Lancret-type theorems in equiaffine space (Q6564736)

The authors prove the following theorems, where \(\kappa_e(s)\) and \(\tau_e(s)\) are the equiaffine curvature functions of an equiaffine curve. \N\NTheorem. A parametrized curve in \({\mathbb R}^3\) is an equiaffine general helix if and only if its equiaffine curvatures satisfy \N\[\N\tau_e(s) = \int^s\kappa_e(u)\mathrm{d}u,\N\] \Nwhere \(s\) is the equiaffine arc length parameter.\N\NTheorem. A parametrized curve in \({\mathbb R}^3\) is an equiaffine slant helix if and only if, up to a translation of s, its equiaffine curvatures satisfy \N\[\Ns\tau_e(s) = \int^su\kappa_e(u)\mathrm{d}u,\N\]\Nwhere \(s\) is the equiaffine arc length parameter.