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Planar C2 cubic spline interpolation under geometric boundary conditions

A Ginnis
2002 Computer Aided Geometric Design  
This paper deals with the problem of C 2 cubic spline interpolation under geometric boundary conditions, that is, fixing the unit-tangent vector and the curvature at the end points of a planar point-set  ...  A robust scheme for the numerical solution of the quadratic system is presented, and the use of the new boundary conditions is illustrated in the context of three examples.  ...  In this work we investigate the problem of C 2 cubic spline interpolation under geometric boundary conditions for point-sets containing more than two points (N 3).  ... 
doi:10.1016/s0167-8396(02)00091-2 fatcat:sntghwttxfflljddsesziotvsi

Pseudospline interpolation for space curves

D. H. Thomas
1976 Mathematics of Computation  
It is the direct analogue of Fowler-Wilson or pseudospline interpolation for plane curves in that local coordinate systems, cubic polynomials of suitable parameters, and mildly nonlinear equations are  ...  A method for interpolating a curve through points in space is described.  ...  a variety of boundary conditions [3] .  ... 
doi:10.1090/s0025-5718-1976-0400651-9 fatcat:xgevyqlvkjfu3p6sxl57auvf7m

Pseudospline Interpolation for Space Curves

D. H. Thomas
1976 Mathematics of Computation  
It is the direct analogue of Fowler-Wilson or pseudospline interpolation for plane curves in that local coordinate systems, cubic polynomials of suitable parameters, and mildly nonlinear equations are  ...  A method for interpolating a curve through points in space is described.  ...  a variety of boundary conditions [3] .  ... 
doi:10.2307/2005430 fatcat:64rurkjtdrfqtpbabjh6x2aadm

Minimal energy surfaces using parametric splines

Gregory E. Fasshauer, Larry L. Schumaker
1996 Computer Aided Geometric Design  
We explore the construction of parametric surfaces which interpolate prescribed 3D scattered data using spaces of parametric splines defined on a 2D triangulation.  ...  This requires introducing spaces of splines on certain generalized triangulations.  ...  Fig. 4a Shows the surface corresponding to the cubic C' parametric spline interpolant.  ... 
doi:10.1016/0167-8396(95)00006-2 fatcat:znmq2aphmfbnxpnwps5lvbfoni

Fitting scattered data on sphere-like surfaces using spherical splines

Peter Alfeld, Marian Neamtu, Larry L. Schumaker
1996 Journal of Computational and Applied Mathematics  
Spaces of polynomial splines defined on planar triangulations are very useful tools for fitting scattered data in the plane.  ...  For interpolation. we discuss macro-element methods and minimal energy splines. and for fitting, we consider discrete least squares and penalized least squares.  ...  As in the planar case, we conjecture that interpolation is also possible with cubic C' splines.  ... 
doi:10.1016/0377-0427(96)00034-9 fatcat:rw5j6e4aonfxfim7pytfsmepyq

A sensitivity approach to optimal spline robot trajectories

A. de Luca, L. Lanari, G. Oriolo
1991 Automatica  
Using smooth interpolating cubic splines as joint space trajectories, the path is parameterized in terms of time intervals between knots.  ...  The proposed approach can be used for optimizing more general objective functions under different types of constraints.  ...  The trajectory is a C2-spline passing through n knots, with boundary conditions on initial and final velocity. The n -1 time intervals between the knots uniquely parameterize the path.  ... 
doi:10.1016/0005-1098(91)90111-e fatcat:rxdzf6upbrcitjs4wdwnxz57ha

Curvature continuous spline surfaces over irregular meshes

Jörg Peters
1996 Computer Aided Geometric Design  
Conccpts and techniques for thc construction of spline surfaces over irregular meshes are developed and made concrete by exhibiting a family of 0 2 surface splines.  ...  These splines extend the B-spline paradigm for the construction of parametric piecewise polynomial surfaces to control_meshe9 with non quadrila.leral cells and mOre or fewer Lhan four cells meeting at  ...  The mesh may model bivariate surfaces with or without boundary and of arbitrary topological genus; for a discussion of boundary conditions see [Peters '93] , [Loop '94] .  ... 
doi:10.1016/0167-8396(95)00017-8 fatcat:kz7vpxskinglhncyheh3xtzyme

New Approximation Methods for Planar Offset and Convolution Curves [chapter]

In-Kwon Lee, Myung-Soo Kim, Gershon Elber
1997 Geometric Modeling: Theory and Practice  
These methods can be used as fundamental tools in various geometric applications such as NC machining and collision detection of planar curved objects.  ...  We present new methods to approximate the offset and convolution of planar curves.  ...  (s)) Various QAC approximations computed from cubic C1(t) and rational cubic C2(s) TMC approximation of two cubic B-spline curves Table 1 . 1 Number of curve subdivisions in the offset approximation  ... 
doi:10.1007/978-3-642-60607-6_7 fatcat:cyfsjd7sdrdnxegm4mcyrnzpqm

