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Enhancing Levin's method for computing quadric-surface intersections

Wenping Wang, Ronald Goldman, Changhe Tu
2003 Computer Aided Geometric Design  
Levin's method produces a parameterization of the intersection curve of two quadrics in the form where a(u) and d(u) are vector valued polynomials, and s(u) is a quartic polynomial.  ...  enhanced version of Levin's method is proposed that, besides classifying the morphology of the intersection curve of two quadrics, produces a rational parameterization of the curve if the curve is singular  ...  We thank Professor Helmut Pottmann and the referee for their helpful comments. We also thank Barry Joe for commenting on an earlier version of the paper.  ... 
doi:10.1016/s0167-8396(03)00081-5 fatcat:dr3pbke2lvbybjpw4vwxbx4t5i

G1 interpolation using piecewise quadric and cubic surfaces

Chandrajit L. Bajaj, Leonard A. Ferrari, Rui J. P. de Figueiredo
1990 Curves and Surfaces in Computer Vision and Graphics  
Algorithms are presented for constructing G t continuous meshes of degree two (quadric) and degree three (cubic) implicitly defined, piecewise algebraic surfaces, which exactly fit any given collection  ...  of points and algebraic space curves, of arbitrary degree.  ...  It follows then from Bezout's theorem 3.1 for surface intersection, that the two quadrics 51 and 52 must meet in a plane curve (either an irreducible conlc or straight lines).  ... 
doi:10.1117/12.19736 fatcat:64kdqd4m65gfjorjr7m7ltv3ye

Solid reconstruction using recognition of quadric surfaces from orthographic views

Jie-Hui Gong, Hui Zhang, Gui-Fang Zhang, Jia-Guang Sun
2006 Computer-Aided Design  
This method can handle objects with interacting quadric surfaces and avoids the combinatorial search for tracing all the quadric surfaces in an intermediate wire-frame by the existing methods.  ...  A B-rep oriented method for reconstructing curved objects from three orthographic views is presented by employing a hybrid wire-frame in place of an intermediate wire-frame.  ...  This work was supported by the 973 Program of China (Grant No. 2004CB719404) and the Program for New Century Excellent Talents in University (Grant No. NCET-04-0088).  ... 
doi:10.1016/j.cad.2006.04.009 fatcat:h4gws3ywkbh5vgmwwweq4x5ggq

Some Applications of Constructive Real Algebraic Geometry [chapter]

Chandrajit L. Bajaj
1994 Algebraic Geometry and its Applications  
In this short article we summarize a number of recent applications of constructive real algebraic geometry to geometric modelling and robotics, that we have been involved with under the tutelage of Abhyankar  ...  The three surface intersection points are shown as the common intersections of the space curves for each pair of surfaces.  ...  Here we consider constructive methods for both local and global real parameterizations of curves and surfaces.  ... 
doi:10.1007/978-1-4612-2628-4_25 fatcat:wuqw3apqqnfdnmvt5xe5s7fdxa

Page 1211 of Mathematical Reviews Vol. , Issue 87c [page]

1987 Mathematical Reviews  
(F-ANGR) 87c:14063 Théoréme de Torelli affine pour les intersections de deux quadriques. [The affine Torelli theorem for intersections of two quadrics] Invent. Math. 80 (1985), no. 3, 375-416.  ...  The main result of this paper deals with affine complete intersections U of two quadrics in affine space of even dimension.  ... 

Rational hypersurface display

Chanderjit L. Bajaj
1990 Proceedings of the 1990 symposium on Interactive 3D graphics - SI3D '90  
The number of hyperpolygons use d are optimal, in that they are the order of the minimu m number required for a smooth Gouraud like shading o f the hypersurfaces .  ...  Abstrac t Algorithms are presented for polygonalizing implicitl y defined, quadric and cubic hypersurfaces in n > 3 dimensional space and furthermore displaying their projections in 3D .  ...  Acknowledgements : I sincerely thank Insung Ihm an d Andrew Royappa for the many hours they spent in fron t of the graphics workstation, helping me visualize in fou r and higher dimensional space .  ... 
doi:10.1145/91385.91428 dblp:conf/si3d/Bajaj90 fatcat:fpdkcsjxmndt5lol3knwo4q4a4

Constraints in constructive solid geometry

Jaroslaw R. Rossignac
1987 Proceedings of the 1986 workshop on Interactive 3D graphics - SI3D '86  
Constraint evaluation results in the activation of methods which compute rigid motions from surface information.  ...  The order in which constraints are evaluated may also be used as a language for specifying the sequence of assembly and set-up operations.  ...  Closed form solution are available for parametric representation of curves of intersection between any two natural quadrics [ 11,101. Vertices may be computed by efficient numeric methods [12] .  ... 
doi:10.1145/319120.319129 dblp:conf/si3d/Rossignac86 fatcat:ehudqr6oknan5mfxj4avz52rbu

Page 417 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 20, Issue [page]

1914 American Mathematical Society. Bulletin of the American Mathematical Society  
treatment of the intersection of two quadrics is too brief and too difficult to be of greatest service.  ...  It seems odd that it was thought necessary to speak in such detail of quadric surfaces after developing so large a number of formulas for surfaces defined by any analytic function, but the outline of the  ... 

