This Article 
 Bibliographic References 
 Add to: 
Uniform B-Spline Curve Interpolation with Prescribed Tangent and Curvature Vectors
Sept. 2012 (vol. 18 no. 9)
pp. 1474-1487
A. Abbas, Dept. of Comput. Sci., Univ. of Balamand, Tripoli, Lebanon
Hongwei Lin, Zhejiang Univ., Hangzhou, China
A. Nasri, American Univ. of Beirut, Beirut, Lebanon
S. Okaniwa, Digital Imaging Div., Casio Comput. Co., Ltd., Shibuya, Japan
Y. Kineri, Dept. of Mech. Eng., Yokohama Nat. Univ., Yokohama, Japan
T. Maekawa, Dept. of Mech. Eng., Yokohama Nat. Univ., Yokohama, Japan
This paper presents a geometric algorithm for the generation of uniform cubic B-spline curves interpolating a sequence of data points under tangent and curvature vectors constraints. To satisfy these constraints, knot insertion is used to generate additional control points which are progressively repositioned using corresponding geometric rules. Compared to existing schemes, our approach is capable of handling plane as well as space curves, has local control, and avoids the solution of the typical linear system. The effectiveness of the proposed algorithm is illustrated through several comparative examples. Applications of the method in NC machining and shape design are also outlined.

[1] A. Abbas, A. Nasri, and T. Maekawa, "Generating B-Spline Curves with Points, Normals and Curvature: A Constructive Approach," The Visual Computer, vol. 26, pp. 823-829, 2010.
[2] F. Yamaguchi, "A Method of Designing Free form Surfaces by Computer Display (1st Report) (in Japanese)," Precision Machinery, vol. 43, no. 2, pp. 168-173, 1977.
[3] F. Yamaguchi, Curves and Surfaces in Computer Aided Geometric Design. Springer-Verlag, 1988.
[4] H. Lin, G. Wang, and C. Dong, "Constructing Iterative Non-Uniform B-Spline Curve and Surface to Fit Data Points," Science in China, vol. 47, no. 3, pp. 315-331, 2004.
[5] T. Maekawa, Y. Matsumoto, and K. Namiki, "Interpolation by Geometric Algorithm," Computer-Aided Design, vol. 39, no. 4, pp. 313-323, 2007.
[6] F. Fan, F. Cheng, and S. Lai, "Subdivision Based Interpolation with Shape Control," Computer Aided Design and Application, vol. 5, nos. 1-4, pp. 539-547, 2008.
[7] S. Gofuku, S. Tamura, and T. Maekawa, "Point-Tangent/Point-Normal B-Spline Curve Interpolation by Geometric Algorithms," Computer-Aided Design, vol. 41, no. 6, pp. 412-422, 2009.
[8] H. Lin, "The Convergence of the Geometric Interpolation Algorithm," Computer-Aided Design, vol. 42, no. 6, pp. 505-508, 2010.
[9] Y. Bazilevs, T.J.R. Hughes, and J.A. Cottrell, "Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement," Computer Methods in Applied Mechanics and Eng., vol. 194, nos. 39-41, pp. 4135-4195, 2005.
[10] N.M. Patrikalakis and T. Maekawa, Shape Interrogation for Computer Aided Design and Manufacturing. Springer-Verlag, 2002.
[11] L. Piegl and W. Tiller, The NURBS Book, second ed. Springer-Verlag, 1997.
[12] H. Prautzsch, W. Boehm, and M. Paluszny, Bézier and B-Spline Techniques. Springer, 2002.
[13] X. Ye, T.R. Jackson, and N.M. Patrikalakis, "Geometric Design of Functional Surfaces," Computer-Aided Design, vol. 28, no. 9, pp. 741-752, 1996.
[14] C. deBoor, K. Höllig, and M. Sabin, "High Accuracy Geometric Hermite Interpolation," Computer Aided Geometric Design, vol. 4, no. 4, pp. 269-278, 1987.
[15] L. Xu and J. Shi, "Geometric Hermite Interpolation for Space Curves," Computer-Aided Geometric Design, vol. 18, no. 9, pp. 817-829, 2001.
[16] G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, fifth ed. Morgan Kaufmann, 2002.
[17] K. Höllig and J. Koch, "Geometric Hermite Interpolation," Computer Aided Geometric Design, vol. 12, no. 6, pp. 567-580, 1995.
[18] K. Höllig and J. Koch, "Geometric Hermite Interpolation with Maximal Order and Smoothness," Computer Aided Geometric Design, vol. 13, no. 8, pp. 681-695, 1996.
[19] A. Nasri and A. Abbas, "Designing Catmull-Clark Subdivision Surfaces with Curve Interpolation Constraints," Computer and Graphics, vol. 26, no. 3, pp. 393-400, 2002.
[20] A. Nasri and M. Sabin, "Taxonomy of Interpolation Constraints on Recursive Subdivision Curves," The Visual Computer, vol. 18, nos. 5/6, pp. 382-403, 2002.
[21] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, second ed. CRC Press, 1997.

Index Terms:
splines (mathematics),computational geometry,interpolation,shape design,uniform B-spline curve interpolation,prescribed tangent,curvature vectors,geometric algorithm,data point sequence,curvature vectors constraints,knot insertion,handling plane,space curves,linear system,NC machining,Spline,Interpolation,Equations,Aerospace electronics,Vectors,Educational institutions,Electronic mail,parametric curve.,B-spline curve,interpolation,tangent,curvature vector
A. Abbas, Hongwei Lin, A. Nasri, S. Okaniwa, Y. Kineri, T. Maekawa, "Uniform B-Spline Curve Interpolation with Prescribed Tangent and Curvature Vectors," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 9, pp. 1474-1487, Sept. 2012, doi:10.1109/TVCG.2011.262
Usage of this product signifies your acceptance of the Terms of Use.