Lines of curvature near singular points of implicit surfaces. (English) Zbl 0791.53007

Garcia, R.; Sotomayor, J.

Bull. Sci. Math., II. Sér. 117, No. 3, 313-331 (1993).

On a quadratic cone, the lines of curvature are the rulings starting at the vertex and the parallel circles. The authors show that the same holds for cubic cones \(x_ 3= f(x_ 1,x_ 2)\) if the cubic term is either \(x_ 1 x_ 2^ 2\) or \(x_ 1^ 2 x_ 2\) but that the second family becomes a family of spirals if the cubic term involves \(x_ 1 x_ 2 x_ 3\), and that the latter is the generic case for algebraic cones of degree \(>2\).

Reviewer: H.Guggenheimer (West Hempstead)

MSC:

53A05

Surfaces in Euclidean and related spaces

Keywords:

lines of curvature; rulings; cubic cones