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Constructing Curvature-continuous Surfaces by Blending [article]

Denis Zorin
2006 Symposium on geometry processing : [proceedings]. Symposium on Geometry Processing  
In this paper we describe an approach to the construction of curvature-continuous surfaces with arbitrary control meshes using subdivision.  ...  spots or curvature singularities at extraordinary vertices.  ...  The blending function B 2 k • Φ −1 k is C 2 -continuous by construction.  ... 
doi:10.2312/sgp/sgp06/031-040 fatcat:fl4pvsxoffdandnq3tqgnyidem

Rapid Generation of C 2 Continuous Blending Surfaces [chapter]

Jian J. Zhang, Lihua You
2002 Lecture Notes in Computer Science  
Most surface-blending methods are able to blend surfaces with tangent continuity.  ...  However, curvature continuity has become increasingly important in geometric modelling and its applications, such as computer animation, computer-aided design and virtual reality.  ...  This method was first proposed by Bloor and Wilson [5] . In the reported literature, however, the PDE based approach has only been used to construct a C 1 continuous blending surface.  ... 
doi:10.1007/3-540-46080-2_10 fatcat:5xozlocfqvbgnago2ckp3x6hva

Curvature Computation for Triangular Meshes based on Local Parametrization

Fuhua (Frank) Cheng, Conglin Huang, Shuhua Lai, Fengtao Fan, Jiaxi Wang, Caimin Zhang
2008 Computer-Aided Design and Applications  
A blending technique is then applied to vicinities of extra-ordinary vertices to ensure continuity and boundedness of curvature at each extra-ordinary vertex.  ...  This blending process does not change the value of the surface at the extra-ordinary points.  ...  The construction of the locally blended Loop surface requires two steps.  ... 
doi:10.3722/cadaps.2008.220-227 fatcat:i2vx5jddr5etdmwl7o5mj4yoh4

An Introduction to Guided and Polar Surfacing [chapter]

Jörg Peters, Kȩstutis Karčiauskas
2010 Lecture Notes in Computer Science  
We demonstrate the challenge of high-quality surface construction by examples since the notion of surface quality lacks an overarching theory.  ...  A second ingredient is separation of shape design from surface representation by creating an initial guide shape and leveraging classical approximation-theoretic tools to construct a final surface compatible  ...  The images were generated with the surface analysis tool BezierView. This work was supported by the National Science Foundation Grant 0728797.  ... 
doi:10.1007/978-3-642-11620-9_20 fatcat:d7ou4xj4sbhoraouioiw2sagdi

G1 surface modelling using fourth order geometric flows

Guoliang Xu, Qing Pan
2006 Computer-Aided Design  
We use two fourth order geometric partial differential equations to efficiently solve several surface modelling problems, including the surface blending, the N -sided hole filling and the free-form surface  ...  fitting with the G 1 boundary continuity.  ...  Surface Blending Given a collection of surface meshes with boundaries, we construct a fair surface to blend the meshes with the specified G 1 continuity at the boundaries.  ... 
doi:10.1016/j.cad.2005.11.002 fatcat:souh5r7tibdrvhgibzkuar344i

Rational fixed radius rolling ball blends between natural quadrics

Heidi E.I. Dahl, Rimvydas Krasauskas
2012 Computer Aided Geometric Design  
In particular, G 1 or G 2 continuity of the curve is inherited by the corresponding canal surface.  ...  The above results are applied in the construction of composite rolling ball blends with internal G 1 or G 2 continuity.  ...  Acknowledgements This paper has been partially financed by the Marie-Curie Initial Training Network SAGA (ShApes, Geometry, Algebra), FP7-PEOPLE contract PITN-GA-2008-214584.  ... 
doi:10.1016/j.cagd.2012.06.008 fatcat:k4yeed77ene5bj5zpnur3pzbmu

Modified subdivision surfaces with continuous curvature

Adi Levin
2006 ACM Transactions on Graphics  
(b) Circular regions around the extraordinary points, in which we modify the Catmull-Clark limit surface for curvature continuity. (c) Isophotes of the modified C 2 surface.  ...  The idea is to blend the limit surface with a low degree polynomial defined over the characteristic map, in the vicinity of each extraordinary point.  ...  Acknowledgement This paper was inspired by a conversation with my father, David Levin.  ... 
doi:10.1145/1141911.1141990 fatcat:cadvn4cz6fe6xnx423mragujum

Discrete surface modelling using partial differential equations

Guoliang Xu, Qing Pan, Chandrajit L. Bajaj
2006 Computer Aided Geometric Design  
We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N -sided hole filling and free-form surface fitting.  ...  The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.  ...  Given a collection surface mesh with boundaries, we construct a fair surface to blend the meshes at the boundaries with specified geometric continuity.  ... 
doi:10.1016/j.cagd.2005.05.004 pmid:19830268 pmcid:PMC2760856 fatcat:74ryy3jcwzeipa6tizdjk6kpm4

