Review:

# Aesthetic Curves and Surfaces in Computer Aided Geometric Design

## Kenjiro T. Miura^{*} and R. U. Gobithaasan^{**}

^{*}Shizuoka University, 3-5-1 Jouhoku, Naka-ku, Hamamatsu, Shizuoka 432-8561, Japan

^{**}University Malaysia Terengganu, 21030 Kuala Terengganu, Malaysia

*Int. J. Automation Technol.*, Vol.8 No.3, pp. 304-316, 2014.

- [1] G. Farin, “Curves and Surfaces for CAGD: A Practical Guide,” Morgan Kaufmann, 2002.
- [2] “Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-Aided Design,” N. S. Sapidis (Ed.), SIAM, 1994.
- [3] R. Levien and C. H. Sequin, “Interpolating splines: Which is the fairest of them all?” Computer-Aided Design and Applications, Vol.6, Issue 1, pp. 91-102, 2009.
- [4] R. U. Gobithaasan and K. T. Miura, “Aesthetic Spiral for Design,” Sains Malaysiana, Vol.40, No.11, pp. 1301-1305, 2011.
- [5] J. M. Ali, R. M. Tookey, J. V. Ball, and A. A. Ball, “The Generalized Cornu Spiral and its Application to Span Generation,” J. of Computational and Applied Mathematics, Vol.102, Issue 1, pp. 37-47, 1999.
- [6] R. U. Gobithaasan, J. M. Ali, and K. T. Miura, “The Elucidation of Planar Aesthetic Curves,” 17th Int. Conf. in Central Europe on Computer graphics, Visualization and Computer Vision, WSCG2009, 138-188.
- [7] K. T. Miura, R. Shirahata, S. Agari, S. Usuki, and R. U. Gobithaasan, “Variational Formulation of the Log-aesthetic Surface and Development of Discrete Durface Filters,” Computer-Aided Design and Applications, Vol.9, Issue 6, pp. 901-914, 2012.
- [8] R. Ziatdinov, N. Yoshida, T. Kim, “Analytic Parametric Equations of Log-aesthetic Curves in Terms of Incomplete Gamma Functions,” Computer Aided Geometric Design, Vol.29, Issue 2, pp. 129-140, 2012.
- [9] D. S. Meek, T. Saito, D. J. Walton, and Y. Yoshida, “Planar Twopoint G1 Hermite Interpolating Log-aesthetic Spirals,” J. of Computational and Applied Mathematics, Vol.236, Issue 17, pp. 4485-4493, 2012.
- [10] P. Joshi and C. Séquin, “Energy Minimizers for Curvature-based Surface Functionals,” Computer Aided Design & Applications, Vol.4, Issue 5, pp. 607-617, 2007.
- [11] H. Moreton and C. Séquin, “Functional Optimization for Fair Surface Design,” Proc. SIGGRAPH’92, 1992.
- [12] C. Séquin, P. Chang, and H. Moreton, “Scale-invariant Functionals for Smooth Curves and Surfaces,” Dagstuhl Seminar on Geometric Modeling, 1993.
- [13] M. Higashi, K. Kaneko, and M. Hosaka, “Generation of High Quality Curve and Surface with Smoothing Varying Curvature,” Eurographics’ 88, pp. 79-92, 1988.
- [14] Y. Mineur, T. Lichah, J. M. Castelain, and H. Giaume, “A Shape Controlled Fitting Method for Bézier Curves,” Computer Aided Geometric Design, Vol.15, Issue 9, pp. 879-891, 1998.
- [15] M. Higashi, I. Kohzen, and J. Nagasaka, “An Interactive CAD System for Construction of Shapes with High-Quality Surface,” IFIP’83, North-Holland, pp. 371-389, 1983.
- [16] G. Farin, “Class A Bezier Curves,” Computer Aided Geometric Design, Vol.23, Issue 7, pp. 573-581, 2006.
