Research Article
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Year 2016, Volume: 4 Issue: 2, 139 - 148, 30.10.2016
https://doi.org/10.36753/mathenot.421467

Abstract

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.

Bézier Curve with a Minimal Jerk Energy

Year 2016, Volume: 4 Issue: 2, 139 - 148, 30.10.2016
https://doi.org/10.36753/mathenot.421467

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.

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.
There are 18 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Hale Erişkin This is me

Ahmet Yücesan

Publication Date October 30, 2016
Submission Date October 3, 2015
Published in Issue Year 2016 Volume: 4 Issue: 2

Cite

APA Erişkin, H., & Yücesan, A. (2016). Bézier Curve with a Minimal Jerk Energy. Mathematical Sciences and Applications E-Notes, 4(2), 139-148. https://doi.org/10.36753/mathenot.421467
AMA Erişkin H, Yücesan A. Bézier Curve with a Minimal Jerk Energy. Math. Sci. Appl. E-Notes. October 2016;4(2):139-148. doi:10.36753/mathenot.421467
Chicago Erişkin, Hale, and Ahmet Yücesan. “Bézier Curve With a Minimal Jerk Energy”. Mathematical Sciences and Applications E-Notes 4, no. 2 (October 2016): 139-48. https://doi.org/10.36753/mathenot.421467.
EndNote Erişkin H, Yücesan A (October 1, 2016) Bézier Curve with a Minimal Jerk Energy. Mathematical Sciences and Applications E-Notes 4 2 139–148.
IEEE H. Erişkin and A. Yücesan, “Bézier Curve with a Minimal Jerk Energy”, Math. Sci. Appl. E-Notes, vol. 4, no. 2, pp. 139–148, 2016, doi: 10.36753/mathenot.421467.
ISNAD Erişkin, Hale - Yücesan, Ahmet. “Bézier Curve With a Minimal Jerk Energy”. Mathematical Sciences and Applications E-Notes 4/2 (October 2016), 139-148. https://doi.org/10.36753/mathenot.421467.
JAMA Erişkin H, Yücesan A. Bézier Curve with a Minimal Jerk Energy. Math. Sci. Appl. E-Notes. 2016;4:139–148.
MLA Erişkin, Hale and Ahmet Yücesan. “Bézier Curve With a Minimal Jerk Energy”. Mathematical Sciences and Applications E-Notes, vol. 4, no. 2, 2016, pp. 139-48, doi:10.36753/mathenot.421467.
Vancouver Erişkin H, Yücesan A. Bézier Curve with a Minimal Jerk Energy. Math. Sci. Appl. E-Notes. 2016;4(2):139-48.

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