On curves kinematically related to a given curve

H. Poritsky
1923 Bulletin of the American Mathematical Society  
In a space S is given a curve C. With any point P on it is associated a position of the moving trihedron formed by the tangent, principal normal, and binormal to the curve at P. We might call an indicatrix trihedron a trihedron whose axes pass through a fixed point, and are parallel to those of the moving trihedron, and denote it by 7. As the point P describes the curve, the indicatrix trihedron will rotate about (lines passing through) its vertex. This motion, or the motion of a space rigidly
more » ... of a space rigidly connected to the indicatrix trihedron, with reference to the space S, we shall denote by [I, S]. The problem solved in this paper is, to find all the curves C' such that their indicatrix trihedra V (when drawn with the vertex as I) will experience a motion [ƒ', S] identical with the motion [I, S]; in other words, to find curves C r , whose points P' can be made to correspond to P so that, as the curves are described by corresponding points, their indicatrix trihedra I, V remain relatively invariant. Certain interesting families of curves C f are shown to exist. A kinematic method of treatment has been adopted. Denote unit vectors along the tangent, principal normal, and binormal of C by i, j, k,f respectively. It is known that the motion [I, S] is completely characterized by the fact that there is no component of rotation along the principal normal j.% The components of the rotation along i, k, if P describes C with unit velocity, may be identified with the torsion r and the curvature K. If the velocity of P is not unity but v -ds/dt (s is arc length, t time), the components of rotations become vr, VK. Thus the rotation vector always lies in the plane ir, through i, k, and parallel to the rectifying plane. It will be noticed that T is a plane of (i.e., fixed relative to) the moving space V. * Presented to the Society, December 28, 1922. t Clarendon type will be used to denote vectors. % Darboux, Théorie des Surfaces, 2d éd., vol. I, p. 13.
doi:10.1090/s0002-9904-1923-03667-7 fatcat:q7lzn5odcnaldlouvdgqlmnlqy