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A Geometrical treatment of the Correspondence between Lines in Threefold Space and Points of a Quadric in Fivefold space

Published online by Cambridge University Press:  24 October 2008

H. W. Turnbull
H. W. Turnbull
Affiliation:
Trinity College.
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Extract

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relation

which immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 5 , July 1925 , pp. 694 - 699
Copyright
Copyright © Cambridge Philosophical Society 1925

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References

* It seems unnecessary here to develop the pure geometrical theory of negative circles in full. It corresponds to the analytical fact that a locus

exists where g, f, c are real and g2 + f2 < c.

† Cf. Baker, , Principles of Geometry, I (Cambridge, 1922), 165–172, 182.Google Scholar

* One system of generators of an ordinary ruled quadrio surface.