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Jianhua Fan, Jörg Peters: Smooth Bi-3 spline surfaces with fewest knots. CAD submitted.

Abstract:Converting a quadrilateral input mesh into a C^1 surface with one bi-3 NURBS patch per facet is a classical challenge. We give explicit local averaging formulas for the spline control points. Where the quadrilateral mesh is not regular, the patches have two internal double knots, the least number and multiplicity to always allow for an unbiased G^1 construction.

Jörg Peters, Jianhua Fan: The projective linear transition map for constructing smooth surfaces. Shape Modeling International 2010

Abstract:We exhibit the essentially unique linear(rational linear) reparameterization for constructing C^s surfaces of genus greater than 0. Conversely, for quadrilaterals and isolated vertices of valence 8, we show constructively for s=1,2 that this map yields a projective linear spline space for surfaces of genus greater or equal to 1. This establishes the reparametrization to be the simplest possible transition map.

Jörg Peters, Jianhua Fan: On the Complexity of Smooth Spline Surfaces from Quad Meshes. Computer Aided Geometric Design (CAGD): Vol.27,no.1,pp 96-105 

Abstract: This paper derives strong relations that boundary curves of a smooth complex of patches have to obey when the patches are computed by local averaging. These relations restrict the choice of reparameterizations for geometric continuity. In particular, when one bicubic tensor-product B-spline patch is associated with each facet of a quadrilateral mesh with n-valent vertices and we do not want segments of the boundary curves forced to be linear, then the relations dictate the minimal number and multiplicity of knots: For general data, the tensor-product spline patches must have at least two internal double knots per edge to be able to model a G1-conneced complex of C1 splines. This lower bound on the complexity of any construction is proven to be sharp by suitably interpreting an existing surface construction. That is, we have a tight bound on the complexity of smoothing quad meshes with bicubic tensor-product B-spline patches.

Jianhua Fan, Jörg Peters: On Smooth Bicubic Surfaces from Quad Meshes. ISVC (1) 2008: 87-96    Slides

Abstract: Determining the least m such that one m×m bi-cubic macro-patch per quadrilateral offers enough degrees of freedom to construct a smooth surface by local operations regardless of the vertex valences is of fundamental interest; and it is of interest for computer graphics due to the impending ability of GPUs to adaptively evaluate polynomial patches at animation speeds.
We constructively show that m = 3 suffices, show that m = 2 is unlikely to always allow for a localized construction if each macro-patch is internally parametrically C 1 and that a single patch per quad is incompatible with a localized construction. We do not specify the GPU implementation.

Elimination in Generically Rigid 3D Geometric Constraint Systems
Jörg Peters , Meera Sitharam , Yong Zhou and Jianhua Fan
Algebraic Geometry and Geometric Modeling, Nice, France, September 27 - 29, 2004

Abstract: Modern geometric constraint solvers use combinatorial graph algorithms to recursively decompose the system of polynomial constraint equations into generically rigid subsystems and then solve the overall system by solving subsystems, from the leave nodes up, to be able to access any and all solutions. Since the overall algebraic complexity of the solution task is dominated by the size of the largest subsystem, such graph algorithms attempt to minimize the fan-in at each recombination stage. Recently, we found that, especially for 3D geometric constraint systems, a further graph-theoretic optimization of each rigid subsystem is both possible, and often necessary to solve wellconstrained systems: a minimum spanning tree characterizes what partial eliminations should be performed before a generic algebraic or numeric solver is called. The weights and therefore the elimination hierarchy defined by this minimum spanning tree computation depend crucially on the representation of the constraints. This paper presents a simple representation that turns many previously untractable systems into easy exercises. We trace a solution family for varying constraint data.


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