafx Lecture Notes Chapter 1€¦ · 320491: Advanced Graphics - Chapter 1 87 Visualization and Computer Graphics Lab Jacobs University Direct edge data structure For triangular

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