Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T13:38:29.467Z Has data issue: false hasContentIssue false
  • English
  • Français

Desargues configurations and their collineation groups

Published online by Cambridge University Press:  24 October 2008

H. S. M. Coxeter
H. S. M. Coxeter
Affiliation:
University of Toronto
Get access Rights & Permissions [Opens in a new window]

Extract

When two triangles, ABC and A′B′C′, are perspective from a point 0, their pairs of corresponding sides meet on a line 0, the axis of perspective. The line OA passes through A′ and some point on 0. These four points have a certain cross ratio which is the same if B or C is used instead of A. The reciprocal cross ratio arises if A′B′C′ is regarded as the first triangle and ABC the second. The complete figure contains ten pairs of perspective triangles, yielding twenty cross ratios. In section 6 a technique, suggested by D. W. Babbage and John Rigby, is used to express these cross ratios in the form

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 2 , September 1975 , pp. 227 - 246
Copyright
Copyright © Cambridge Philosophical Society 1975

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

REFERENCES

(1)Baser, H. F.Principles of geometry, I (Cambridge, 1929).Google Scholar
(2)Burnside, W.Theory of groups of finite order (2nd ed., Cambridge, 1911).Google Scholar
(3)Cayley, Arthur, Sur quelques théorèmes de la géométrie de position. J. Reine und Angew. Math. 31 (1846), 213–226.Google Scholar
(4)Coxeter, H. S. M.The map-colouring of unorientable surfaces, Duke Math. J., 10 (1943), 293–304.CrossRefGoogle Scholar
(5)Coxeter, H. S. M.Configurations and maps, Reports of a Mathematical Colloquium, (2), 8 (1948), 18–38.Google Scholar
(6)Coxeter, H. S. M.Self-dual configurations and regular graphs, Bull. Amer. Math. Soc., 56 (1950), 413–455.CrossRefGoogle Scholar
(7)Coxeter, H. S. M.Non-Euclidean geometry (5th ed., Toronto, 1965).CrossRefGoogle Scholar
(8)Coxeter, H. S. M.Twelve geometric essays (Carbondale, III., 1968).Google Scholar
(9)Coxeter, H. S. M.Introduction to geometry (2nd ed., New York, 1969).Google Scholar
(10)Coxeter, H. S. M.Projective geometry (2nd ed., Toronto, 1973).Google Scholar
(11)Dorwart, H. L.The geometry of incidence (Prentice-Hall, 1966).Google Scholar
(12)Edge, W. L.31-point geometry. Math. Gaz. 39 (1955), 113–121.CrossRefGoogle Scholar
(13)Fano, Gino. Sui postulati fondamentali della geometria proiettiva, 30 (1892), 106–132.Google Scholar
(14)Graves, J. T.On the functional symmetry exhibited in the notation of certain geometrical porisms, when they are stated merely with reference to arrangements of points, Philos. Mag. (3), 15 (1893), 129–136.Google Scholar
(15)Heyting, Arend. Axiomatic projective geometry (Groningen, 1963).Google Scholar
(16)Poncelet, J. V.Traité des propriétés projectives des figures, I, II (2nd ed., Paris, 1865, 1866).Google Scholar
(17)Von Staudt, K. G. C.Geometrie der Lage (Nürnberg, 1847).Google Scholar
(18)Veblen, Oswald and Young, J. W.Projective geometry, I (New York, 1938).Google Scholar