A survey of curve and surface methods in CAGD
Computer Aided Geometric Design
Box splines
Piecewise smooth surface reconstruction
SIGGRAPH '94 Proceedings of the 21st annual conference on Computer graphics and interactive techniques
A unified approach to subdivision algorithms near extraordinary vertices
Computer Aided Geometric Design
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Free-form deformations with lattices of arbitrary topology
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Interpolating Subdivision for meshes with arbitrary topology
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Progressive simplicial complexes
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
A multi-resolution topological representation for non-manifold meshes
Proceedings of the seventh ACM symposium on Solid modeling and applications
A new solid subdivision scheme based on box splines
Proceedings of the seventh ACM symposium on Solid modeling and applications
Subdivision Methods for Geometric Design: A Constructive Approach
Subdivision Methods for Geometric Design: A Constructive Approach
Proceedings of the conference on Visualization '01
Representation of non-manifold objects
SM '03 Proceedings of the eighth ACM symposium on Solid modeling and applications
An Interpolatory Subdivision for Volumetric Models over Simplicial Complexes
SMI '03 Proceedings of the Shape Modeling International 2003
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Despite the growing interest in subdivision surfaces within the computer graphics and geometric processing communities, subdivision approaches have been receiving much less attention in solid modeling. This paper presents a powerful new framework for a subdivision scheme that is defined over a simplicial complex in any n-D space. We first present a series of definitions to facilitate topological inquiries during the subdivision process. The scheme is derived from the double (k + 1)-directional box splines over k-simplicial domains. Thus, it guarantees a certain level of smoothness in the limit on a regular mesh. The subdivision rules are modified by spatial averaging to guarantee C1 smoothness near extraordinary cases. Within a single framework, we combine the subdivision rules that can produce 1-, 2-, and 3-manifold in arbitrary n-D space. Possible solutions for non-manifold regions between the manifolds with different dimensions are suggested as a form of selective subdivision rules according to user preference. We briefly describe the subdivision matrix analysis to ensure a reasonable smoothness across extraordinary topologies, and empirical results support our assumption. In addition, through modifications, we show that the scheme can easily represent objects with singularities, such as cusps, creases, or corners. We further develop local adaptive refinement rules that can achieve level-of-detail control for hierarchical modeling. Our implementation is based on the topological properties of a simplicial domain. Therefore, it is flexible and extendable. We also develop a solid modeling system founded on our theoretical framework to show potential benefits of our work in industrial design, geometric processing, and other applications.