Darboux trihedron

A trihedron associated with a point on a surface and defined by a triple of vectors, given by the normal unit vector $ \mathbf n $ to the surface and two mutually orthogonal principal unit tangent vectors $ \mathbf r _ {1} $ and $ \mathbf r _ {2} $ to the surface such that

$$ \mathbf n = \mathbf r _ {1} \times \mathbf r _ {2} . $$

The properties of the surface can be described in terms of displacement of the Darboux trihedron when its base point describes the surface. A systematic use of the Darboux trihedron in the study of surfaces led G. Darboux [1] to the moving-frame method.

References

[1]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887)

Comments

A Darboux trihedron is also called a Darboux frame. It is also introduced in affine differential geometry, cf. [a1].

References

[a1]	H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963)