- [17] J. Cao and G. Wang, “A Note on Class A Bzier curves,” Computer Aided Geometric Design, Vol.25, Issue 7, pp. 523-528, 2008.
- [18] T. Oya, E. Hanafusa, H. Amemiya, and H. Aoyama, “Application of Class A Curves and Surfaces to Aesthetic Design,” 2013 Japan Society of Precision Engineering Autumn Meeing, 2013.
- [19] M. Livio, “The Golden Ratio, Broadway Books, New York, 2002.
- [20] T. Takanashi, “Aesthetic Design Methodology,” David press, Tokyo, 2002. (in Japanese)
- [21] H. Makino, “Clothoidal Interpolation – A New Tool for High-speed Continuous Path Control,” Annals of the CIRP, Vol.37, Issue 1, pp. 25-28, 1988.
- [22] D. S. Meek and D. J.Walton, “The Use of Cornu Spirals in Drawing Planar Curves of Controlled Curvature,” J. of Computational and Applied Mathematics, Vol.25, pp. 69-78, 1989.
- [23] D. J.Walton and D. S. Meek, “
*G*^{1}interpolation with a single Cornu spiral segment,” J. of Computational and Applied Mathematics, Vol.223, pp. 86-96, 2009. - [24] D. S. MeeK and R. S. D. Thomas, “A Guided Clothoid Spline,” Computer Aided Geometric Design, Vol.8, Issue 2, pp. 163-174, 1991.
- [25] G. Li, X. Li, and H. Li, “3D Discrete Clothoid Splines,” CGI’01, 321, 2001.
- [26] G. Li, X. Li, and H. Li, “Discrete Clothoid Spline Surfaces on Open Meshes,” Int. J. of Image and Graphics, Vol.1, No.4, 575, 2001.
- [27] R. J. Cripps, “Algorithms to Support Point-based Cadcam,” Int. J. of Machine Tools and Manufacture, Vol.43, Issue 4, pp. 425-432, 2003.
- [28] K. T. Miura, “Unit Quaternion Integral Curve: A New Type of Fair Free-form Curve,” Computer Aided Geometric Design, Vol.17, Issue 1, pp. 39-58, 2000.
- [29] K. T. Miura, “Unit Quaternion Integral Surface,” Transaction of IPSJ, 41, 3, 39-58, 2000. (in Japanese)
- [30] T. Harada, N. Mori, and K. Sugiyama, “Curves’ physical characteristics and self-affine properties,” Bulletin of Japanese Society for the Science of Design, Vol.42, Issue 3, pp. 30-40, 1995. (in Japanese)
- [31] K. T. Miura, J. Sone, A. Yamashita, and T. Kaneko, “Derivation of a general formula of aesthetic curves,” In Proc. of the Eighth Int. Conf. on Humans and Computers (HC2005), pp. 166-171, 2005.
- [32] K. T. Miura, “A General Equation of Aesthetic Curves and its Selfaffinity,” Computer-Aided Design & Applications, Vol.3, Issues 1-4, pp. 457-464, 2006.
- [33] N. Yoshida and T. Saito, “Interactive aesthetic curve segments,” The Visual Computer (Pacific Graphics), Vol.22, Issues 9-11, pp. 896-905, 2006.
- [34] K. T. Miura, M. Fujisawa, J. Sone, and K. G. Kobayashi, “The Aesthetic Space Curve,” Humans and Computers, 101-106, 2006.
- [35] N. Yoshida and T. Saito, “Classification of Aesthetic Space Curves,” SIAM Conf. on Geometric Design and Computing, 2007.
- [36] K. T. Miura, S. Agari, Y. Kawata, M. Fujisawa, and F. Cheng, “Input of log-aesthetic curve segments with inflection end points and generation of log-aesthetic curves with G2 continuity,” Computer-Aided Design & Applications, Vol.5, Issues 1-4, pp. 77-85, 2008.
