Curves in Space

Abstract

AbstractThe next simplest object of differential geometry after a plane curve is a curve in space, three- or many-dimensional. In addition to curvature, a curve in three-dimensional space has one more characteristic, torsion. A space curve lies in one plane if and only if its torsion identically vanishes. A curve in n-dimensional space is characterized by numbers ϰ1, …, ϰn−1, which generalize curvature and torsion. Again, a curve lies in one hyperplane if and only if ϰn−1 = 0 at all points of this curve.