Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T07:39:48.904Z Has data issue: false hasContentIssue false
  • English
  • Français

On Pedal Quadrics in Non-Euclidean Hyperspace

Published online by Cambridge University Press:  24 October 2008

J. P. Gabbatt
J. P. Gabbatt
Affiliation:
Peterhouse.
Get access Rights & Permissions [Opens in a new window]

Extract

1. The following are well-known theorems of elementary geometry: Given any euclidean plane triangle, A0 A1 A2, and any pair of points, X, Y, isogonally conjugate q. A0 A1 A2; then the orthogonal projections of X, Y on the sides of A0 A1 A2 lie on a circle, the pedal circle of the point-pair. If either of the points X,- Y describe a (straight) line, m, then the other describes a conic circumscribing A0 A1 A2, and the pedal circle remains orthogonal to a fixed circle, J; thus the pedal circles in question are members of an ∞2 linear system of circles of which the circle J and the line at infinity constitute the Jacobian. In particular, if the line m meet Aj Ak at Lt (i, j, k = 0, 1, 2), then the circles on Ai Li as diameter, which are the pedal circles of the point-pairs Ai, Li, are coaxial; the remaining circles of the coaxial system being the director circles of the conics, inscribed in the triangle A0 A1 A2, which touch the line m. If Mi denote the orthogonal projection on m of Ai, and Ni the orthogonal projection on Aj Ak of Mi, then the three lines Mi Ni meet at a point (Neuberg's theorem), viz. the centre of the circle J. Analogues for three-dimensional space of most of these theorems are also known ‖.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 5 , July 1925 , pp. 751 - 758
Copyright
Copyright © Cambridge Philosophical Society 1925

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

* Steiner, , Annalen de Math. 19, (1828–1829), 37–64.Google Scholar

† Substantially due to Lemoyne, T., Nouv. Ann. (4) 4, (1904), 400–402.Google Scholar

‡ Bodenmiller; see Gudermann, , Grundriss d. anal. Sphärik, Köln, (1830), 138.Google Scholar

§ Neuberg, , Nouv. Correspondance math. 2, (1876), 189Google Scholar; ibid. 4, (1878), 379–382; Arch. d. Math. u. Phys. (3) 3, (1902), 89–93.Google Scholar

‖ See Servais, , Bull. Sc. Acad. r. de Belgique (5) 8, (1922), 50–66Google Scholar; Mathesis 36, 1922), 87–89.Google Scholar

Proc. Camb. Phil. Soc. 21, (1923), 340–342.Google Scholar

* Proc. Camb. Phil. Soc. 21, (1923), 763–771.Google Scholar

† Abh. säcks. Ges. Wiss. 31, (1909), 335–367.Google Scholar

* See Segre, , Encyk. d. math. Wiss. III c 7, 863–4, footnote 288.Google Scholar

* See Schlāfli, , J. f. Math. 65, (1866), 189–197Google Scholar; Berzolari, , Rend. Circ. Mat. Palermo, 20, (1905), 229–247CrossRefGoogle Scholar; Brusotti, ibid. 248–255.

† Instead of ∞n−2.

‡ The plane euclidean case of this theorem is discussed by Robinson, R. T., Math. Gazette, 12, (1925), 377–383.CrossRefGoogle Scholar