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A Property Characteristic of Quadrics of Revolution and General Cylinders

Published online by Cambridge University Press:  03 November 2016

E. D. Camier
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Extract

The locus of the centres of spherical curvature of a singly infinite family of geodesics which pass through a regular point O on a surface S, one in each direction in the tangent plane there, is, in general, a twisted curve. It will be proved that the only real surfaces, at all points of which (excluding umbilics) this locus is a plane curve, are quadrics of revolution and general cylinders.

Type
Research Article
Information
The Mathematical Gazette , Volume 30 , Issue 290 , July 1946 , pp. 141 - 143
Copyright
Copyright © Mathematical Association 1946

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References

page note 141 * Weatherburn, Differential Geometry, I, Art. 6.

page note 141 † Darboux, Théorie des Surfaces, Art. 510. When the curve is a geodesic, and the expression for “Laguerre’s function” gives the equivalent of (2) since

page note 142 * Darboux, Art. 513.

page note 142 † Blaschke, Vorsles. ü Differentialgeometrie, I, p. 142.

page note 143 * Blaschke, p. 140.