On Holditch’s theorem. (English) Zbl 1440.53002
Let \(C\) be a convex curve, and a chord \(h\) of length \(a+b\) be divided into parts of lengths \(a\) and \(b\) by a point \(A\). Let \(C_{a,b}\) denote a curve traced out by the point \(A\) when the chord \(h\) slides around with both endpoints on \(C\). H. Holditch [“Geometrical theorem”, Quart. J. Pure Appl. Math. 2, 38 (1858)] proved that the area of a ring domain bounded by \(C\) and \(C_{a,b}\) is equal to \(\pi ab\).
Holditch’s construction, in which one ring domain was considered, is modified by the authors. In this modification, they deal with a family of pairs of ring domains and obtain a natural geometric generalization. They derive some Crofton-type formula for a ring domain as an application.
Holditch’s construction, in which one ring domain was considered, is modified by the authors. In this modification, they deal with a family of pairs of ring domains and obtain a natural geometric generalization. They derive some Crofton-type formula for a ring domain as an application.
Reviewer: Ergin Bayram (Samsun)
MSC:
| 53A04 | Curves in Euclidean and related spaces |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
| 53C65 | Integral geometry |
References:
| [1] | Bender, W., The Holditch curve tracer, Math. Mag., 54, 128-129 (1981) · Zbl 0466.51011 · doi:10.1080/0025570X.1981.11976912 |
| [2] | Bonnesen, T.; Fenchel, W., Theorie der konvexen Körper (1948), New York: Chelsea Publishing Company, New York · JFM 60.0673.01 |
| [3] | Broman, A., Holditch’s theorem is somewhat deeper than Holditch thought in 1858, Nordisk Mathematisk Tidskrift, 27, 89-100 (1979) · Zbl 0415.53003 |
| [4] | Broman, A., Holditch’s theorem, Math. Mag., 54, 99-108 (1981) · Zbl 0468.51005 · doi:10.1080/0025570X.1981.11976908 |
| [5] | Cieślak, W., Koshi, S., Zaja̧c, J.: On integral formulas for convex domains. Acta Math. Hungar. 62, 277-283 (1993) · Zbl 0801.52003 |
| [6] | Cieślak, W.; Miernowski, A.; Mozgawa, W., Isoptics of a strictly convex curve, Lect. Notes Math., 1481, 28-35 (1991) · Zbl 0739.53001 · doi:10.1007/BFb0083625 |
| [7] | Cieślak, W.; Miernowski, A.; Mozgawa, W., Isoptics of a strictly convex curve II, Rend. Sem. Mat. Univ. Padova, 96, 37-49 (1996) · Zbl 0881.53003 |
| [8] | Cooker, MJ, An extension of Holditch’s theorem on the area within a closed curve, Math. Gaz., 82, 183-188 (1998) · doi:10.2307/3620400 |
| [9] | Cooker, MJ, On sweeping out an area, Math. Gaz., 83, 69-73 (1999) · doi:10.2307/3618685 |
| [10] | Erişir, T.; Güngör, MA; Tosun, M., A new generalization of the Steiner formula and the Holditch theorem, Adv. Appl. Clifford Algebr., 26, 1, 97-113 (2016) · Zbl 1343.53013 · doi:10.1007/s00006-015-0559-4 |
| [11] | Günş, R.; Keleş, S., Closed spherical motions and Holditch’s theorem, Mech. Mach. Theory, 29, 755-758 (1994) · doi:10.1016/0094-114X(94)90117-1 |
| [12] | Hacisalihoğlu, HH; Adelbaky, A., Holditch’s theorem for one parameter closed motions, Mech. Mach. Theory, 32, 235-239 (1997) · doi:10.1016/S0094-114X(96)00048-1 |
| [13] | Holditch, H., Geometrical theorem, Quart. J. Pure Appl. Math., 2, 38 (1858) |
| [14] | Kiliç, E.; Keleş, S., On Holditch’s theorem and polar inertia momentum, Commun. Fac. Sci. Univ. Ankara Ser. A Math. Stat., 43, 41-47 (1996) · Zbl 0868.53001 |
| [15] | Monterde, J.; Rochera, D., Holditch’s ellipse unveiled, Am. Math. Mon., 124, 5, 403-421 (2017) · Zbl 1391.52006 · doi:10.4169/amer.math.monthly.124.5.403 |
| [16] | Proppe, H.; Stancu, A.; Stern, RJ, On Holditch’s theorem and Holditch curves, J. Convex Anal., 24, 1, 239-259 (2017) · Zbl 1373.53001 |
| [17] | Santaló, L., Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and Its Applications (1976), Reading: Addison-Wesley, Reading · Zbl 0342.53049 |
| [18] | Wells, D., The Penguin Dictionary of Curious and Interesting Geometry, 103 (1991), London: Penguin, London · Zbl 0856.00005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
