Abstract
Holditch’s theorem is an old result on the area generated by a moving chord for closed planar curves. Some generalizations of this result have been given before, but none of these follows the same natural construction of the plane but done in the space. In this work, the notion of Holditch surface is defined, some properties of these surfaces are proved and they are used to generalize Holditch’s theorem for closed space curves naturally. Moreover, an approximation for the area of interest is given. Finally, it is showed that the only minimal non-planar Holditch surface is the helicoid.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnol’d, V.I.: The geometry of spherical curves and quaternion algebra. Russ. Math. Surv. 50(1), 1–68 (1995). https://doi.org/10.1070/RM1995v050n01ABEH001662
Broman, A.: Holditch’s theorem. Math. Mag. 54(3), 99–108 (1981). https://doi.org/10.2307/2689793
do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood Cliffs, N.J. (1976). Translated from the Portuguese
Hacar Benítez, M.A.: Numerosas aplicaciones de un teorema olvidado de geometría. Revista de Obras Públicas 127(3180), 415–428 (1980)
Holditch, H.: Geometrical theorem. Q. J. Pure Appl. Math. 2, 38 (1858)
Monterde, J., Rochera, D.: Holditch’s ellipse unveiled. Am. Math. Monthly 124(5), 403–421 (2017). https://doi.org/10.4169/amer.math.monthly.124.5.403
Pottmann, H.: Holditch–Sicheln. Arch. Math. (Basel) 44(4), 373–378 (1985). https://doi.org/10.1007/BF01235783
Pottmann, H.: Zum Satz von Holditch in der euklidischen Ebene. Elem. Math. 41(1), 1–6 (1986)
Santaló, L.A.: Area bounded by the curve generated by the end of a segment whose other end traces a fixed curve, and application to the derivation of some theorems on ovals (in Spanish). Math. Notae 4, 213–226 (1944)
Vidal Abascal, E.: Area generated on a surface by an arc of a geodesic when one of its ends describes a fixed curve and length of the curve described by the other end (in Spanish). Revista Mat. Hisp. Am. (4) 7, 132–142 (1947)
Acknowledgements
We sincerely wish to thank the referee, whose detailed and clear comments have been very useful to improve the layout and the writing of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work has been partially supported by the Spanish Ministry of Economy, Industry and Competitiveness with Grant MTM2015-64013-P and by the Generalitat Valenciana (and ESF) under the VALi+d/2016/392 Grant.
Rights and permissions
About this article
Cite this article
Monterde, J., Rochera, D. Holditch’s Theorem in 3D Space. Results Math 74, 110 (2019). https://doi.org/10.1007/s00025-019-1035-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-019-1035-6