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Concept# Darboux frame

Summary

In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space. It is named after French mathematician Jean Gaston Darboux.
Let S be an oriented surface in three-dimensional Euclidean space E3. The construction of Darboux frames on S first considers frames moving along a curve in S, and then specializes when the curves move in the direction of the principal curvatures.
At each point p of an oriented surface, one may attach a unit normal vector u(p) in a unique way, as soon as an orientation has been chosen for the normal at any particular fixed point. If γ(s) is a curve in S, parametrized by arc length, then the Darboux frame of γ is defined by
(the unit tangent)
(the unit normal)
(the tangent normal)
The triple T, t, u defines a positively oriented orthonormal basis attached to each point of the curve: a natural moving frame along the embedded curve.
Note that a Darboux frame for a curve does not yield a natural moving frame on the surface, since it still depends on an initial choice of tangent vector. To obtain a moving frame on the surface, we first compare the Darboux frame of γ with its Frenet–Serret frame. Let
(the unit tangent, as above)
(the Frenet normal vector)
(the Frenet binormal vector).
Since the tangent vectors are the same in both cases, there is a unique angle α such that a rotation in the plane of N and B produces the pair t and u:
Taking a differential, and applying the Frenet–Serret formulas yields
where:
κg is the geodesic curvature of the curve,
κn is the normal curvature of the curve, and
τr is the relative torsion (also called geodesic torsion) of the curve.
This section specializes the case of the Darboux frame on a curve to the case when the curve is a principal curve of the surface (a line of curvature).

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