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On Pedal Quadrics in Non-Euclidean Hyperspace

Published online by Cambridge University Press:  24 October 2008

J. P. Gabbatt
Affiliation:
Peterhouse.

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
Copyright
Copyright © Cambridge Philosophical Society 1925

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References

* Steiner, , Annalen de Math. 19, (18281829), 3764.Google Scholar

Substantially due to Lemoyne, T., Nouv. Ann. (4) 4, (1904), 400402.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), 8993.Google Scholar

See Servais, , Bull. Sc. Acad. r. de Belgique (5) 8, (1922), 5066Google Scholar; Mathesis 36, 1922), 8789.Google Scholar

Proc. Camb. Phil. Soc. 21, (1923), 340342.Google Scholar

* Proc. Camb. Phil. Soc. 21, (1923), 763771.Google Scholar

Abh. säcks. Ges. Wiss. 31, (1909), 335367.Google Scholar

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

* See Schlāfli, , J. f. Math. 65, (1866), 189197Google Scholar; Berzolari, , Rend. Circ. Mat. Palermo, 20, (1905), 229247CrossRefGoogle 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), 377383.CrossRefGoogle Scholar