Bézier Curve with a Minimal Jerk Energy

Year 2016, Volume: 4 Issue: 2, 139 - 148, 30.10.2016

Hale Erişkin Ahmet Yücesan

https://doi.org/10.36753/mathenot.421467

Cited By: 11

https://izlik.org/JA35TH87LZ

Abstract

We provide a method in order to determine a Bézier curve with a minimal jerk energy by means of
associated matrices. By way of an application, we show that the unknown control points of the Bézier
curve having a minimal jerk energy can be written as a linear combination of the known control points.
Furthermore, for such a Bézier curve we obtain a general form of its matrix represention.

Keywords

Bézier curve , minimal jerk energy , , control points

References

• [1] Ahn, Y. J., Hoffmann, C., Rosen, P., Geometric constraints on quadratic Bézier curves using minimal length and energy. J. Comput. Appl. Math. 255(2014), 887-897.
• [2] Brunnett, G., Hagen, H., Santarelli, P., Variational design of curves and surfaces. Surv. Math. Indust. 3(1993), no. 3, 1-27.
• [3] Brunnett, G., Kiefer, J., Interpolation with minimal-energy splines. Comput. Aided Design 26(1994), no.2, 137-144.
• [4] Eberly, D., A relationship between minimum bending energy and degree elevation for Bézier curves. http://www.geometrictools.com/Documentation/BézierCurveBendingElevation.pdf
• [5] Farin, G., Curves and surfaces for CAGD: A Practical Guide, fifth ed. Morgan Kaufmann, San Francisco, 2002.
• [6] Farin, G., Class a Bézier curves. Comput. Aided Geom. Design 23(2006), no.7, 573-581.
• [7] Gravesen, J., Differential geometry and design of shape and motion. http://www2.mat.dtu.dk/people/J.Gravesen/cagd.pdf
• [8] Hagen, H., Bézier-curves with curvature and torsion continuity. Rocky Mtn. J. of Math. 16(1986), no.3, 629-638.
• [9] Meier, H., Nowacki, H., Interpolating curves with gradual changes in curvature. Comput. Aided Geom. Design 4(1987), no.4, 297-305.
• [10] Moreton, H. P., Séquin, C. H., Minimum variation curves and surfaces for computer aided geometric design. In: Designing Fair Curves and Surfaces-Shape Quality in Geometric Modeling and Computer Aided Design. SIAM, Philadelphia, USA, 1994.
• [11] Roulier, J., Bézier curves of positive curvature. Comput. Aided Geom. Design 5(1988), no.1, 59-70.
• [12] Saxena, A., Sahay, B., Computer aided engineering design. Anamaya Publishers, 2005.
• [13] Tawiwat, V., Jumnong, P., Combining minimum energy and minimum direct jerk of linear dynamic systems. World Academy of Science, Engineering and Technology, 47(2008), 252-257.
• [14] Veltkamp, R. C., Wesselink, W., Modeling 3D curves of minimal energy. In: Eurographics 95, Maastricht, the Netherlands, 1995, 97-110.
• [15] Weinstock, R., Calculus of variations with applications to physics&engineering. Dover Publications, Inc 1974.
• [16] Xu, G., Wang, G., Chen, W., Geometric construction of energy-minimizing Bézier curves. Sci. China Inf. Sci. 54(2011), no. 7, 1395-1406.
• [17] Yong, J. H., Cheng, F., Geometric Hermite curves with minimum strain energy. Comput. Aided Geom. Design 21(2004), no.3, 281-301.
• [18] Zhang, C. M., Zhang, P. F., Cheng, F., Fairing spline curves and surfaces by minimizing energy. Comput. Aided Design 33(2001), no.13, 913-923.