View source for Bertrand curves

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''conjugate curves, Bertrand pair''

Two space curves $L$ and $L^*$ with common principal normals. Let $k_1$ and $k_2$ be the curvature and the torsion of $L$ respectively. For the curves $L$ and $L^*$ to be conjugate it is necessary and sufficient that

$$ak_1\sin\omega+ak_2\cos\omega=\sin\omega$$

is true. Here $a$ is a constant, and $\omega$ is the angle between the tangent vectors of $L$ and $L^*$. The name Bertrand curve is also given to a curve $L$ for which there exists a conjugate curve $L^*$. They were introduced by [[Joseph Bertrand|J. Bertrand]] in 1850.

====Comments====
Bertrand's original paper is [[#References|[a2]]]. A general reference is [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Bertrand,   "Mémoire sur la théorie des courbes à double courbure"  ''Liouvilles Journal'' , '''15'''  (1850)</TD></TR></table>