## Least squares fitting of piecewise algebraic curves.(English)Zbl 1143.65317

Summary: A piecewise algebraic curve is defined as the zero contour of a bivariate spline. We present a new method for fitting $$C^{1}$$ piecewise algebraic curves of degree 2 over type-2 triangulation to the given scattered data. By simultaneously approximating points, associated normals and tangents, and points constraints, the energy term is also considered in the method. Moreover, some examples are presented.

### MSC:

 65D10 Numerical smoothing, curve fitting 65D17 Computer-aided design (modeling of curves and surfaces) 65D07 Numerical computation using splines
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### References:

 [1] C. L. Bajaj, J. Chen, R. J. Holt, and A. N. Netravali, “Energy formulations of A-splines,” Computer Aided Geometric Design, vol. 16, no. 1, pp. 39-59, 1999. · Zbl 0908.68176 [2] C. L. Bajaj and G. Xu, “A-splines: local interpolation and approximation using Gk-continuous piecewise real algebraic curves,” Computer Aided Geometric Design, vol. 16, no. 6, pp. 557-578, 1999. · Zbl 0997.65013 [3] J. C. Carr, R. K. Beatson, J. B. Cherrie, et al., “Reconstruction and representation of 3D objects with radial basis functions,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’01), pp. 67-76, New York, NY, USA, 2001. [4] B. Jüttler and A. Felis, “Least-squares fitting of algebraic spline surfaces,” Advances in Computational Mathematics, vol. 17, no. 1-2, pp. 135-152, 2002. · Zbl 0997.65028 [5] B. Jüttler, “Least-squares fitting of algebraic spline curves via normal vector estimation,” in The Mathematics of Surfaces, pp. 263-280, Springer, London, UK, 2000. · Zbl 0967.65011 [6] I.-K. Lee, “Curve reconstruction from unorganized points,” Computer Aided Geometric Design, vol. 17, no. 2, pp. 161-177, 1999. · Zbl 0939.68154 [7] V. Pratt, “Direct least-squares fitting of algebraic surfaces,” ACM SIGGRAPH Computer Graphics, vol. 21, no. 4, pp. 145-152, 1987. [8] G. Taubin, “Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 1115-1138, 1991. · Zbl 05112706 [9] T. Tasdizen, J.-P. Tarel, and D. B. Cooper, “Improving the stability of algebraic curves for applications,” IEEE Transactions on Image Processing, vol. 9, no. 3, pp. 405-416, 2000. · Zbl 0962.94001 [10] J. Wang, Z. W. Yang, and J. S. Deng, “Blending surfaces with algebraic tensor-product B-spline surfaces,” Journal of University of Science and Technology of China, vol. 36, no. 6, pp. 598-603, 2006 (Chinese). [11] Z. Yang, J. Deng, and F. Chen, “Fitting point clouds with active implicit B-spline curves,” The Visual Computer, vol. 21, no. 8-10, pp. 831-839, 2005. [12] R.-H. Wang, “The dimension and basis of spaces of multivariate splines,” Journal of Computational and Applied Mathematics, vol. 12-13, pp. 163-177, 1985. · Zbl 0578.41017 [13] R.-H. Wang, Multivariate Spline Functions and Their Applications, vol. 529 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. · Zbl 1002.41001 [14] R.-H. Wang and C.-G. Zhu, “Cayley-Bacharach theorem of piecewise algebraic curves,” Journal of Computational and Applied Mathematics, vol. 163, no. 1, pp. 269-276, 2004. · Zbl 1070.14034 [15] R.-H. Wang and C.-G. Zhu, “Nöther-type theorem of piecewise algebraic curves,” Progress in Natural Science, vol. 14, no. 4, pp. 309-313, 2004. · Zbl 1083.14525 [16] C.-G. Zhu and R.-H. Wang, “Lagrange interpolation by bivariate splines on cross-cut partitions,” Journal of Computational and Applied Mathematics, vol. 195, no. 1-2, pp. 326-340, 2006. · Zbl 1097.65023 [17] Y. Yuan and W. Sun, Optimization Theory and Methods, Science Press, Beijing, China, 2001.
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