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Planar G2 Hermite interpolation with some fair, C-shaped curves

D.S. Meek, D.J. Walton
2002 Journal of Computational and Applied Mathematics  
The planar G 2 Hermite interpolation problem is to ÿnd a planar curve matching planar G 2 Hermite data. In this paper, a C-shaped interpolating curve made of one or two spirals is sought.  ...  The new curve is compared to the cubic BÃ ezier curve and to a curve made from a G 2 join of a pair of quadratics.  ...  Acknowledgements The authors acknowledge the ÿnancial support of the Natural Sciences and Engineering Research Council of Canada for this research.  ... 
doi:10.1016/s0377-0427(01)00385-5 fatcat:tkgkcrvrbrdh3pis5t7upobtde

Geometric Hermite interpolation by logarithmic arc splines

Xunnian Yang
2014 Computer Aided Geometric Design  
This paper considers the problem of G 1 curve interpolation using a special type of discrete logarithmic spirals. A "logarithmic arc spline" is defined as a set of smoothly connected circular arcs.  ...  Given two points together with two unit tangents at the points, interpolation of logarithmic arc splines with a user specified winding angle is formulated into finding the positive solutions to a vector  ...  One reviewer's suggestion of the use of Descartes rule of signs simplified the proof of Theorem 3.1 considerably. This work is supported by NNSF of China grants (11290142, 61272300, 60970077).  ... 
doi:10.1016/j.cagd.2014.09.001 fatcat:qttmsrpspvcyxnnpr2tuiy2m74

Page 1517 of Mathematical Reviews Vol. , Issue 95c [page]

1995 Mathematical Reviews  
D. (3-MB-A; Winnipeg, MB) Hermite interpolation with a pair of spirals. (English summary) Comput. Aided Geom. Design 10 (1993), no. 6, 491-507.  ...  When can we find a smooth curve consisting of a pair of spirals joining these points with the associated tangents and curvatures?  ... 

Geometric Hermite interpolation by a family of intrinsically defined planar curves

Weidong Wu, Xunnian Yang
2016 Computer-Aided Design  
This paper proposes techniques of interpolation of intrinsically defined planar curves to Hermite data.  ...  By choosing low order polynomials for representing the curvature radius functions, the interpolating curves can be spirals that have monotone curvatures or fair curves with small numbers of curvature extremes  ...  This work is supported by the National Natural Science Foundation of China grants (11290142, 61272300).  ... 
doi:10.1016/j.cad.2016.04.001 fatcat:s6o4z2diwzhsrnwsulftc6q33y

Planar interpolation with a pair of rational spirals

T.N.T. Goodman, D.S. Meek
2007 Journal of Computational and Applied Mathematics  
Two situations where the planar, two-point G 2 Hermite interpolation problem can be solved with a pair of rational spiral segments are outlined here.  ...  Spirals are curves of one-signed, monotone increasing or decreasing curvature. Spiral segments have the advantage that the minimum and maximum curvatures are at their endpoints.  ...  The second author appreciates the hospitality of the University of Dundee, Dundee, Scotland, and acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada.  ... 
doi:10.1016/j.cam.2006.02.003 fatcat:wirvbtsbmndyhbvoau53woui7i

A two-point G1 Hermite interpolating family of spirals

D.S. Meek, D.J. Walton
2009 Journal of Computational and Applied Mathematics  
It is shown that there is a unique G 1 Hermite interpolating spiral that passes through a given point in an allowable region.  ...  A one-parameter family of spirals that can match planar, two-point G 1 Hermite data is presented.  ...  The authors appreciate the financial support of the National Science and Engineering Research Council of Canada for this research.  ... 
doi:10.1016/j.cam.2007.12.027 fatcat:vfw6r7mgefayfjpo7h2yzwnaee

Spiral arc spline approximation to a planar spiral

D.S. Meek, D.J. Walton
1999 Journal of Computational and Applied Mathematics  
The resulting collection of biarcs forms a G 1 arc spline that interpolates the entire set of G 1 Hermite data.  ...  An arc spline approximation to a smooth planar curve can be found by reading G 1 Hermite data from the curve and ÿtting a biarc between each pair of data points.  ...  Acknowledgements The authors acknowledge the ÿnancial support of the Natural Sciences and Engineering Research Council of Canada for this research.  ... 
doi:10.1016/s0377-0427(99)00072-2 fatcat:j5vcovxe3nfjpg4ahiyrm5rq4m

