Integrability for the Derivative Formulas of Rotation Minimizing Frame in Euclidean 3-Space and Its Applications

Abstract

We analyze integrability for the derivative formulas of the rotation minimizing frame in theEuclidean 3-space from a viewpoint of rotations around axes of the natural coordinate system.We give a theorem that presents only one component of the indirect solution of the rotationminimizing formulas. Using this theorem, we find a lemma which states the necessary conditionfor the indirect solution to be a steady solution. As an application of the lemma, the naturalrepresentation of the position vector field of a smooth curve whose the rotation minimizing vectorfield (or the Darboux vector field) makes a constant angle with a fixed straight line in space isobtained. Also, we realize that general helices using the position vector field consist of slant helicesand Darboux helices in the sense of Bishop.