Interpolation of Surfaces with Line of Curvature

Year 2021, , 239 - 246, 29.10.2021

Ergin Bayram

https://doi.org/10.36890/iejg.890585

Abstract

We introduce a method to construct parametric surfaces interpolating given finite points and a curve as a line of curvature in 3-dimensional Euclidean space. We present an existence theorem of a $C^{0}$-Hermite interpolation of surfaces possessing the given data. We show that every parameter curve of a constructed surface is a circular helix if the given curve is a circular helix. The method is validated with illustrative examples.

Keywords

$C^{0}$-Hermite interpolation, line of curvature, parametric surface

References

• [1] Alliez P., Cohen-Steiner D., Devillers O., Lévy B., Desbrum M. Anisotropic polygonal pemeshing. In: Proceeding of ACM SIGGRAPH.485-493 (2003).
• [2] Bayram E., Güler F., Kasap E.: Parametric representation of a surface pencil with a common asymptotic curve. Computer Aided Design. 44, 637-643 (2012).
• [3] Bayram E.: Surface pencil with a common adjoint curve. Turkish Journal of Mathematics. 44, 1649-1659 (2020).
• [4] Bayram E., Ergün E., Kasap E.: Surface family with a common natural asymptotic lift. Journal of Science and Arts. 2, 117-124 (2015).
• [5] Do Carmo M. P.: Differential geometry of curves and surfaces. Prentice-Hall Inc., Englewood Cliffs, New Jersey (1976).
• [6] Güler F., Bayram E., Kasap E.: Offset surface pencil with a common asymptotic curve. International Journal of Geometric Methods in Modern Physics. 15 (11), 1850195 (2018).
• [7] Lee C. L., Lee J. W., Yoon D.W.: Interpolation of surfaces with geodesic. Journal of the Korean Mathematical Society. 57 (4), 957-971 (2020).
• [8] Li C. Y., Wang R. H., Zhu C. G.: Parametric representation of a surface pencil with a common line of curvature. Computer Aided Design. 43, 1110-1117 (2011).
• [9] Maekawa T., Wolter F. E., Patrikalakis N. M.: Umbilics and lines of curvature for shape interrogation. Computer Aided Geometric Design. 13 (2), 133-161 (1996).
• [10] O’Neill B.: Elementary differential geometry. Elsevier Inc., (1966).
• [11] Patrikalakis N. M., Maekawa T.: Shape interrogation for computer aided design and manufacturing. Springer-Verlag, Heidelberg (2002).
• [12] Wang G.J., Tang K., Tai C.L.: Parametric representation of a surface pencil with a common spatial geodesic. Computer Aided Design. 36, 447-459 (2004).
• [13] Willmore T. J.: An introduction to differential geometry. Dover Publications (2012).