Minimal Mean-Curvature-Variation Surfaces and Their Applications in Surface Modeling [chapter]

Guoliang Xu, Qin Zhang
2006 Lecture Notes in Computer Science  
Using this Euler-Lagrange equation, we construct a sixth-order geometric flow (named as minimal mean-curvature-variation flow), which is solved numerically by a divided-difference-like method.  ...  We apply our equation to solving several surface modeling problems, including surface blending, N-sided hole filling and point interpolating.  ...  From these figures, we can observe that the surface produced by MMCVF is mean-curvature continuous at the blending boundaries, while the surfaces produced by SDF and WF are not.  ... 
doi:10.1007/11802914_25 fatcat:j7rawjw2azbdvgqnm27ga3geka

Constructing G2 Continuous Curve on Freeform Surface with Normal Projection

Wang Xiaoping, An Luling, Zhou Laishui, Zhang Liyan
2010 Chinese Journal of Aeronautics  
First, a G 2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface.  ...  This article presents a new method for G 2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and curvature vector, respectively  ...  Fig.3 illustrates the G 2 continuous blending curve of two curves on surface S, i.e., the blending curve segment 01 and curves 0 have common tangent direction and common curvature vectors at P 0 , 1 ,  ... 
doi:10.1016/s1000-9361(09)60197-7 fatcat:u2szuawb35d6dfs3f6os4yqqmq

Streamline Modeling [chapter]

Kenjiro T. Miura, Naoki Sakiyama, Toru Kaneko
2001 Geometric Modelling  
The popular conventional surface construction method is to generate free-form surfaces by determining positions of their control points for Bezier, NURBS and other free-form surface formulations.  ...  blend the tangent vectors of these streamlines to generate isoparametric lines of a free-form surface.  ...  The main idea of their surface construction shceme is to blend positions of points called control ones to define surface shapes.  ... 
doi:10.1007/978-0-387-35490-3_16 fatcat:6opli4cl2fh7fargtcnfexocpm

surface modeling using minimal mean-curvature-variation flow

Guoliang Xu, Qin Zhang
2007 Computer-Aided Design  
We apply our equation to solving several surface modeling problems, including surface blending, N-sided hole filling and point interpolating, with G 2 continuity.  ...  Using this Euler-Lagrange equation, we construct a sixthorder geometric flow, which is solved numerically by a divided-difference-like method.  ...  From these figures, we can observe that the surface produced by MMCVF is mean-curvature continuous at the blending boundaries, while the surfaces produced by SDF and WF are not.  ... 
doi:10.1016/j.cad.2007.02.007 fatcat:zrwo2r6ozjea3mgf4wspbtg5pi

Semi-structured B-spline for blending two B-spline surfaces

Hongwei Lin, Yunyang Xiong, Hongwei Liao
2014 Computers and Mathematics with Applications  
We formulated the B-spline surface blending problem as an optimization problem with continuity constraints, and the continuity between the base and blending surfaces can reach G 2 or C 2 .  ...  By assigning suitable knot vectors to the head and tail skinned curves, the semi-structured B-spline surface can blend two B-spline surfaces smoothly without disturbing them at all.  ...  Acknowledgment This paper is supported by Natural Science Foundation of China (Nos. 61379072, 60970150, 60933008).  ... 
doi:10.1016/j.camwa.2014.07.013 fatcat:buxe3b2su5hojmg7fvywse37l4

Intuitive procedure for constructing geometrically complex objects using cyclides

Y.L. Srinivas, Debasish Dutta
1994 Computer-Aided Design  
In the past, cyclide surfaces have been used effectively for the variable radius blending of natural quadric intersections.  ...  However, attempts to use cyclides for constructing realistic, freeform composite surfaces met with rather limited success.  ...  In fact, *A double-cyclide blend is a blending technique in which two pieces from (different) cyclides, joined in a tangent-continuous manner, are used to produce a smooth blending surface.  ... 
doi:10.1016/0010-4485(94)90078-7 fatcat:6ultcr6bbrexdfjzfbtayyxwrm

Curvature continuous blend surfaces [chapter]

Günther Greiner, Hans-Peter Seidel
1993 Modeling in Computer Graphics  
We describe a method to generate blend surfaces which t with continuous curvature to the primary surfaces.  ...  Among all the bicubic tensor splines which give a curvature continuous blend surface, the one is chosen which minimizes a bilinear functional.  ...  In this case, one can only prescribe the boundary values not the normal or even the curvature. Thus, generating blend surfaces by this method will only yield continuous surfaces.  ... 
doi:10.1007/978-3-642-78114-8_19 dblp:conf/ifip5-10/GreinerS93 fatcat:z7r7mzujjjhstk3nocahtwtkfm
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