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

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 $$>2$$.

### MSC:

 53A05 Surfaces in Euclidean and related spaces

### Keywords:

lines of curvature; rulings; cubic cones