Efficient, fair interpolation using Catmull-Clark surfaces
SIGGRAPH '93 Proceedings of the 20th annual conference on Computer graphics and interactive techniques
A unified approach to subdivision algorithms near extraordinary vertices
Computer Aided Geometric Design
Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values
Proceedings of the 25th annual conference on Computer graphics and interactive techniques
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Piecewise smooth subdivision surfaces with normal control
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Progressive geometry compression
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
CHARMS: a simple framework for adaptive simulation
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Subdivision Methods for Geometric Design: A Constructive Approach
Subdivision Methods for Geometric Design: A Constructive Approach
Bezier and B-Spline Techniques
Bezier and B-Spline Techniques
A Method for Analysis of C1-Continuity of Subdivision Surfaces
SIAM Journal on Numerical Analysis
Proceedings of the conference on Visualization '01
Composite primal/dual √3-subdivision schemes
Computer Aided Geometric Design
Discrete multiscale vector field decomposition
ACM SIGGRAPH 2003 Papers
Taming liquids for rapidly changing targets
Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation
Edge subdivision schemes and the construction of smooth vector fields
ACM SIGGRAPH 2006 Papers
Edge subdivision schemes and the construction of smooth vector fields
ACM SIGGRAPH 2006 Papers
Rotational symmetry field design on surfaces
ACM SIGGRAPH 2007 papers
Design of tangent vector fields
ACM SIGGRAPH 2007 papers
Discrete quadratic curvature energies
Computer Aided Geometric Design
Generalized Streak Lines: Analysis and Visualization of Boundary Induced Vortices
IEEE Transactions on Visualization and Computer Graphics
N-symmetry direction field design
ACM Transactions on Graphics (TOG)
Collision-free construction of animated feathers using implicit constraint surfaces
ACM Transactions on Graphics (TOG)
Modeling smooth shape using subdivision on differential coordinates
Computer-Aided Design
Proceedings of the 7th international conference on Curves and Surfaces
The chain collocation method: A spectrally accurate calculus of forms
Journal of Computational Physics
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Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete differential 0- resp. 2-forms, and complete the picture by introducing edge-based subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are intrinsic to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a joint triple and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.