ASYMTPTOTIC LINE OF A SURFACE

 Notion studied by Dupin. From the Greek asumptôtos: not falling together.

 Differential equation: where is the normal vector of the surface at M, i.e. (cancellation of the second fundamental form).

The asymptotic lines of a surface can be defined in the three following equivalent ways:

DEF 1: they are the curves traced on the surface that are tangent at each point to one of the asymptotic directions (i.e. one of the directions where the curvature is zero, or also one of the asymptotes of the Dupin indicatrix conic relative to this point, or its axis when it is reduced to two parallel lines).

DEF 2: they are the curves traced on the surface with zero normal curvature (i.e. the curvature of the section of the surface by the plane containing the tangent to the curve and the normal at the surface).

DEF 3: they are the curves traced on the surface such that, at each point, the plane tangent to the surface is an osculating plane of the curve.

The asymptotic lines only pass through hyperbolic points (through which pass two asymptotic lines) or parabolic points (through which pass only one line) of the surface.

Examples:
- the lines included in the surface are asymptotic lines.
- for a developable surface, the asymptotic lines are the generatrices, and them only.
- A surface is minimal iff two orthogonal asymptotic lines pass by all points.