Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-12-01T01:11:35.894Z Has data issue: false hasContentIssue false
  • English
  • Français

On infinitesimal triangles

Published online by Cambridge University Press:  26 February 2010

J. A. Tyrrell
J. A. Tyrrell
Affiliation:
King's College, London.
Get access

Extract

An ordered triangle in the plane S2 is defined as a sextuple (PlP2, P3; l1, l2, l3) consisting of three points Pi and three lines lj restricted by the relations of incidence Pi Ì lj (i ¹ j) If we map unrestricted sextuples by the points of a V12—Segre product of six planes—we obtain an image-manifold Ω6 for ordered triangles as the appropriate subvariety of V12. The variety Ω6 possesses an ordinary double threefold ɸ3 whose points map the totally degenerate triangles (i.e. those for which P1 ═ P2 ═ P3 and l1 ═ l2 ═ l3); Ω6 is therefore unsuitable as a basis for the construction of an enumerative calculus for triangles, for equivalence theory is as yet developed satisfactorily only on non-singular varieties.

Type
Research Article
Information
Mathematika , Volume 7 , Issue 1 , June 1960 , pp. 10 - 14
Copyright
Copyright © University College London 1960

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

1.Schubert, H., “Anzahlgeometrische Behandlung des Dreiecks”, Math. Ann., 17 (1880) 153–212.CrossRefGoogle Scholar
2.Semple, J. G., “The triangle as a geometric variable”, Mathematika, 1 (1954), 80–88.CrossRefGoogle Scholar
3.Tyrrell, J. A., “On the enumerative geometry of triangles”, Mathematika, 6 (1959), 158–164.CrossRefGoogle Scholar
4.Segre, B., Some properties of differentiable varieties and transformations (Berlin, 1957).CrossRefGoogle Scholar