A parametric algorithm for drawing pictures of solid objects bounded by quadric surfaces

Joshua Z. Levin
1976 Proceedings of the 3rd annual conference on Computer graphics and interactive techniques - SIGGRAPH '76  
Woon suggests two methods for calculating points on a quadric surface intersection curve ( QSIC).  ...  For non-planar intersections , the curve lies in a quadric surface which has a "base curv e" which is either a line , a parabola , a hyperbola , or an box " , which is a cube or r ectangula r par all epiped  ... 
doi:10.1145/563274.563320 dblp:conf/siggraph/Levin76 fatcat:wmskkxqcojbuzfkaub6sx7o4v4

A parametric algorithm for drawing pictures of solid objects bounded by quadric surfaces

Joshua Z. Levin
1976 Computer graphics  
Woon suggests two methods for calculating points on a quadric surface intersection curve ( QSIC).  ...  For non-planar intersections , the curve lies in a quadric surface which has a "base curv e" which is either a line , a parabola , a hyperbola , or an box " , which is a cube or r ectangula r par all epiped  ... 
doi:10.1145/965143.563320 fatcat:dgaptn4rwvcg7nenfxzniwpaoi

Chapter 2: Surface Fitting Using Implicit Algebraic Surface Patches [chapter]

Chandrajit L. Bajaj
1992 Topics in Surface Modeling  
C k Least-Squares Approximate Surface Fit: Coustruct a real algebraic surface S, which C k -1 interpolates a collection of points Pi in m..1 and given space curves Cj in m.3 as before, with associated  ...  Fil wilh SlIrface Patches: Construct a mesh of real algebraic surface patches S;, which C k inlNpolaLcs a collection of points Pi in m. 3 and given space curves C j in m. 3 , with associated "IlOl'lllal  ...  It follows then frolll Bezout's theorem 4.1 for surface intersection, that the two quadrics 51 and S'l must meet in ,L plane curve (either an irreducible conic or straight lines).  ... 
doi:10.1137/1.9781611971644.ch2 fatcat:pyxpssexaff6pczvuz7cy7vhxm

Intersecting quadrics

S. Lazard, L. M. Peñaranda, S. Petitjean
2004 Proceedings of the twentieth annual symposium on Computational geometry - SCG '04  
We present the first complete, exact and efficient C++ implementation of a method for parameterizing the intersection of two implicit quadrics with integer coefficients of arbitrary size.  ...  Unlike existing implementations, it correctly identifies and parameterizes all the connected components of the intersection in all cases, returning parameterizations with rational functions whenever such  ...  Until recently, the only known general method for computing a parametric representation of the intersection between two arbitrary quadrics was that of J. Levin [7] .  ... 
doi:10.1145/997817.997880 dblp:conf/compgeom/LazardPP04 fatcat:6mekaucuu5ekhod7n2ponfawme

Book Review: Poliedri, Curve e Superficie secondo i Metodi della Geometria descrittiva

Virgil Snyder
1914 Bulletin of the American Mathematical Society  
treatment of the intersection of two quadrics is too brief and too difficult to be of greatest service.  ...  [May, intersections. The method of Monge is given the preference, but other methods are given in outline.  ... 
doi:10.1090/s0002-9904-1914-02513-3 fatcat:h6wigtbqyneghcp2cbvknn4t3i

Smoothing polyhedra using implicit algebraic splines

Chandrajit L. Bajaj, Insung Ihm
1992 Computer graphics  
Individual faces of a polyhedron are replaced by low degree implicit algebraic surface patches with local support.  ...  This paper presents efficient algorithms for generating families of curved solid objects with boundaty topology related to an input polyhedron.  ...  Acknowledgements : We are grateful to Vinod Anupam, Andrew Royappa and Dan Schikore for their assistance in the implementation of the smoothing algorithms.  ... 
doi:10.1145/142920.134014 fatcat:7whadzf2z5hnfcbmaz5cmpiu5m

Conchoid surfaces of quadrics

David Gruber, Martin Peternell
2013 Journal of symbolic computation  
The conchoid surface F d of a surface F with respect to a fixed reference point O is a surface obtained by increasing the distance function with respect to O by a constant d.  ...  This contribution studies conchoid surfaces of quadrics in Euclidean R 3 and shows that these surfaces admit real rational parameterizations.  ...  Acknowledgments This work has been partially supported by the 'Ministerio de Economia y Competitividad' under the project MTM2011-25816-C02-01.  ... 
doi:10.1016/j.jsc.2013.07.003 fatcat:fltp23pn2rcjhg3ihphfj6m2g4
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