3D Euler spirals for 3D curve completion

Gur Harary, Ayellet Tal
2012 Computational geometry  
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.  ...  A. Gilboa and the Zinman Institute of Archaeology at the University of Haifa for providing the archaeological models.  ... 
doi:10.1016/j.comgeo.2011.10.001 fatcat:r6lkxhbmbna4bhtk65ejuwwbw4

3D Euler spirals for 3D curve completion

Gur Harary, Ayellet Tal
2010 Proceedings of the 2010 annual symposium on Computational geometry - SoCG '10  
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.  ...  A. Gilboa and the Zinman Institute of Archaeology at the University of Haifa for providing the archaeological models.  ... 
doi:10.1145/1810959.1811025 dblp:conf/compgeom/HararyT10a fatcat:mcdobufxmbdfrar3fa7eggi2ge

G1 interpolation with a single Cornu spiral segment

D.J. Walton, D.S. Meek
2009 Journal of Computational and Applied Mathematics  
Cornu spiral segments are used in applications such as the geometric design of highways and railways, robot path planning, and shape completion.  ...  An improved method for doing so using a Cornu spiral segment is discussed.  ...  Acknowledgements The authors acknowledge with thanks the suggestions of the anonymous reviewer which helped to improve the paper.  ... 
doi:10.1016/j.cam.2007.12.022 fatcat:eldytdz3sras5oiz6hl73rtwym

A further generalisation of the planar cubic Bézier spiral

D.J. Walton, D.S. Meek
2012 Journal of Computational and Applied Mathematics  
Spiral segments are useful in the design of fair curves.  ...  They are important in CAD/CAM applications, the design of highway and railway routes, trajectories of mobile robots and other similar applications.  ...  Acknowledgment This work was supported by The Natural Sciences and Engineering Research Council of Canada.  ... 
doi:10.1016/j.cam.2012.01.024 fatcat:opz6onzz7ncq7azjcxqsnsfdja

Coaxing a planar curve to comply

D.S. Meek
2002 Journal of Computational and Applied Mathematics  
We concentrate on the ÿrst of these, interpolation methods.  ...  A long-standing problem in computer graphics is to ÿnd a planar curve that is shaped the way you want it to be shaped. A selection of various methods for achieving this goal is presented.  ...  Pair of spirals A single spiral can match a rather restricted set of two-point G 2 data; a pair of spirals can match a wider range of G 2 data.  ... 
doi:10.1016/s0377-0427(01)00478-2 fatcat:jr43wuflt5ftborljhiyfdeoum

Planar two-point G1 Hermite interpolating log-aesthetic spirals

D.S. Meek, T. Saito, D.J. Walton, N. Yoshida
2012 Journal of Computational and Applied Mathematics  
This paper proves that for any member of the family, a unique segment of that spiral can be found that matches given two-point G 1 Hermite data.  ...  A family of planar log-aesthetic spirals that include a point of zero curvature is used in this paper. The two-point G 1 Hermite data that is considered has some restrictions on the angles.  ...  Acknowledgements The authors wish to express appreciation for the financial support of the Natural Sciences and Engineering Research Council of Canada.  ... 
doi:10.1016/j.cam.2012.04.021 fatcat:ln3zknjihbhrvo6jneecp7zose

Visualizing 3D Euler spirals

Gur Harary, Ayellet Tal
2010 Proceedings of the 2010 annual symposium on Computational geometry - SoCG '10  
This video describes a new type of 3D curves, which generalizes the family of 2D Euler spirals.  ...  The utility of these spirals for curve completion applications is demonstrated. This video accompanies the paper presented in [4] .  ...  Roundness: If C interpolates two point-tangent pairs lying on a circle, then C is a circle. 3D CURVE COMPLETION 3D curve features convey important information about the shape.  ... 
doi:10.1145/1810959.1810977 dblp:conf/compgeom/HararyT10 fatcat:4ez5ye5ykbcp3l4f2sc6tss2kq

Page 8453 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
Hermite interpolation problem is to find a planar curve matching planar G? Hermite data. In this paper, a C-shaped interpolating curve made of one or two spirals is sought.  ...  join of a pair of quadratics.  ... 
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