Comparing offset curve approximation methods

G. Elber, In-Kwon Lee, Myung-Soo Kim
1997 IEEE Computer Graphics and Applications  
): The regularity condition of C(t) guarantees that (x (t), y (t)) = (0, 0) and N (t) is well defined on the curve C(t).  ...  Offset of Planar Curves Given a regular parametric curve, C(t) = (x(t), y(t)), in the plane, its offset curve C d (t) by a constant radius d, is defined by: where N (t) is the unit normal vector of C(t  ...  by the local cubic B-spline interpolation method, Pham.  ... 
doi:10.1109/38.586019 fatcat:ihv5bch5zvc4deeutv7yzhxg4u

Triangular NURBS and their dynamic generalizations

Hong Qin, Demetri Terzopoulos
1997 Computer Aided Geometric Design  
They provide an elegant and unified representation scheme for all piecewise continuous polynomial surfaces over planar triangulations.  ...  Fixing the weights to unity reduces triangular NURBS to triangular B-splines.  ...  Acknowledgments We would like to thank Professor Hans-Peter Seidel of Universit~it Erlangen for kindly making available the software for triangular B-spline surfaces that he developed with Mr.  ... 
doi:10.1016/s0167-8396(96)00062-3 fatcat:5pyhbzjurzh2hggwdeaai4dro4

Dynamic manipulation of triangular B-splines

Hong Qin, Demetri Terzopoulos
1995 Proceedings of the third ACM symposium on Solid modeling and applications - SMA '95  
We demonstrate several applications of dynamic triangular B-splines, including interactive sculpting using forces and physical parameters, the fitting of unstructured data, and solid rounding with geometric  ...  Triangular B-splines provide a unified representation scheme for all piecewise polynomials.  ...  Acknowledgments We are grateful to Professor Hans-Peter Seidel for kindly making available the software for triangular B-spline surfaces that he developed with Philip Fong.  ... 
doi:10.1145/218013.218084 dblp:conf/sma/QinT95 fatcat:aadkvyhr7rf2zhnzruekddjqny

Triangular Bernstein-Bézier patches

Gerald Farin
1986 Computer Aided Geometric Design  
These auxiliary points can be interpreted as the B spline vertices of cubic C 2 spline curves, but it is not known if the d x, could be utilized for the definition of C 2 splines over arbitrary triangulations  ...  These are identically zero except over the triangles pertaining to the data point under consideration. These cardinal functions are called v-splines (for vertex-splines) by [Chui '85] . 3.  ... 
doi:10.1016/0167-8396(86)90016-6 fatcat:4eofrjn7tbdgjchktn2akwt7pa

Unwarping of unidirectionally distorted EPI images

J. Kybic, P. Thevenaz, A. Nirkko, M. Unser
2000 IEEE Transactions on Medical Imaging  
Echo-planar imaging (EPI) is a fast nuclear magnetic resonance imaging (MRI) method.  ...  The spline model gives us good accuracy thanks to its high approximation order. The short support of splines leads to a fast algorithm.  ...  spline [30] , [31] , i.e., a piecewise cubic function that is twice continuously differentiable, with knots (the boundaries between polynomial pieces) at points .  ... 
doi:10.1109/42.836368 pmid:10784280 fatcat:fjzgosqbg5h5zck6vofh3jb5du

Handling shape and contact location uncertainty in grasping two-dimensional planar objects

Vassilios N. Christopoulos, Paul Schrater
2007 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems  
Focusing on two-dimensional planar objects and two finger grasps for simplicity, we present a principled approach for selecting contact points by analyzing the risk of force closure failure.  ...  Right: Extracted boundary of a typical sample from the bottle database using model-based object segmentation technique (blue discontinuous line) and natural cubic spline interpolation (red continuous line  ...  Finally, the N points are interpolated using smoothing "natural" cubic splines (illustrated in Fig. 4 ) which can be converted to B-spline form via a linear transformation. B.  ... 
doi:10.1109/iros.2007.4399509 dblp:conf/iros/ChristopoulosS07 fatcat:w6icprbrlneetlot2ue6hvequu

Smoothing polyhedra using implicit algebraic splines

Chandrajit L. Bajaj, Insung Ihm
1992 Computer graphics  
These quintic patches replace the @ contacts of planar facets with C' continuity along all irtterpatch boundaries.  ...  Polyhedron "smoothing" is an efficient construction scheme for generating complex boundary models of solid physical objects.  ...  Prior work on splines have traditionally worked with a given planar triangulation using a polynomial finction basis [1, 32, 35] .  ... 
doi:10.1145/142920.134014 fatcat:7whadzf2z5hnfcbmaz5cmpiu5m
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