Art and Design, Vol. 1, Issue 1, Sep  2018, Pages 14-22; DOI: 10.31058/ 10.31058/

Fitting Hyperboloid and Hyperboloid Structures

Art and Design, Vol. 1, Issue 1, Sep  2018, Pages 14-22.

DOI: 10.31058/

Sebahattin Bektas 1*

1 Geomatics Engineering, Faculty of Engineering, Ondokuz Mayis University, Samsun, Turkey

Received: 29 November 2017; Accepted: 20 December 2017; Published: 17 January 2018

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In this paper, we present the design of hyperboloid structures and techniques for hyperboloid fitting which are based on minimizing the sum of the squares of the algebraic distances between the noisy data and the hyperboloid. Algebraic fitting methods solve the linear least squares (LS) problem, and are relatively straightforward and fast. Fitting orthogonal hyperboloid is a difficult issue. Usually, it is impossible to reach a solution with classic LS algorithms, because they are often faced with the problem of convergence. Therefore, it is necessary to use special algorithms e.g. nonlinear least square algorithms. Hyperboloid has a complex geometry as well as hyperboloid structures have always been interested. The two main reasons, apart from aesthetic considerations, are strength and efficiency.


Hyperboloid Structure, Design of Hyperboloid, Fitting Hyperboloid, Least Squares Problem


© 2017 by the authors. Licensee International Technology and Science Press Limited. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


[1] Andrews J.; Séquin C.H. Type-Constrained Direct Fitting of Quadric Surfaces. Computer-Aided Design & Applications, 2013, 10(a), bbb-ccc.

[2] Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987, 210-211.

[3] Bektas, S. Orthogonal distance from an ellipsoid. Boletim de Ciências Geodésicas, 2014, 20(4), DOI: .

[4] Bektas, S. Least squares fitting of ellipsoid using orthogonal distances. Boletim de Ciencias Geodesicas, 2015, 21(2), 329-339.

[5] Bektas, S. Orthogonal (shortest) distance from an hyperboloid. International Journal of Research in Engineering and Applied Sciences, 2017, 7(5), 37-45. Available online: (accessed on Jun 3, 2017).

[6] Chernov, N.; Ma, H. Least squares fitting of quadratic curves and surfaces. Computer Vision, 2011, 285-302.

[7] E.L. Hall; J.B.K. Tio; C.A. McPherson; F.A. Sadjadi. Measuring Curved Surfaces for Robot Vision. IEEE Computer, 1982, 15(12), 42-54.

[8] Eberly, D, 2008 “Least Squares Fitting of Data”,Geometric Tools,LLC,

[9] Eberly D, 2013. Distance from a Point to an Ellipse, an Ellipsoid, or a Hyperellipsoid. Geometric Tools, LLC. Bertoni B. Multi-dimensional Ellipsoidal Fitting. Preprint SMU-HEP-10-14. Available online: (accessed on 29 November 2017).

[10] Feltens, J. Vector method to compute the Cartesian (X, Y, Z) to geodetic (, λ, h) transformation on a triaxial ellipsoid. Journal of Geod. 2009, 83, 129-137.

[11] Hilbert, D.; Cohn-Vossen, S. The Second-Order Surfaces. §3 in Geometry and the Imagination. New York: Chelsea, 1999, 12-19.

[12] Ligas, M. Cartesian to geodetic coordinates conversion on a triaxial ellipsoid. Journal of Geod. 2012, 86, 249-256.

[13] Ray, A.; Srivastava D.C. Non-linear least squares ellipse fitting using the genetic algorithm with applications to strain analysis. Journal of Structural Geology, 2008, 30, 1593-1602.

[14] X. Cao, N. Shrikhande, and G. Hu, “Approximate orthogonal distance regression method for fitting quadric surfaces to range data”, Pattern Recognition Letters, Vol. 15, 1994, pp. 781-796

[15] Zhang, Z. Parameter estimation techniques: a tutorial with application to conic fitting. Image and Vision Computing, 1997, 15, 59-76.

[16] Zisserman, A. 2013 “C25 Optimization,8 Lectures” Hilary Term 2013, 2 Tutorial Sheets, Lectures 3-6 (BK)

[17] Available online: (accessed on 10 May 2017).

[18] The Hyperboloid and its Applications to Engineering. Available online: (accessed on 10 May 2017).

[19] Applications of Hyperbolas. Available online: (accessed on 10 May 2017).

[20] Available online: (accessed on 10 May 2017).

[21] Available online: (accessed on 10 May 2017).

[22] Available online: (accessed on 10 May 2017).

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