# CONNECTIVITY MAPS FOR SUBDIVISION SURFACES

### Ali Mahdavi Amiri, Faramarz Samavati

#### Abstract

In this paper, we introduce a hierarchical indexing for adjacency queries specially for applying subdivision schemes on some simple spherical and toroidal polyhedrons as the base model for the content creation process. The indexing method is created from integer coordinates of regular 2D domains (connectivity maps) resulting from unfolding polyhedrons. All connectivities are found using algebraic relationships of the connectivity map’s indices; therefore, no additional data structure is required and operations are extremely fast and efficient. Although connectivity relationships of the polyhedrons are as simple as the first resolution, the models created by our proposed method is not restricted to the subdivided polyhedrons. Using our connectivity based method, complex objects can be created by adding sharp features and holes and applying deformation and remeshing techniques. We demonstrate capacities and the efficiency of the method with several example results and compare its speed with that of the half-edge data structure.

#### References

- Bunnell, M. (2005). GPU Gems 2: Programming Techniques for High-Performance Graphics and GeneralPurpose Computation, chapter Adaptive Tessellation of Subdivision Surfaces with Displacement Mapping, pages 33-40. Addison Wesley.
- Catmull, E. and Clark, J. (1998). Recursively generated Bspline surfaces on arbitrary topological meshes, pages 183-188. ACM, New York, NY, USA.
- DeRose, T., Kass, M., and Truong, T. (1998). Subdivision surfaces in character animation. In Proceedings of the 25th annual conference on Computer graphics and interactive techniques, SIGGRAPH 7898, pages 85-94. ACM.
- Gargantini, I. (1982). An effective way to represent quadtrees. Commun. ACM, 25(12):905-910.
- GeoWeb (2011). Pyxis innovation.
- Goodchild, M. F. (2006). Discrete global grids for digital earth. In Proceedings of 1st International Conference on Discrete Global Grids, March, 2000.
- Gu, X., Gortler, S. J., and Hoppe, H. (2002). Geometry images. ACM Trans. Graph., 21(3):355-361.
- Halstead, M., Kass, M., and DeRose, T. (1993). Efficient, fair interpolation using catmull-clark surfaces. In SIGGRAPH 7893: Proceedings of the 20th annual conference on Computer graphics and interactive techniques, pages 35-44. ACM.
- Hoppe, H., DeRose, T., Duchamp, T., Halstead, M., Jin, H., McDonald, J., Schweitzer, J., and Stuetzle, W. (1994). Piecewise smooth surface reconstruction. In Proceedings of the 21st annual conference on Computer graphics and interactive techniques, SIGGRAPH 7894, pages 295-302. ACM.
- Hormann, K., Lévy, B., and Sheffer, A. (2007). Mesh parameterization: theory and practice. In SIGGRAPH 7807: ACM SIGGRAPH 2007 courses, page 1. ACM.
- Igarashi, T., Matsuoka, S., and Tanaka, H. (2007). Teddy: a sketching interface for 3d freeform design. In ACM SIGGRAPH 2007 courses, SIGGRAPH 7807. ACM.
- Kettner, L. (1998). Designing a data structure for polyhedral surfaces. In SCG 7898: Proceedings of the fourteenth annual symposium on Computational geometry, pages 146-154. ACM.
- Loop, C. (1987). Smooth subdivision surfaces based on triangles. Department of mathematics, University of Utah.
- Losasso, F., Hoppe, H., Schaefer, S., and Warren, J. (2003). Smooth geometry images. In Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing, SGP 7803, pages 138-145. Eurographics Association.
- Maya (2011). Autodesk inc.
- Nasri, A., Karam, W. B., and Samavati, F. (2009). Sketchbased subdivision models. In Proceedings of the 6th Eurographics Symposium on Sketch-Based Interfaces and Modeling (SBIM'09), pages 53-60. ACM.
- Nealen, A., Igarashi, T., Sorkine, O., and Alexa, M. (2007). Fibermesh: designing freeform surfaces with 3d curves. In ACM SIGGRAPH 2007 papers, SIGGRAPH 7807. ACM.
- Nealen, A., Pett, J., Alexa, M., and Igarashi, T. (2009). Gridmesh: Fast and high quality 2d mesh generation for interactive 3d shape modeling. In Shape Modeling and Applications, 2009. SMI 2009. IEEE International Conference on, pages 155 -162.
- Olsen, L., Samavati, F., Sousa, M., and Jorge, J. (2009). Sketch-based modeling: A survey. Computers & Graphics, 33:85-103.
- Peters, J. (2000). Patching catmull-clark meshes. In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 255-258.
- Praun, E. and Hoppe, H. (2003). Spherical parameterization and remeshing. In SIGGRAPH 7803: ACM SIGGRAPH 2003 Papers, pages 340-349. ACM.
- Pusch, R. and Samavati, F. (2010). Local constraint-based general surface deformation. In Proceedings of the International Conference on Shape Modeling and Applications (SMI 2010), pages 256-260. IEEE Computer Society.
- Samet, H. (1985). Data structures for quadtree approximation and compression. Commun. ACM, 28(9):973- 993.
- Samet, H. (1990). Applications of spatial data structures: computer graphics, image processing, and GIS. Addison-Wesley.
- Samet, H. (2005). Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann Publishers Inc.
- Sharf, A., Lewiner, T., Shamir, A., Kobbelt, L., and CohenOr, D. (2006). Competing fronts for coarse-to-fine surface reconstruction. In Eurographics 2006 (Computer Graphics Forum), volume 25, pages 389-398, Vienna. Eurographics.
- Shiue, L.-J., Jones, I., and Peters, J. (2005). A realtime gpu subdivision kernel. ACM Trans. Graph., 24:1010- 1015.
- Taubin, G. (1995). A signal processing approach to fair surface design. In Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, SIGGRAPH 7895, pages 351-358. ACM.
- Weiler, K. (1985). Edge-based data structures for solid modeling in curved-surface environments. Computer Graphics and Applications, IEEE, 5(1):21 -40.
- Zorin, D., Schröder, P., and Sweldens, W. (1997). Interactive multiresolution mesh editing. In Proceedings of the 24th annual conference on Computer graphics and interactive techniques, SIGGRAPH 7897, pages 259- 268. ACM Press/Addison-Wesley Publishing Co.

#### Paper Citation

#### in Harvard Style

Mahdavi Amiri A. and Samavati F. (2012). **CONNECTIVITY MAPS FOR SUBDIVISION SURFACES** . In *Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2012)* ISBN 978-989-8565-02-0, pages 26-37. DOI: 10.5220/0003814200260037

#### in Bibtex Style

@conference{grapp12,

author={Ali Mahdavi Amiri and Faramarz Samavati},

title={CONNECTIVITY MAPS FOR SUBDIVISION SURFACES},

booktitle={Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2012)},

year={2012},

pages={26-37},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003814200260037},

isbn={978-989-8565-02-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2012)

TI - CONNECTIVITY MAPS FOR SUBDIVISION SURFACES

SN - 978-989-8565-02-0

AU - Mahdavi Amiri A.

AU - Samavati F.

PY - 2012

SP - 26

EP - 37

DO - 10.5220/0003814200260037