Double and Type (3,0) Minkowski Pythagorean Hodograph Curves

Abstract

In present paper, Double Minkowski Pythagorean Hodograph (DMPH) curves and type (3,0) Minkowski Pythagorean Hodograph (MPH) curves. Firstly, we obtained the conditions for a MPH curve to be a DMPH curve. Then, we examined these conditions in split quaternion form. Finally, a special class of seventh degree MPH curves is characterized and illustrative examples are given.

Keywords

• Double Minkowski Pythagorean hodograph curve
• Minkowski-Hopf map
• Type (n,m) curve

Double and Type (3,0) Minkowski Pythagorean Hodograph Curves

Abstract

In present paper, Double Minkowski Pythagorean Hodograph (DMPH) curves and type (3,0) Minkowski Pythagorean Hodograph (MPH) curves are studied. Firstly, the conditions for a MPH curve to be a DMPH curve are obtained. Then, these conditions are examined in split quaternion form. Finally, a special class of seventh degree MPH curves is characterized and illustrative examples are given.

Keywords

• Double Minkowski Pythagorean Hodograph Curve
• Minkowski-Hopf Map
• Type (n,m) Curve

References

1. Choi, H. I., Lee, D. S., Moon, H. P. 2002. Clifford Algebra, Spin Representation and Rational Parameterization of Curves and Surfaces. Advances in Computational Mathematics, 17(1-2), 5-48.
2. Dospra, P. 2015. Quaternion Polynomials and Rational Rotation-Minimizing Frame Curves. Ph.D. Thesis, Agricultural University of Athens, 2015.
3. Farouki, R. T., Sakkalis, T. 1990. Pythagorean Hodographs. IBM Journal of Research and Development, 34(5), 736-752.
4. Farouki, R. T. 1994. The Conformal Map z→z² of the Hodograph Plane. Computer Aided Geometric Design, 11(4), 363-390.
5. Farouki, R. T., Sakkalis, T. 1994. Pythagorean-Hodograph Space Curves. Advances in Computational Mathematics, 2(1), 41-66.
6. Farouki, R. T. 2008. Pythagorean-Hodograph Curves. Springer.
7. Han, C. Y. 2008. Nonexistence of Rational Rotation-Minimizing Frames on Cubic Curves. Computer Aided Geometric Design, 25(4-5), 298-304
8. Inoguchi, J. I. 1998. Timelike Surfaces of Constant Mean Curvature in Minkowski 3-Space, Tokyo Journal of Mathematics, 21(1), 141-152.

9. Ramis, Ç. 2013. PH Curves and Applications. M.S. Thesis, Ankara University.
10. Yazla, A., Sariaydin, M. T. 2019. Applications of the Fermi-Walker derivative. Journal of Science and Arts, 19(3), 545-560.
11. Yazla, A., Sariaydin, M. T. 2020. On Surfaces Constructed by Evolution According to Quasi Frame. Facta Universitatis, Series: Mathematics and Informatics, 605-619.
12. Yazla, A., Sariaydin, M. T. Modeling with Double Minkowski Pythagorean Hodograph Curves. (Submitted)