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CURVATURE LINE of a surface

Notion studied by Monge in 1776 and Dupin in 1813.
Other name: principal curve.

Differential equation:  where  is the normal vector of the surface at M,
i.e. , which can be written: .
Using the Monge notations:  (see the notations).

The curvature lines of a surface have three equivalent definitions:

DEF #1: they are the curves traced on the surface that are tangent at each point to one of the principal directions (i.e. the direction in which the curvature is maximal or minimal, i.e. one of the axes of the Dupin indicatrix conic relative to this point)

DEF #2: they are the curves traced on the surface with zero geodesic torsion.

DEF #3: they are the curves traced on the surface such that the associated normal surface is developable, in other words, such that the normals to the surface along the curve have an envelope (then, the cuspidal edge of the normal surface is the locus of the centers of curvature of the principal sections tangent to the curvature line).

By every point that is not an umbilic nor a meplat pass two orthogonal curvature lines, one corresponding to a maximal curvature while the other corresponds to a minimal curvature.
In the domain of the surface formed by the hyperbolic points, the curvature lines are the bisectors of the asymptotic lines .

The Joachimstal theorem states that given two surfaces (S1) and (S2) intersecting along a curve (C), two among the three following properties imply the third.
    - the angle between the two surfaces along (C) is constant
    - (C) is a curvature line of (S1)
    - (C) is a curvature line of (S2)

We can derive from this, for example, that a plane intersects a surface along a curvature line iff this plane intersects the surface at a constant angle.

We can also derive the Dupin theorem: in a triple orthogonal system of surfaces, i.e. a family of surfaces such that by each point of each surface pass exactly two other surfaces of the family so that these three surfaces are pairwise orthogonal at this point, the surfaces intersect along their curvature lines.

Under a similarity, and even under a conformal transformation, but not under an affine transformation, the curvature lines are sent on curvature lines.

Examples of curvature lines:
 - in a plane or a sphere, any line is a curvature line.
 - for a surface of revolution, the two families of curvature lines are composed of the meridian and the parallel lines.
 - for a developable surface (therefore, in particular, for cones and cylinders), the two families of curvature lines are the generatrices and their orthogonal trajectories.
 - more generally, for a Monge surface, the curvature lines are the generatrices and the directrices.
 - the surfaces the curvature lines of which are circles or straight lines are the Dupin cyclides.
 - curvature lines of the quadrics: see on the pages corresponding to each type of quadric.
Let us highlight that the family of homofocal quadrics   () forms a triple orthogonal system. They are ellipsoids for , one-sheeted hyperboloids for  and two-sheeted hyperboloids for . This determines the curvature lines of the surfaces.

See also at focal.

Ellipsoid with its curvature lines, by Patrice Jeener, with his gracious authorization.

Periodic minimal surface, with its curvature lines

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© Robert FERRÉOL  2018