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 rand.

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