3-boyutlu Öklid uzayında Conchoid eğri ve yüzeyleri

Year 2018, Volume: 20 Issue: 2, 467 - 481, 01.12.2018

Betül Bulca S. Neslihan Oruç Kadri Arslan

https://doi.org/10.25092/baunfbed.485640

Cited By: 1

Abstract

Bu çalışmada ilk olarak düzlemde conchoid eğrileri çalışılmıştır. Conchoid eğrisinin eğriliği hesaplanıp bazı sonuçlar verilmiştir. Ayrıca 3-boyutlu Öklid uzayında conchoid eğrisiyle elde edilen dönel yüzeyler ele alınmıştır. Bu yüzeylerin Gauss ve ortalama eğrilikleri hesaplanmış, bunlarla ilgili örnekler verilip grafikleri çizdirilmiştir. Son olarak 3-boyutlu Öklid uzayında conchoidal yüzeyler üzerinde durulmuş ve conchoidal yüzeylerin flat ve minimal olma şartlarına bakılmıştır. Conchoidal yüzey örnekleri de verilip grafikleri çizdirilmiştir.

Keywords

Conchoid, Pascal Limaçonu, Gauss eğrilik, ortalama eğrilik

Conchoid curves and surfaces in Euclidean 3-Space

Abstract

In this study firstly, we study with conchoid curves in Euclidean plane E2. We calculate the curvature of the conchoid curve and give some results. Furthermore, we consider the surface of revolution given with the conchoid curve in Euclidean 3-space E3. The Gaussian and mean curvature is calculated of these surfaces. Also we give some examples and plot their graphics. Finally we study conchoidal surface in Euclidean 3-space. We give some results for the conchoidal surface to become flat and minimal. We give an example and plot the garphics of the conchoidal surfaces.

Keywords

Conchoid, Limaçons Pascal, Gaussian curvature, mean curvature

References

• Albano, A. and Roggero, M., Conchoidal transform of two plane curves, AAECC, 21, 309-328, (2010).
• Azzam, R.M.A., Limacon of Pascal locus of the complex refractive indices of interfaces with maximally flat reflectance-versus-angle curves for incident unpolarized light, J. Opt. Soc. Am. Opt. Imagen Sci. Vis., 9, 957-963, (1992).
• Dede, M., Spacelike Conchoid curves in the Minkowski plane, Balkan Journal of Mathematics, 1, 28-34, (2013).
• Glaeser, G., Stachel, H. and Odehnal, B., The Universe of Conics, Springer Spektrum, Berlin Heidelberg, (2016).
• Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, CCR Press, (1997).
• Gruber, D. and Peternell, M., Conchoid surfaces of quadrics, Journal of Symbolic Computation, 59, 36-53, (2013).
• Kang, M., Hip joint center location by fitting conchoid shape to the acetabular rim images, Conference Proceedings Engineering in Medicine and Biology Society, 6, 4477--4480, (2004).
• Kerrick, A.H., The limacon of Pascal as a basis for computed and graphic methods of determining astronomic positions, J. Inst. Navigat. 6, 5, 310-316, (1959).
• Lin., W., Yu, Z., Yuang, E.K.N., and Luk, K.M., Conchoid of Nicomedes and Limaçon of Pascal as electrode of static field and a waveguide of high frecuency wave, Prog. Electromagnet. Res. Symp. PIER, 30, 273-284, (2001).
• Lockwood, E.H., A Book of Curves, Cambrdidge University Press, (1961).
• Odehnal, B., Generalized Conchoids, KoG, 21, 35-46, (2017).
• O'Neill, B., Elementary Differential Geometry, Academic Press, USA, (1997).
• Peternell, M., Gotthart, L., Sendra, J. and Sendra, J. R., Offsets, conchoids and pedal surfaces, Journal of Geometry, 106, 321-339, (2015).
• Peternell, M., Gruber, D. and Sendra, J., Conchoid surfaces of rational ruled surfaces, Computer Aided Geometric Design, 28, 427-435, (2011).
• Peternell, M., Gruber, D. and Sendra, J., Conchoid surfaces of spheres, Computer Aided Geometric Design, 30, 35-44, (2013).
• Sendra, J. R. and Sendra, J., An algebraic analysis of conchoids to algebraic curves, AAECC, 19, 413-428, (2008).
• Sendra, J. and Sendra, J.R., Rational parametrization of conchoids to algebraic curves, AAECC, 21, 285-308, (2010).
• Sultan, A., The Limaçon of Pascal: Mechanical Generating Fluid Processing, J. of Mechanical Engineering Science, 219, 8, 813-822, (2005).
• Szmulowicz, F., Conchoid of Nicomedes from reflections and refractions in a cone, Am. J. Phys., 64, 467-471, (1996).