- [37] S. Agari, K. T. Miura, M. Fujisawa, T. Nisikawa, and T. Hada, “Input of the Compound-rhythm Log-aesthetic curve and its Applications for styling Design,” J. of Japan Society of Mechanical Engineers, Vol.75, No.756, pp. 2159-2164, 2009. (in Japanese)
- [38] R. U. Gobithaasan, R. Karpagavalli, and K. T. Miura, “Shape Analysis of Generalized Log-Aesthetic Curves,” Int. J ofMath. Analysis, Vol.7, No.36, pp. 1751-1759, 2013.
- [39] R. U. Gobithaasan, R. Karpagavalli, and K. T. Miura, “Drawable Region of the Generalized Log Aesthetic Curves,” J. of Applied Mathematics, Vol.2013, Article ID 732457, 7 pages, 2013.
- [40] R. U. Gobithaasan, L. P. Yee, and K. T. Miura, “A Generalized Log Aesthetic Space Curve,” ACM Int. Conf. Proc. Series,145-149, 2012.
- [41] I. Kanaya, Y. Nakano, and K. Sato, “Simulated Designer’s Eyes – Classification of Aesthetic Surfaces –,” Proc. VSMM, 289-296, 2003.
- [42] A. Gray, E. Abbena, and S. Salamon, “Modern Differential Geometry of Curves and Surfaces with Mathematica,” Chapman & Hall, 2006.
- [43] T. Takagi, “Mathematics on Shapes,” Asakura Shoten, Tokyo, 1992. (in Japanese)
- [44] B. B. Mandelbrot, “The Fractal Geometry of Nature,” W.H. Freeman and Company, 1983.
- [45] K. T. Miura, S. Agari, T. Akie, N. Yoshida, and T. Saito, “Discrete Log-aesthetic Filter,” Computer-Aided Design & Applications, Vol.6, Issue 4, pp. 501-512, 2009.
- [46] F. Lan, H. Tamai, and H. Makino, “Interpolation of Arbitrary Point Sequence by Triple Clothoid Curves,” J. of the Japan Society for Precision Engineering, Vol.76, No.10, pp. 1194-1199, 2010, (in Japanese).
- [47] K. T. Miura, M. Yagi, Y. Kawata, and M. Fujisawa, “Input of aesthetic curve segments with inflection end points and generation of aesthetic curves with G2 continuity,” Proc. Graphics and CAD/Visual Computing Joint Symp. 2007, 297-302. (in Japanese)
- [48] T. Harada, F. Yoshimoto, and M. Moriyama, “An Aesthetic Curve in the Field of Industrial Design,” Procs. IEEE Symp. on Visual Language 1999, 38-47.
- [49] K. T. Miura, D. Shibuya, R. U. Gobithaasan, and S. Usuki, “Designing Log-aesthetic Splines with
*G*^{2}Continuity,” Computer Aided Design & Application, Vol.10, Issue 6, pp. 1021-1032, 2013. - [50] T. Hagiwara and T. Harada, “An Algorithm of a Log-aesthetic Curved Surface Generation and a Development of the Curved Surfaces Generation System with VR,” Graphics and CAD, No.2009-CG-134, 13-16, 2009. (in Japanese)
- [51] M. do Carmo, “Differential Geometry of Curves and Surfaces,” Prentice hall, Englewood Cliffs, 1976.
- [52] R. Schneider and L. Kobbelt, “Geometric Fairing of Irregular Meshes for Free-form Surface Design,” Computer Aided Geometric Design, Vol.18, Issue 4, pp. 359-379, 2001.
- [53] K. T. Miura, S. Usuki, and R. U. Gobithaasan, “Variational Formulation of the Log-Aesthetic Curve,” 14th Int. Conf. on Humans and Computers, Mar. 9-10, 2012, pp. 215-219.
- [54] K. Kenmotsu, “Surfaces with Constant Mean Curvature,” American Mathematical Society, 2003.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.