3D Euler spirals for 3D curve completion

Shape completion is an intriguing problem in geometry processing with applications in CAD and graphics. This paper defines a new type of 3D curve, which can be utilized for curve completion. It can be considered as the extension to three dimensions of the 2D Euler spiral. We prove several properties of this curve - properties that have been shown to be important for the appeal of curves. We illustrate its utility in two applications. The first is ''fixing'' curves detected by algorithms for edge detection on surfaces. The second is shape illustration in archaeology, where the user would like to draw curves that are missing due to the incompleteness of the input model.

[1]  Jacqueline M. Fulvio,et al.  Visual extrapolation of contour geometry. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Ariel Shamir,et al.  Relief analysis and extraction , 2009, ACM Trans. Graph..

[3]  R. Bellman.  Calculus of Variations (L. E. Elsgolc) , 1963 .

[4]  Hua Li,et al.  3D discrete clothoid splines , 2001, Proceedings. Computer Graphics International 2001.

[5]  D. Mumford.  Elastica and Computer Vision , 1994 .

[6]  Georg Glaeser,et al.  Open Geometry: OpenGL + Advanced Geometry with Disk , 1999 .

[7]  T. J. Sharrock.  Biarcs in three dimensions , 1987 .

[8]  Subodh Kumar,et al.  Repairing CAD models , 1997, Proceedings. Visualization '97 (Cat. No. 97CB36155).

[9]  Adam Finkelstein,et al.  Suggestive contours for conveying shape , 2003, ACM Trans. Graph..

[10]  W. Eric L. Grimson,et al.  Shape Encoding and Subjective Contours , 1980, AAAI.

[11]  Rida T. Farouki,et al.  Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable , 2007, Geometry and Computing.

[12]  Ayellet Tal,et al.  3D Euler spirals for 3D curve completion , 2010, Comput. Geom..

[13]  Luciano Silva,et al.  A 3D reconstruction pipeline for digital preservation , 2009, CVPR.

[14]  Marc Alexa,et al.  Context-based surface completion , 2004, ACM Trans. Graph..

[15]  Ayellet Tal,et al.  The Natural 3D Spiral , 2011, Comput. Graph. Forum.

[16]  C. A. Neff,et al.  Hermite interpolation by Pythagorean hodograph quintics , 1995 .

[17]  Carlo H. Séquin,et al.  Functional optimization for fair surface design , 1992, SIGGRAPH.

[18]  T. Banchoff,et al.  Differential Geometry of Curves and Surfaces , 2010 .

[19]  Raph Levien,et al.  The Euler spiral: a mathematical history , 2008 .

[20]  S. Ullman,et al.  Filling-in the gaps: The shape of subjective contours and a model for their generation , 1976, Biological Cybernetics.

[21]  Ayellet Tal,et al.  Visualizing 3D Euler spirals , 2010, SoCG '10.

[22]  W. K. Chiu,et al.  Direct 5-axis tool-path generation from point cloud input using 3D biarc fitting , 2008 .

[23]  Georg Glaeser,et al.  Open Geometry: OpenGL® + Advanced Geometry , 1998, Springer: New York.

[24]  Tim Weyrich,et al.  A system for high-volume acquisition and matching of fresco fragments: reassembling Theran wall paintings , 2008, SIGGRAPH 2008.

[25]  Frédo Durand,et al.  Apparent ridges for line drawing , 2007, ACM Trans. Graph..

[26]  Micha Sharir,et al.  Filling gaps in the boundary of a polyhedron , 1995, Comput. Aided Geom. Des..

[27]  D. Walton,et al.  Clothoid spline transition spirals , 1992 .

[28]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[29]  Holly Rushmeier,et al.  Eternal Egypt: Experiences and Research Directions , 2006 .

[30]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[31]  Karan Singh,et al.  Sketching piecewise clothoid curves , 2008, SBM'08.

[32]  Dereck S. Meek,et al.  A controlled clothoid spline , 2005, Comput. Graph..

[33]  Ilan Shimshoni,et al.  Demarcating curves for shape illustration , 2008, SIGGRAPH 2008.

[34]  Hans-Peter Seidel,et al.  Fast and robust detection of crest lines on meshes , 2005, SPM '05.

[35]  Trimble,et al.  Fragments of the City: Stanfordʹs Digital Forma Urbis Romae Project , 2022 .

[36]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[37]  Berthold K. P. Horn.  The Curve of Least Energy , 1983, TOMS.

[38]  Benjamin B. Kimia,et al.  Euler Spiral for Shape Completion , 2003, International Journal of Computer Vision.

[39]  Max Born,et al.  Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen , 1906 .

[40]  Raph Levien,et al.  The elastica: a mathematical history , 2008 .

[41]  D. Walton,et al.  G 1 interpolation with a single Cornu spiral segment , 2008 .

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