3D CURVES, or "skew" curves

FRENCH VERSION
See the notations below.

 A B C DEFGH IJKLM NOPQR STUVWXYZ

LINE (TOPOGRAPHIC/):
CONTOUR/, CREST/, SLOPE/, THALWEG/

TRACTORY

NOTATIONS : curve under consideration.

M: current point on the curve. direct orthonormal frame, with axes Ox , Oy and Oz. : Cartesian coordinates of M. : cylindrical coordinates of M . or : spherical coordinates of M ( is the longitude, is the latitude and the colatitude). , speed vector, V: algebraic speed. , acceleration vector.

(T): tangent, supported by .
(P): osculating plane, supported by and .

(N): principal normal, orthogonal to (T), in the osculating plane.

(B): binormal, orthogonal to (P), supported by .

s: curvilinear abscissa

(  ) : tangent (unit) vector.

V: absolute speed ( ). : normal (unit) vector, supports the principal normal; the plane (M, , ) is the osculating plane at M. : center of curvature at M. : binormal vector = (unit) ( = base, or Frenet trihedron). : radius of curvature, always nonnegative. is the angle between and , thus between two infinitely close tangents; j is the angle of curvature; it represents the length of the path on which travels the end of the tangent vector attached to a fixed point. : radius of torsion of a skew curve. , defined by is the angle between two infinitely close osculating planes; the sign convention we used, called Darboux convention, is such that the right-handed curves have a positive torsion; its sign does not depend on the travel sense along the curve; is the torsion angle; it represents the length of the path on which travels the end of the binormal vector attached to a fixed point.

We have the Frenet formulas:   . : curvature ;  ( measures the intensity of the variation of the tangente) . : torsion ( measures the intensity of the variation of the osculating plane).
Hence the condensed writing of Frenet formulas: .

Cartesian system of equations, parametrization: characterization in terms of x, y and z.

Cylindrical system of equations, parametrization: characterization in terms of , and z.

Spherical system of equations, parametrization: characterization in terms of r and .

See the notations for curves traced on a surface on the page dealing with surfaces.