Abstract
Bézier curves are a type of curves used in computer aided design and related fields. The curves can be defined with the help of De Casteljau algorithm, which is one of the most basic elements of curve and surface design, and Bernstein polynomials, which facilitate theoretical developments. A rational Bézier curve may be evaluated by applying the de Casteljau algorithm to both numerator and denominator and finally dividing through. The curves are defined by suitable control points and corresponding scalar weights. In this work, we constitute the asymptotic orthonormal frame field of a spacelike quadratic rational Bézier curve at all points on 2 and 3-dimensional lightlike cones which are degenerate surfaces in Minkowski 3 and 4-spaces. We get the formulas of curvatures for a spacelike quadratic rational Bézier curve 2 and 3-dimensional lightlike cones.
References
- [1] G. Farin, Curves and Surfaces for CAGD: a practical guide, Morgan Kaufmann Pub., USA, 2002.
- [2] H. Liu, “Curves in the Lightlike Cone,” Contributions to Algebra and Geometry, vol. 45, no. 1, pp. 291-303, 2004.
- [3] H. Liu, and M. Qingxian, “Representation Formulas of Curves in a Two-and Three-dimensional Lightlike Cone,” Results in Mathematics, vol. 59 no.3 pp. 437-451, 2011.
- [4] R. López, “Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space,” International Electronic Journal of Geometry, vol. 7, no.1, pp. 44-107, 2014.
- [5] D. Marsh, Applied Geometry for Computer Graphics and CAD, Springer Science and Business Media, USA, 2005.
- [6] B. O’Neill, Semi-Riemann Geometry with Applictions to Relativity, Academic Pres., New York, 1983.
- [7] B. O’Neill, Elementary Differential Geometry, Academic Pres., New York, 2006. [8] H.K. Samancı, “Minkowski 3-uzayında Timelike Rasyonel Bézier Eğrilerinin Eğrilikleri Üzerine,” Bitlis Üniversitesi Fen Bilimleri Dergisi, c. 7, s. 2, ss. 243-255, 2018.
- [9] H.K. Samancı, “Some Geometric Properties of the Spacelike Bézier Curve with a Timelike Principal Normal in Minkowski 3-space,” Cumhuriyet Science Journal, vol. 39, no. 1, pp. 71-79, 2018.
- [10] G. Özkan Tükel, A. Yücesan, “Elastic Curves in a Two-dimensional Lightlike Cone,” International Electronic Journal of Geometry, vol. 8, no.2, pp. 1-8, 2015.
- [11] T. Turhan, A. Yılmaz Ceylan and G. Özkan Tükel, “Rational Bézier Curves on 2-dimensional Anti de Sitter Space,” International Asian Congress on Contemporary Sciences-V, Nakhchivan, Azerbaijan, 2021, pp. 467-473.
- [12] A. Yılmaz Ceylan, T. Turhan, and G. Özkan Tükel, “On Non-null Rational Bézier Curves on 2-dimensional de Sitter Space”, 4th International Conference on Mathematics ”An İstanbul Meeting for World Mathematicians,” İstanbul, Turkey, 2020, pp. 132.
- [13] A. Yılmaz Ceylan, T. Turhan, and G. Özkan Tükel, “On the Geometry of Rational Bézıer Curves,” Honam Mathematical Journal, vol. 43, no. 1, pp. 88-99, 2021.
Abstract
Bézier eğrileri, bilgisayar destekli tasarım ve bununla ilişkili alanlarda kullanılan bir eğri türüdür. Bu eğriler eğri ve yüzey tasarımının en temel unsurlarından biri olan De Casteljau algoritması ve teorik gelişmeleri kolaylaştıran Bernstein polinomları yardımıyla tanımlanabilir. Rasyonel bir Bézier eğrisi, hem paya hem de paydaya de Casteljau algoritması uygulanarak ve son olarak bölünerek değerlendirilebilir. Bu eğriler, uygun kontrol noktaları ve karşılık gelen skaler ağırlıklarla tanımlanır. Bu çalışmada, Minkowski 3 ve 4-uzaylarında dejenere yüzeyler olan 2 ve 3-boyutlu lightlike konilerinde bir spacelike kuadratik rasyonel Bézier eğrisinin bütün noktalarında asimptotik ortonormal çatı alanını oluşturduk. Spacelike kuadratik rasyonel Bézier eğrisi 2 ve 3-boyutlu lightlike koniler için eğrilik formüllerini elde ettik.
References
- [1] G. Farin, Curves and Surfaces for CAGD: a practical guide, Morgan Kaufmann Pub., USA, 2002.
- [2] H. Liu, “Curves in the Lightlike Cone,” Contributions to Algebra and Geometry, vol. 45, no. 1, pp. 291-303, 2004.
- [3] H. Liu, and M. Qingxian, “Representation Formulas of Curves in a Two-and Three-dimensional Lightlike Cone,” Results in Mathematics, vol. 59 no.3 pp. 437-451, 2011.
- [4] R. López, “Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space,” International Electronic Journal of Geometry, vol. 7, no.1, pp. 44-107, 2014.
- [5] D. Marsh, Applied Geometry for Computer Graphics and CAD, Springer Science and Business Media, USA, 2005.
- [6] B. O’Neill, Semi-Riemann Geometry with Applictions to Relativity, Academic Pres., New York, 1983.
- [7] B. O’Neill, Elementary Differential Geometry, Academic Pres., New York, 2006. [8] H.K. Samancı, “Minkowski 3-uzayında Timelike Rasyonel Bézier Eğrilerinin Eğrilikleri Üzerine,” Bitlis Üniversitesi Fen Bilimleri Dergisi, c. 7, s. 2, ss. 243-255, 2018.
- [9] H.K. Samancı, “Some Geometric Properties of the Spacelike Bézier Curve with a Timelike Principal Normal in Minkowski 3-space,” Cumhuriyet Science Journal, vol. 39, no. 1, pp. 71-79, 2018.
- [10] G. Özkan Tükel, A. Yücesan, “Elastic Curves in a Two-dimensional Lightlike Cone,” International Electronic Journal of Geometry, vol. 8, no.2, pp. 1-8, 2015.
- [11] T. Turhan, A. Yılmaz Ceylan and G. Özkan Tükel, “Rational Bézier Curves on 2-dimensional Anti de Sitter Space,” International Asian Congress on Contemporary Sciences-V, Nakhchivan, Azerbaijan, 2021, pp. 467-473.
- [12] A. Yılmaz Ceylan, T. Turhan, and G. Özkan Tükel, “On Non-null Rational Bézier Curves on 2-dimensional de Sitter Space”, 4th International Conference on Mathematics ”An İstanbul Meeting for World Mathematicians,” İstanbul, Turkey, 2020, pp. 132.
- [13] A. Yılmaz Ceylan, T. Turhan, and G. Özkan Tükel, “On the Geometry of Rational Bézıer Curves,” Honam Mathematical Journal, vol. 43, no. 1, pp. 88-99, 2021.
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Articles |
| Authors | |
| Publication Date | December 31, 2021 |
| Published in Issue | Year 2021 |