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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
(*S*_{1}) and (*S*_{2})
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
(*S*_{1})

- (*C*) is a curvature line of
(*S*_{2})

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