Let be an interval, and let be an arclength parameterization of an oriented space curve, assumed to be regular (http://planetmath.org/Curve), and free of points of inflection. Let , , denote the corresponding moving trihedron, and the corresponding curvature (http://planetmath.org/CurvatureOfACurve) and torsion functions (http://planetmath.org/Torsion). The following differential relations, called the Serret-Frenet equations, hold between these three vectors.
Equation (1) follows directly from the definition of the normal (http://planetmath.org/MovingFrame) and from the definition of the curvature (http://planetmath.org/CurvatureAndTorsion), . Taking the derivative of the relation
Taking the derivative of the relation
By the definition of torsion (http://planetmath.org/CurvatureAndTorsion), we have
This proves equation (2). Finally, taking derivatives of the relations
This proves equation (3).
It is also convenient to describe the Serret-Frenet equations by using matrix notation. Let (see - special orthogonal group), the mapping defined by
In this formulation, the above relation is also known as the structure equations of an oriented space curve.
|Date of creation||2013-03-22 12:15:13|
|Last modified on||2013-03-22 12:15:13|
|Last modified by||rmilson (146)|