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The osculating of a certain curve in [4]

Published online by Cambridge University Press:  20 January 2009

W. L. Edge
Affiliation:
Mathematical Institute, 20 Chambers Street, Edinburgh EH1 1HZ
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The equation of the osculating plane at a point on the complete irreducible curve of intersection of two algebraic surfaces in [3] was found by Hesse (5, p. 283); the plane, having to contain the tangent of the curve, belongs to the pencil spanned by the tangent planes of the two surfaces, and it is a question of determining which plane of the pencil to choose. The equation also appears in the books of Salmon (6, p. 378) and Baker (1, p. 206). The analogous problem for the osculating solid at a point on the complete irreducible curve of intersection of three algebraic primals, or threefolds, in [4] does not appear to have been considered. The simplest instance is the octavic curve C of intersection of three quadrics, and this has the special interest of being a canonical curve; moreover the quadrics are of the same order, and so can be replaced by any three linearly independent members of the net which they determine, a replacement of which it may be prudent to take advantage with a view to simplifying the algebra. It is a question of determining which solid to choose among the tangent solids to the quadrics of the net at a point on C, but while Hesse's methods serve to carry one a certain distance there seems no obvious way of pushing them to a conclusion. It is then natural, with a view to reaching a conclusion, to choose a net of quadrics that, through having some particular property, is more amenable.

Type
Articles
Copyright
Copyright © Edinburgh Mathematical Society 1971

References

REFERENCES

(1)Baker, H. F., Principles of Geometry Vol. 5 (Cambridge 1933).Google Scholar
(2)Baker, H. F., Principles of Geometry Vol. 6 (Cambridge 1933).Google Scholar
(3)Edge, W. L., Humbert's plane sextics of genus 5, Proc. Cambridge Philos. Soc. 47 (1951), 483495.CrossRefGoogle Scholar
(4)Edge, W. L., The tacnodal form of Humbert's sextic, Proc. Royal Soc. Edinburgh Sect. A 68 (1970), 257269.Google Scholar
(5)Hesse, O., Über die Wendepunkte der algebraischen ebenen Kurven und die Schmiegungs-Ebenen der Kurven von doppelter Krummung, welche durch den Schnitt zweier algebraischen Oberflächen entstehen, Journal für die reine und angewandte Mathematik 41 (1851), 272284;Google Scholar
Gesammelte Werke (Munich, 1897), 263278.Google Scholar
(6)Salmon, G., A treatise on the analytic geometry of three dimensions Vol. 1 (Dublin 1914).Google Scholar
(7)Severi, F., Vorlesungen über algebraische Geometrie (Leipzig 1921).CrossRefGoogle Scholar
(8)Semple, J. G. and Roth, L., Introduction to algebraic geometry (Oxford 1949).Google Scholar