Old problems and new challenges in subdivision

Abstract Subdivision has been an active research topic for about 40 years, and it has seen commercial exploitation in animated Computer Graphics. The question has to be asked: ‘Is there anything left to do research-wise?’ This note contains the authors’ responses. Yes, much has been learnt and does not need to be relearnt, but there are still results that we cannot prove, properties of specific schemes that we cannot compute exactly, and there are still ways in which we need to improve the technology to make it useful in the more demanding contexts of CAD, CAM, and CAE. We have approached the subject by looking first at curves, then at surfaces, and finally at the trivariate structures which are required for CAE. For each dimension we identify first what is already established, and then proceed to suggest directions for further research. Three different labels indicate, for each useful piece of work, the relative difficulty and hence the prospects of success: Download : Download high-res image (24KB) Download : Download full-size image Download : Download high-res image (25KB) Download : Download full-size image Download : Download high-res image (40KB) Download : Download full-size image Of course, these ratings are just our personal estimates. We will only know for sure how hard the questions were when the work has been done. Further, we do not claim completeness of the list of topics discussed here. Also the list of references selects only a few samples from the vast literature.

[1]  Andreas Weinmann,et al.  Nonlinear Subdivision Schemes on Irregular Meshes , 2010 .

[2]  Malcolm A. Sabin,et al.  Analysis and Design of Univariate Subdivision Schemes , 2010, Geometry and Computing.

[3]  M. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1978 .

[4]  U. Reif,et al.  C1-continuity of the generalized four-point scheme , 2009 .

[5]  E. Catmull,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[6]  Jörg Peters,et al.  Shape characterization of subdivision surfaces--case studies , 2004, Comput. Aided Geom. Des..

[7]  Ulrich Reif,et al.  On the volume of sets bounded by refinable functions , 2016, Appl. Math. Comput..

[8]  Ulrich Reif,et al.  Curvature integrability of subdivision surfaces , 2001, Adv. Comput. Math..

[9]  J. Peters,et al.  Shape characterization of subdivision surfaces: basic principles , 2004 .

[10]  Nicola Guglielmi,et al.  Exact Computation of Joint Spectral Characteristics of Linear Operators , 2011, Found. Comput. Math..

[11]  Chandrajit L. Bajaj,et al.  A subdivision scheme for hexahedral meshes , 2002, The Visual Computer.

[12]  U. Reif.  TURBS—Topologically Unrestricted Rational B-Splines , 1998 .

[13]  Leif Kobbelt,et al.  A variational approach to subdivision , 1996, Comput. Aided Geom. Des..

[14]  D. Levin,et al.  Subdivision schemes in geometric modelling , 2002, Acta Numerica.

[15]  Neil A. Dodgson,et al.  A Circle-Preserving Variant of the Four-Point Subdivision Scheme , 2012 .

[16]  Ulrich Reif,et al.  Generalized Lane-Riesenfeld algorithms , 2013, Comput. Aided Geom. Des..

[17]  Jörg Peters,et al.  Subdivision Surfaces , 2002, Handbook of Computer Aided Geometric Design.

[18]  Jörg Peters,et al.  Guided spline surfaces , 2009, Comput. Aided Geom. Des..

[19]  C. Micchelli,et al.  Stationary Subdivision , 1991 .

[20]  Philipp Grohs,et al.  Smoothness Analysis of Subdivision Schemes on Regular Grids by Proximity , 2008, SIAM J. Numer. Anal..

[21]  M. Ortiz,et al.  Subdivision surfaces: a new paradigm for thin‐shell finite‐element analysis , 2000 .

[22]  Scott Schaefer,et al.  Smooth subdivision of tetrahedral meshes , 2004, SGP '04.

[23]  Bernhard Mößner,et al.  On the joint spectral radius of matrices of order 2 with equal spectral radius , 2010, Adv. Comput. Math..

[24]  Johannes Wallner.  Smoothness Analysis of Subdivision Schemes by Proximity , 2006 .

[25]  M. Sabin,et al.  Hölder Regularity of Geometric Subdivision Schemes , 2014, 1401.6341.

[26]  Vincent D. Blondel,et al.  Computationally Efficient Approximations of the Joint Spectral Radius , 2005, SIAM J. Matrix Anal. Appl..

[27]  U. Reif,et al.  A tree-based approach to joint spectral radius determination , 2014 .