Shape Operator via Darboux Frame Curvatures and Its Applications

Yıl 2019, Cilt: 12 Sayı: 2, 182 - 187, 03.10.2019

Bahar Uyar Düldül Mustfa Düldül

https://doi.org/10.36890/iejg.628075

Öz

We present the shape operator’s matrix of a surface along a surface curve. By using the obtained

matrix, we give a short proof of the Beltrami-Enneper theorem. Also, we give a new method for 

determining the well-known geodesic curves of a plane, a sphere and a circular cylinder by using 

the relation between the ordinary curvatures of the geodesic and the curvatures of the surface.

Anahtar Kelimeler

Shape operator, Darboux frame, curvature, geodesic curve

Kaynakça

• [1] Gray, A., Abbena, E. and Salamon, S., Modern differential geometry of curves and surfaces with Mathematica, 3rd Edition, Chapman Hall CRC, 2006.
• [2] O’Neill, B., Elementary Differential Geometry, Academic Press, 1966.
• [3] Uyar Düldül, B. and Çalışkan, M., On the geodesic torsion of a tangential intersection curve of two surfaces in R3. Acta Math. Univ. Comenian. 82 (2013), 177-189.
• [4] Ye, X. and Maekawa, T., Differential geometry of intersection curves of two surfaces. Comput. Aided Geom. Design 16 (1999), 767-788.