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On the real twisted cubic curves which are geodesic upon a quadric surface

Published online by Cambridge University Press:  24 October 2008

H. W. Richmond
H. W. Richmond
Affiliation:
King's College
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Extract

1. Geodesic curves, even of simple surfaces such as quadrics, are almost always transcendental: occasionally they are algebraic: still more rarely do they belong to simple and familiar types. The form of the differential equation referred to proves conclusively that even such integrals as are expressible by elementary functions must be quite exceptional.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 1 , February 1924 , pp. 42 - 48
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* See Salmon, , Analytical Geometry of Three Dimension, 1912 edition, vol. I, §§397–420Google Scholar for a general account of geodesics on quadrics, and specially the footnote to § 412 for the differential equation in terms of confocal coordinates.

* I believe it to be the fact that a quadric cone cannot have on it a real twisted cubic which is geodesic, and a proper quadric cannot have on it a quartic curve of genus one (i.e. elliptic) which is geodesic.