A survey of partial differential equations in geometric design

Computer-aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand, since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling, since they offer a number of features from which these areas can benefit. This work summarizes the uses given to PDE surfaces as a surface generation technique together with some other applications to computer graphics.

[1]  Hassan Ugail,et al.  On harmonic and biharmonic Bézier surfaces , 2004, Comput. Aided Geom. Des..

[2]  Hassan Ugail,et al.  Efficient shape parametrisation for automatic design optimisation using a partial differential equation formulation , 2003 .

[3]  Guillermo Sapiro,et al.  Image inpainting , 2000, SIGGRAPH.

[4]  Lihua You,et al.  Fast Surface Modelling Using a 6th Order PDE , 2004, Comput. Graph. Forum.

[5]  George Celniker,et al.  Deformable curve and surface finite-elements for free-form shape design , 1991, SIGGRAPH.

[6]  Chandrajit L. Bajaj,et al.  Anisotropic diffusion of surfaces and functions on surfaces , 2003, TOGS.

[7]  Hong Qin,et al.  Dynamic PDE surfaces with flexible and general geometric constraints , 2000, Proceedings the Eighth Pacific Conference on Computer Graphics and Applications.

[8]  Ronald Fedkiw,et al.  Animation and rendering of complex water surfaces , 2002, ACM Trans. Graph..

[9]  Kunwoo Lee,et al.  Proceedings of the sixth ACM symposium on Solid modeling and applications , 2001 .

[10]  Malcolm I. G. Bloor,et al.  Using partial differential equations to generate free-form surfaces , 1990, Comput. Aided Des..

[11]  George Celniker,et al.  Linear constraints for deformable non-uniform B-spline surfaces , 1992, I3D '92.

[12]  Hong Qin,et al.  Direct Manipulation and Interactive Sculpting of PDE Surfaces , 2000, Comput. Graph. Forum.

[13]  Michael J. Wilson,et al.  Generating blend surfaces using partial differential equations , 1989 .

[14]  Malcolm I. G. Bloor,et al.  Efficient parametrization of generic aircraft geometry , 1995 .

[15]  Duc Quang Nguyen,et al.  Physically based modeling and animation of fire , 2002, ACM Trans. Graph..

[16]  Malcolm I. G. Bloor,et al.  Functionality in Solids Obtained from Partial Differential Equations , 1993, Geometric Modelling.

[17]  Leif Kobbelt,et al.  Geometric fairing of irregular meshes for free-form surface design , 2001, Comput. Aided Geom. Des..

[18]  Haixia Du.  Interactive shape design using volumetric implicit PDEs , 2003, SM '03.

[19]  David C. Gossard,et al.  Variational geometry in computer-aided design , 1981, SIGGRAPH '81.

[20]  Taku Komura,et al.  Topology matching for fully automatic similarity estimation of 3D shapes , 2001, SIGGRAPH.

[21]  Hong Qin,et al.  D-NURBS: A Physics-Based Framework for Geometric Design , 1996, IEEE Trans. Vis. Comput. Graph..

[22]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[23]  Hassan Ugail,et al.  Interactive design using higher order PDEs , 2004, The Visual Computer.

[24]  Hassan Ugail.  Parametric Design and Optimisation of Thin-Walled Structures for Food Packaging , 2003 .

[25]  Ronald Fedkiw.  Simulating Natural Phenomena for Computer Graphics , 2002 .

[26]  S. Osher,et al.  Solving variational problems and partial differential equations mapping into general target manifolds , 2004 .

[27]  Hong Qin,et al.  A shape design system using volumetric implicit PDEs , 2004, Comput. Aided Des..

[28]  Horst Nowacki,et al.  Construction of fair surfaces over irregular meshes , 2001, SMA '01.

[29]  Guillermo Sapiro,et al.  Variational Problems and Partial Differential Equations on Implicit Surfaces: Bye Bye Triangulated Surfaces? , 2003 .

[30]  Qing Pan,et al.  Discrete surface modelling using partial differential equations , 2006, Comput. Aided Geom. Des..

[31]  Malcolm I. G. Bloor,et al.  Spectral approximations to PDE surfaces , 1996, Comput. Aided Des..

[32]  Alan H. Barr,et al.  Global and local deformations of solid primitives , 1984, SIGGRAPH.

[33]  John C. Platt,et al.  Elastically deformable models , 1987, SIGGRAPH.

[34]  Hassan Ugail.  On the Spine of a PDE Surface , 2003, IMA Conference on the Mathematics of Surfaces.

[35]  Hong Qin,et al.  Integrating physics-based modeling with PDE solids for geometric design , 2001, Proceedings Ninth Pacific Conference on Computer Graphics and Applications. Pacific Graphics 2001.

[36]  Hassan Ugail.  Spine Based Shape Parameterisation for PDE Surfaces , 2003, Computing.

[37]  Ronald Fedkiw.  Simulating Natural Phenomena , 2003 .

[38]  Hassan Ugail,et al.  A general 4th-order PDE method to generate Bézier surfaces from the boundary , 2006, Comput. Aided Geom. Des..

[39]  Andrew P. Witkin,et al.  Energy constraints on parameterized models , 1987, SIGGRAPH.

[40]  Malcolm I. G. Bloor,et al.  Functionality in blend design , 1991, Comput. Aided Des..

[41]  Ronald Fedkiw,et al.  Visual simulation of smoke , 2001, SIGGRAPH.

[42]  C. W. Dekanski,et al.  The generation of propeller blade geometries using the PDE method , 1995 .

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

[44]  Hassan Ugail,et al.  Techniques for interactive design using the PDE method , 1999, TOGS.

[45]  Malcolm I. G. Bloor,et al.  Representing PDE surfaces in terms of B-splines , 1990, Comput. Aided Des..

[46]  Hong Qin,et al.  Dynamic PDE-based surface design using geometric and physical constraints , 2005, Graph. Model..

[47]  Lihua You,et al.  PDE blending surfaces with C2 continuity , 2004, Comput. Graph..

[48]  Malcolm I. G. Bloor,et al.  An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch , 2005, Comput. Aided Geom. Des..