A Linear Variational System for Modelling From Curves

We present a linear system for modelling 3D surfaces from curves. Our system offers better performance, stability and precision in control than previous non‐linear systems. By exploring the direct relationship between a standard higher‐order Laplacian editing framework and Hermite spline curves, we introduce a new form of Cauchy constraint that makes our system easy to both implement and control. We introduce novel workflows that simplify the construction of 3D models from sketches. We show how to convert existing 3D meshes into our curve‐based representation for subsequent editing and modelling, allowing our technique to be applied to a wide range of existing 3D content.

[1]  Mark Pauly,et al.  Positional, Metric, and Curvature Control for Constraint‐Based Surface Deformation , 2009, Comput. Graph. Forum.

[2]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[3]  Levent Burak Kara,et al.  Supporting Early Styling Design of Automobiles Using Sketch-based 3D Shape Construction , 2008 .

[4]  Daniel Cohen-Or,et al.  iWIRES: an analyze-and-edit approach to shape manipulation , 2009, ACM Trans. Graph..

[5]  Eitan Grinspun,et al.  Discrete quadratic curvature energies , 2006, Comput. Aided Geom. Des..

[6]  Henning Biermann,et al.  Piecewise smooth subdivision surfaces with normal control , 2000, SIGGRAPH.

[7]  John F. Hughes,et al.  SmoothSketch: 3D free-form shapes from complex sketches , 2006, SIGGRAPH '06.

[8]  Yotam I. Gingold,et al.  Shading-based surface editing , 2008, ACM Trans. Graph..

[9]  Charles T. Loop.  Generalized B-spline surfaces of arbitrary topological type , 1992 .

[10]  Brian Wyvill,et al.  ShapeShop: sketch-based solid modeling with BlobTrees , 2006, SBM.

[11]  Daniel Cohen-Or,et al.  Least-squares meshes , 2004, Proceedings Shape Modeling Applications, 2004..

[12]  Marc Alexa,et al.  A sketch-based interface for detail-preserving mesh editing , 2007, SIGGRAPH Courses.

[13]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[14]  Adi Levin,et al.  Interpolating nets of curves by smooth subdivision surfaces , 1999, SIGGRAPH.

[15]  Ravin Balakrishnan,et al.  ILoveSketch: as-natural-as-possible sketching system for creating 3d curve models , 2008, UIST '08.

[16]  Jonathan Richard Shewchuk,et al.  Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator , 1996, WACG.

[17]  Leif Kobbelt,et al.  An intuitive framework for real-time freeform modeling , 2004, ACM Trans. Graph..

[18]  James Demmel,et al.  A Supernodal Approach to Sparse Partial Pivoting , 1999, SIAM J. Matrix Anal. Appl..

[19]  John Sullivan,et al.  Minimizing the Squared Mean Curvature Integral for Surfaces in Space Forms , 1992, Exp. Math..

[20]  Hans-Peter Seidel,et al.  Ridge-valley lines on meshes via implicit surface fitting , 2004, ACM Trans. Graph..

[21]  Satoshi Matsuoka,et al.  Teddy: A Sketching Interface for 3D Freeform Design , 1999, SIGGRAPH Courses.

[22]  Marc Alexa,et al.  A sketch-based interface for detail-preserving mesh editing , 2005, SIGGRAPH '05.

[23]  Guoliang Xu,et al.  G2 surface modeling using minimal mean-curvature-variation flow , 2007, Comput. Aided Des..

[24]  Faramarz F. Samavati,et al.  Image-assisted modeling from sketches , 2010, Graphics Interface.

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

[26]  Marc Alexa,et al.  FiberMesh: designing freeform surfaces with 3D curves , 2007, ACM Trans. Graph..

[27]  Mark Meyer,et al.  Harmonic coordinates for character articulation , 2007, ACM Trans. Graph..

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

[29]  Nathan A. Carr,et al.  Repoussé: automatic inflation of 2D artwork , 2008, SBM'08.

[30]  David Bommes,et al.  Efficient Linear System Solvers for Mesh Processing , 2005, IMA Conference on the Mathematics of Surfaces.