Differential geometry notes 4
by Emin Orhan
Formulae of Frenet: These describe the derivatives of the moving trihedron of a curve in terms of the moving trihedron itself:
Note that the coefficients inside the skew-symmetric matrix need not be constant, hence the system described by this equation is not necessarily linear.
Definition 16.1: A vector is called a rotation vector of a rotation if it has the following properties:
- has the direction of the axis of rotation.
- The sense of is such that the rotation has the clockwise sense if one looks from the initial point of to its terminal point.
- The magnitude of equals the angular velocity of the rotation, that is, the velocity of points at distance 1 from the axis of rotation.
Theorem 16.2: The rotation vector of the trihedron of a curve of class with non-vanishing curvature, when a point moves along with constant velocity 1 is given by the expression:
This vector is called the vector of Darboux.
The curves traced out by the terminal points of , and on the unit sphere as the trihedron of a curve moves are called the tangent, binormal and principal normal indicatrix respectively. The linear elements of these curves , and are given by:
Hence, we can write (Equation of Lancret).
Shape of a curve in the neighborhood of any of its points: We can Taylor expand the representation of the curve around (note that the zero point can be chosen arbitrarily):
We have , , and . We can choose the coordinate system such that , , .
Plugging these expressions in the Taylor expansion, we get:
where and are the values of the curvature and torsion at . If we keep only the leading terms, we get: