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On the Singularities of Plane Curves

Published online by Cambridge University Press:  20 November 2018

Tibor Bisztriczky
Tibor Bisztriczky*
Affiliation:
University of Calgary, Calgary, Alberta
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Let Γ be a differentiable curve in a real projective plane P2 met by every line of P2 at a finite number of points. The singular points of Γ are inflections, cusps (cusps of the first kind) and beaks (cusps of the second kind). Let n1(Γ), n2(Γ) and n3(Γ) be the number of these points in Γ respectively. Then Γ is non-singular if

otherwise, Γ is singular.

We wish to determine when T is singular and then find the minimum value of n(Γ). A history and an analysis of this problem were presented in [1] and [2]. It was shown that we may assume that Γ is a curve of even order (even degree if Γ is algebraic), met by every line in P2. Then if Γ does not contain any multiple points or if Γ contains only a certain type of multiple point, Γ is singular. Presently, we complete this investigation

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 4 , 01 August 1986 , pp. 947 - 968
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Bisztriczky, T., On the singularities of almost-simple plane curves, Pac. J. Math. 109 (1983), 257–273.Google Scholar
2. Bisztriczky, T., On the singularities of simple plane curves, Mich. Math. J. 32 (1985), 141–151.Google Scholar
3. Brunn, H., Sâtze ùher zwei getrennte Eikörper, Math. Ann. 104 (1931), 300–324.Google Scholar
4. Park, R., Topics in direct differential geometry, Can. J. Math. 24 (1972), 98–148.Google Scholar