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On the Projective Centres of Convex Curves

Published online by Cambridge University Press:  20 November 2018

Paul Kelly  and
E. G. Straus
Paul Kelly
Affiliation:
University of California, Santa Barbara
E. G. Straus
Affiliation:
Los Angeles
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We consider a closed curve C in the projective plane and the projective involutions which map C into itself. Any such mapping γ, other than the identity, is a harmonic homology whose axis η we call a projective axis of C and whose centre p we call an interior or exterior projective centre according as it is inside or outside C. The involutions are the generators of a group Γ, and the set of centres and the set of axes are invariant under Γ. The present paper is concerned with the type of centre sets which can exist and with the relationship between the nature of C and its centre set.

If C is a conic, then every point which is not on C is a projective centre. Conversely, it was shown by Kojima (4) that if C has a chord of interior centres, or a full line of exterior centres, then C is a conic.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 568 - 581
Copyright
Copyright © Canadian Mathematical Society 1960

References

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3. Kneser, H., Eine Erweiterung des Begriffes “konvexer Kôrper”, Math. Ann., 82 (1921), 287–296.Google Scholar
4. Kojima, T., On characteristic properties of the conic and the quadric, Sci. Rep. Tohoku Univ., 8 (1919), 67–68.Google Scholar