The total absolute curvature and the total absolute torsion of open curves in the Euclidean spaces

摘要

The total absolute curvature of smooth curves in E~3 is defined as the total integral of the curvature and the total absolute torsion is defined as the total integral of the absolute value of the torsion. Many works have been done to study the shape of the curves which minimize these integrals in the past, but mostly for closed curves. We study this problem for open curves satisfying reasonable boundary conditions. In the present paper, we overview our previous works and compare the minimization problems of the total absolute curvature and the total absolute torsion. We see that the extremal curve often has nonsmooth points in the open curve case, in contrast with the closed curve case. This leads us to extending the concepts of the total absolute curvature and the total absolute torsion to piecewise linear curves. Most part of our work is devoted to finding a way of reducing the total absolute curvature or the total absolute torsion of a piecewise linear curve.