TY  - JOUR
T1  - On The Some Special Biquadratic Bezier Surfaces in E^3
TT  - E^3'te Bazı Özel Bikuadratik Bezier Yüzeyleri Üzerine
AU  - Yurttançıkmaz, Semra
AU  - Kılıçoglu, Şeyda
PY  - 2025
DA  - September
Y2  - 2025
DO  - 10.21597/jist.1643500
JF  - Journal of the Institute of Science and Technology
JO  - J. Inst. Sci. and Tech.
PB  - Iğdır Üniversitesi
WT  - DergiPark
SN  - 2536-4618
SP  - 1110
EP  - 1123
VL  - 15
IS  - 3
LA  - en
AB  - In this study, biquadratic Bezier surfaces were investigated. These surfaces have a very important place in the scientific world. The importance of biquadratic Bézier surfaces lies in their ability to provide a smooth, flexible, and computationally efficient method for modeling surfaces. They fill the gap between simpler bilinear surfaces and more complex bicubic surfaces, offering a great solution in areas where both simplicity and precision are needed. While not as commonly used in high-end 3D modeling as their higher-order counterparts, they remain a useful tool for surface design, particularly when the trade-off between quality and computational efficiency is critical. In this study, firstly the matrix representations of biquadratic Bezier surfaces are given. Then, it is stated how to find the control points of some special surface types given as biquadratic Bezier surfaces, and the control points of parabolic cylinder, elliptic paraboloid and hyperbolic paraboloid surfaces are calculated.
KW  - Biquadratic bezier Surfaces
KW  - Elliptic paraboloid
KW  - Hyperbolic Paraboloid
KW  - Parabolic cylinder
N2  - Bu çalışmada bikuadratik Bezier yüzeyleri incelenmiştir. Bu yüzeyler bilim dünyasında çok önemli bir yere sahiptir. Bikuadratik Bézier yüzeylerinin önemi, yüzeyleri modellemek için düzgün, esnek ve hesaplama açısından verimli bir yöntem sağlama yeteneklerinde yatmaktadır. Daha basit bilineer yüzeyler ile daha karmaşık bikübik yüzeyler arasındaki boşluğu doldururlar ve hem basitliğin hem de hassasiyetin gerekli olduğu alanlarda harika bir çözüm sunarlar. Üst düzey 3B modellemede yüksek mertebeden benzerleri kadar yaygın olarak kullanılmasalar da, özellikle kalite ve hesaplama verimliliği arasındaki denge kritik olduğunda, yüzey tasarımı için yararlı bir araç olmaya devam etmektedirler. Bu çalışmada, öncelikle bikuadratik Bezier yüzeylerinin matris gösterimleri verilmiştir. Daha sonra, bikuadratik Bezier yüzeyleri olarak verilen bazı özel yüzey tiplerinin kontrol noktalarının nasıl bulunacağı belirtilmiş ve parabolik silindir, eliptik paraboloid ve hiperbolik paraboloid yüzeylerinin kontrol noktalarının nasıl hesaplandığı açıklanmıştır.
CR  - Anonymous. 2013. Practical Guide to Bezier Surfaces. URL: https://www.gamedev.net/tutorials/programming/math-and-physics/practical-guide-to-bezier-surfaces-r3170/.
CR  - Farin, G. (2002). Curves and Surfaces for CAGD. (5th ed.). Academic Press.
CR  - Hagen, H. (1986). Bezier-curves with curvature and torsion continuity. Rocky Mountain J. Math., 16(3), 629-638.
CR  - Han, Xi-An, YiChen Ma, & XiLi Huang. (2008). A novel generalization of Bézier curve and surface. Journal of Computational and Applied Mathematics, 217(1), 180-193.
CR  - Hu, Gang, Junli Wu, & Xinqiang Qin. (2018). A new approach in designing of local controlled developable H-Bézier surfaces. Advances in Engineering Software, 121, 26-38.
CR  - Kılıçoğlu, Ş. & Şenyurt, S. (2022). On the Matrix Representation of 5th order Bézier Curve and derivatives in E³. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistic, 71(1), 133-152.
CR  - Kılıçoğlu, Ş. (2023). On Approximation of Helix by 3rd, 5th and 7th Order Bézier curves in E³. Thermal Science, 26(2), 525-538.
CR  - Kılıçoğlu, Ş. (2023). On approximation sine wave with the 5^{th} and 7^{th} order Bézier paths in E². Thermal Science, 26(2), 539-550.
CR  - Kılıçoğlu, Ş. & Yurttançıkmaz S. (2023). How to approximate cosine curve with 4^{th} and 6^{th} order Bézier curve in plane?. Thermal Science, 26(2), 559-570.
CR  - Michael, S. (2003). Bezier curves and surfaces. Lecture 8, Floater, Oslo.
CR  - Sederberg, T.W., Anderson, D.C. & Goldman, R.N. (1984). Implicit representation of parametric curves and surfaces. Computer Vision, Graphics, and Image Processing, 28(1), 72-84.
CR  - Weir, M.D.,  Hass, J. & Giordiano F.R. (2005). Thomas' Calculus. (11th ed.). Pearson Addison Wesley.
CR  - Weisstein, E.W. (2008). Hyperbolic Paraboloid. URL: http://mathworld.wolfram.com/HyperbolicParaboloid html.
CR  - Zhang, Jiwen. (1999). C-Bézier curves and surfaces. Graphical Models and Image Processing, 61(1), 2-15.
CR  - Zhang, H. & Jieqing, F. (2006). Bezier Curves and Surfaces (2). State Key Lab of CAD&CG Zhejiang University.
CR  - Zheng, J., Sederberg, T.W. (2003). Gaussian and mean curvature of rational Bézier patches. Computer Aided Geometric Design, 2(6), 297-301.
UR  - https://doi.org/10.21597/jist.1643500
L1  - https://dergipark.org.tr/tr/download/article-file/4625817
ER  -