On the Singularities of Plane Curves

Bisztriczky, Tibor

Canadian journal of mathematics, Tome 38 (1986) no. 4, pp. 947-968

Résumé

Let Γ be a differentiable curve in a real projective plane P 2 met by every line of P 2 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 n 1(Γ), n 2(Γ) and n 3(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 P 2. 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

Détail

DOI : 10.4153/CJM-1986-047-3

Bibliographie

[1] 1. Bisztriczky, T., On the singularities of almost-simple plane curves, Pac. J. Math. 109 (1983), 257–273. Google Scholar

[2] 2. Bisztriczky, T., On the singularities of simple plane curves, Mich. Math. J. 32 (1985), 141–151. Google Scholar

[3] 3. Brunn, H., Sâtze ùher zwei getrennte Eikörper, Math. Ann. 104 (1931), 300–324. Google Scholar

[4] 4. Park, R., Topics in direct differential geometry, Can. J. Math. 24 (1972), 98–148. Google Scholar