Bertrand curves

conjugate curves, Bertrand pair

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

$$a{k}_1\sin \omega +a{k}_2\cos \omega =\sin \omega$$

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

Comments[edit]

Bertrand's original paper is [a2]. A general reference is [a1].

References[edit]