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

(G): curve under consideration.

M: current point on the curve.

(O, , , ) direct orthonormal frame, with axes Ox , Oy and Oz.

( ): Cartesian coordinates of M.

( ): cylindrical coordinates of M .

(r, q, l) or (r, q, j): spherical coordinates of M (q is the longitude, l is the latitude and j 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; : torsion.

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

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

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

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