Normal curvature

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 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

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

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