Subdivision Surfaces for CAD

AbstractSubdivision surfaces refer to a class of modelling schemes that define an object through recursive subdivision starting from an initial control mesh. Similar to B-splines, the final surface is defined by the vertices of the initial control mesh. These surfaces were initially conceived as an extension of splines in modelling objects with a control mesh of arbitrary topology. They exhibit a number of advantages over traditional splines. Today one can find a variety of subdivision schemes for geometric design and graphics applications. This paper provides an overview of subdivision surfaces with a particular emphasis on schemes generalizing splines. Some common issues on subdivision surfaces modelling are addressed. Several key topics, such as scheme construction, property analysis and parametric evaluation, are discussed. Some other important topics are also summarized for potential future research and development.

[1]  Malcolm Sabin,et al.  Subdivision Surfaces , 2002, Handbook of Computer Aided Geometric Design.

[2]  N. Dyn,et al.  A butterfly subdivision scheme for surface interpolation with tension control , 1990, TOGS.

[3]  Joe Warren,et al.  Subdivision Methods for Geometric Design: A Constructive Approach , 2001 .

[4]  Scott Schaefer,et al.  A factored approach to subdivision surfaces , 2004, IEEE Computer Graphics and Applications.

[5]  Ulf Labsik,et al.  Interpolatory √3‐Subdivision , 2000 .

[6]  Ahmad H. Nasri,et al.  Interpolating meshes of boundary intersecting curves by subdivision surfaces , 2000, The Visual Computer.

[7]  D. Zorin,et al.  4-8 Subdivision , 2001 .

[8]  Tony DeRose,et al.  Subdivision surfaces in character animation , 1998, SIGGRAPH.

[9]  Tony DeRose,et al.  Efficient, fair interpolation using Catmull-Clark surfaces , 1993, SIGGRAPH.

[10]  Jos Stam,et al.  Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values , 1998, SIGGRAPH.

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

[12]  Peter Schröder,et al.  A unified framework for primal/dual quadrilateral subdivision schemes , 2001, Comput. Aided Geom. Des..

[13]  M. A. Sabin,et al.  Cubic Recursive Division With Bounded Curvature , 1991, Curves and Surfaces.

[14]  Jean Schweitzer,et al.  Analysis and application of subdivision surfaces , 1996 .

[15]  Hartmut Prautzsch,et al.  Smoothness of subdivision surfaces at extraordinary points , 1998, Adv. Comput. Math..

[16]  Denis Zorin,et al.  A Method for Analysis of C1 -Continuity of Subdivision Surfaces , 2000, SIAM J. Numer. Anal..

[17]  Jos Stam,et al.  On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree , 2001, Comput. Aided Geom. Des..

[18]  D. Zorin,et al.  A unified framework for primal/dual quadrilateral subdivision schemes , 2001 .

[19]  Jörg Peters,et al.  The simplest subdivision scheme for smoothing polyhedra , 1997, TOGS.

[20]  Weiyin Ma,et al.  Smooth multiple B-spline surface fitting with Catmull%ndash;Clark subdivision surfaces for extraordinary corner patches , 2002, The Visual Computer.

[21]  Malcolm Sabin,et al.  Recent Progress in Subdivision: a Survey , 2005, Advances in Multiresolution for Geometric Modelling.

[22]  Peter Schröder,et al.  Composite primal/dual -subdivision schemes , 2003, Comput. Aided Geom. Des..

[23]  Hujun Bao,et al.  Interpolatory v2-Subdivision Surfaces , 2004, GMP.

[24]  H. Ehlers.  LECTURERS , 1948, Statistics for Astrophysics.

[25]  Tony DeRose,et al.  Piecewise smooth surface reconstruction , 1994, SIGGRAPH.

[26]  Ulrich Reif,et al.  A unified approach to subdivision algorithms near extraordinary vertices , 1995, Comput. Aided Geom. Des..

[27]  G. Umlauf.  Analyzing the Characteristic Map of Triangular Subdivision Schemes , 2000 .

[28]  Malcolm A. Sabin,et al.  Non-uniform recursive subdivision surfaces , 1998, SIGGRAPH.

[29]  A. A. Ball,et al.  Recursively generated B-spline surfaces , 1984 .

[30]  J. Peters,et al.  Analysis of Algorithms Generalizing B-Spline Subdivision , 1998 .

[31]  H. Bao,et al.  Interpolatory /spl radic/2-subdivision surfaces , 2004, Geometric Modeling and Processing, 2004. Proceedings.

[32]  A. A. Ball,et al.  Conditions for tangent plane continuity over recursively generated B-spline surfaces , 1988, TOGS.

[33]  Hujun Bao,et al.  √2 Subdivision for quadrilateral meshes , 2004, The Visual Computer.

[34]  Henning Biermann,et al.  Sharp features on multiresolution subdivision surfaces , 2001, Proceedings Ninth Pacific Conference on Computer Graphics and Applications. Pacific Graphics 2001.

[35]  Charles T. Loop,et al.  Smooth Subdivision Surfaces Based on Triangles , 1987 .

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

[37]  Denis Z orin.  Smoothness of Stationary Subdivision on Irregular Meshes , 1998 .

[38]  George Merrill Chaikin,et al.  An algorithm for high-speed curve generation , 1974, Comput. Graph. Image Process..

[39]  I. Daubechies,et al.  Regularity of Irregular Subdivision , 1999 .