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On the uninodal quartic curve

Published online by Cambridge University Press:  20 January 2009

H. W. Richmond
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In comparison with the general plane quartic on the one hand, and the curves having either two or three nodes on the other, the uninodal curve has been neglected. Many of its properties may of course be deduced from those of the general quartic in the limiting case when an oval shrinks to a point or when two branches approach and ultimately unite. The modifications of properties of the bitangents are shewn more clearly by Geiser's method, in which these lines are obtained by projecting the lines of a cubic surface from a point on the surface. As the point moves up to and crosses a line on the surface, the quartic acquires a node and certain pairs of bitangents obviously coincide, viz. those obtained by projecting two lines coplanar with that on which the point lies. A nodal quartic curve and its double tangents may also be obtained by projecting a cubic surface which has a conical point from an arbitrary point on the surface. Each of these three methods leads us to the conclusion that, when a quartic acquires a node, twelve of the double tangents coincide two and two and become six tangents from the node, and the other sixteen remain as genuine bitangents: the twelve which coincide are six pairs of a Steiner complex.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 1 , Issue 1 , May 1927 , pp. 31 - 38
Copyright
Copyright © Edinburgh Mathematical Society 1927

References

1 But see Hilton, , Plane Algebraic Curves, pp. 298–303. Clebsch (Vorlesungen), Brill (Math. Annalen, 6 (1873), p. 66) and others have considered the curve in connection with hyperelliptic functions. But a treatment by the elementary methods which are applied successfully to the general quartic is wanting.Google Scholar

2 Math. Annalen, 1 (1869) p, 129, Hilton, p. 345.CrossRefGoogle Scholar

3 See Quarterly Journal of Mathematics, 26 (1893), 5–26.Google Scholar

4 The six points of contact lie also on a conic.

5 If all are negative, A, B, C, D, E, F are pure imaginary quantities, and S2, S4 are real.