Also known as the Serret-Frenet formulas, these vector differential equations relate inherent properties of a parametrized curve. In matrix form, they can be written
Frenet, F. "Sur les courbes à double courbure." Thèse. Toulouse, 1847. Abstract in J. de Math.17, 1852.Gray,
A. Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, p. 186, 1997.Kreyszig, E. "Formulae
of Frenet." §15 in Differential
Geometry. New York: Dover, pp. 40-43, 1991.Serret, J. A.
"Sur quelques formules relatives à la théorie des courbes à
double courbure." J. de Math.16, 1851.
![[T^.; N^.; B^.]=[0 kappa 0; -kappa 0 tau; 0 -tau 0][T; N; B],](/images/equations/FrenetFormulas/NumberedEquation1.svg)
is the unit tangent vector, 
is the unit normal vector,
is the unit binormal
vector, 
is the torsion, 
is the curvature, and 
denotes 
.
