Curves and Surfaces

We review the definition of hierarchical spline spaces and their application to finite element methods. Then we discuss how hierarchical techniques can be implemented using the FEMB program package. Subdivision algorithms play a crucial role and lead to a very simple program structure. A numerical example illustrates the substantial gains in accuracy for the adaptive strategy, in particular for higher degree B-splines.

[1]  J. Stillwell.  Mathematics and Its History , 2020, Undergraduate Texts in Mathematics.

[2]  Felix Klein.  Elementary Mathematics from an Advanced Standpoint: Geometry , 1941 .

[3]  Jacques Liandrat,et al.  On the stability of the PPH nonlinear multiresolution , 2005 .

[4]  Robert R. Sokal,et al.  A statistical method for evaluating systematic relationships , 1958 .

[5]  Helmut Pottmann,et al.  Laguerre minimal surfaces, isotropic geometry and linear elasticity , 2009, Adv. Comput. Math..

[6]  Ralph R. Martin,et al.  Reverse engineering of geometric models - an introduction , 1997, Comput. Aided Des..

[7]  Ian R. Porteous,et al.  Geometric differentiation for the intelligence of curves and surfaces , 1994 .

[8]  Generic affine differential geometry of plane curves , 1998 .

[9]  Helmut Pottmann,et al.  Applications of Laguerre geometry in CAGD , 1998, Comput. Aided Geom. Des..

[10]  Albert Cohen,et al.  Tensor product multiresolution analysis with error control for compact image representation , 2002, Signal Process..

[11]  Sergio Amat,et al.  Exact error bounds for the reconstruction processes using interpolating wavelets , 2009, Math. Comput. Simul..

[12]  Rimvydas Krasauskas,et al.  Bilinear Clifford-Bézier Patches on Isotropic Cyclides , 2012, MMCS.

[13]  Leif Kobbelt,et al.  Structure Recovery via Hybrid Variational Surface Approximation , 2005, Comput. Graph. Forum.

[14]  Guillermo Sapiro,et al.  Affine-invariant symmetry sets , 1997 .

[15]  Bernd Hamann,et al.  Surface Reconstruction from Unorganized Point Data with Quadrics , 2008 .

[16]  Raphael L. Levien,et al.  From Spiral to Spline: Optimal Techniques in Interactive Curve Design , 2009 .

[17]  Helmut Pottmann,et al.  Discrete Surfaces in Isotropic Geometry , 2007, IMA Conference on the Mathematics of Surfaces.

[18]  Helmut Pottmann,et al.  A Laguerre geometric approach to rational offsets , 1998, Comput. Aided Geom. Des..

[19]  Ariel Shamir,et al.  A survey on Mesh Segmentation Techniques , 2008, Comput. Graph. Forum.

[20]  Bernd Hamann,et al.  Segmenting point-sampled surfaces , 2010, The Visual Computer.

[21]  B. Welford.  Note on a Method for Calculating Corrected Sums of Squares and Products , 1962 .

[22]  Erwin Kreyszig,et al.  Differential geometry , 1991 .

[23]  Jacques Liandrat,et al.  A fully adaptive multiresolution scheme for image processing , 2007, Math. Comput. Model..

[24]  Robert Osserman.  The Four-or-More Vertex Theorem , 1985 .

[25]  Pep Mulet,et al.  Point-Value WENO Multiresolution Applications to Stable Image Compression , 2010, J. Sci. Comput..

[26]  Les A. Piegl,et al.  Symbolic operators for NURBS , 1997, Comput. Aided Des..