Characterizations of a helicoid and a catenoid
In the present article, we consider a parametric surface generated by the Frenet frame of a curve, and study the minimality condition for the surface. As a result, we give characterizations of a helicoid and a catenoid. Finally we show some examples of minimal surfaces generated by a circle and a helix.
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- [1] Y.-X. Hao, R.-H. Wang and C.-J. Li, Minimal quasi-Bézier surface, Appl. Math.
Model. 36, 5751–5757, 2012.
- [2] E. Kasap and F.T. Akyildiz, Surfaces with common geodesic in Minkowski 3-space,
Appl. Math. Comput. 177, 260–270, 2006.
- [3] E. Kasap, F.T. Akyildiz and K. Orbay, A generalization of surfaces family with common spatial geodesic, Appl. Math. Comput. 201, 781–789, 2008.
- [4] C.-Y. Li, R.-H. Wang and C.-G. Zhu, Designing approximation minimal parametric
surfaces with geodesics Appl. Math. Model. 37, 6415–6424, 2013.
- [5] C.M.C. Riverros and A.M.V. Corro, Geodesics in minimal surfaces, Math. Notes,
101, 497–514, 2017.
- [6] J. Sánchez-Reyes, On the construction of minimal surfaces from geodesics, Appl.
Math. Model. 40, 1676–1682, 2016.
- [7] J. Sánchez-Reyes and R. Dorado, Constrained design of polynomial surfaces from
geodesic curves, Comput. Aided Design. 40, 49–55, 2008.
- [8] G.-J. Wang, K. Tang and C.-L. Tai, Parametric representation of a surface pencil
with a common spatial geodesic, Comput. Aided Design. 36, 447–459, 2004.
- [9] G. Xu and G.-Z. Wang, Quintic parametric polynomial minimal surfaces and thier
properties, Diff. Geom. Appl. 28, 697–704, 2010.
- [10] D.W. Yoon, Approximation of minimal surfaces in Minkowski 3-space, preprint.