On the Definition of Curve's Frame at the Points of inflection

Abstract

AbstractThis paper deals with the problem of defining a frame along the curve at the point of inflection such that the intrinsic properties of the curve are reflected in the defined frame and the frame can be used as a local frame of the body moving on the curve. It is natural to use the Frenet Serret Frames of the curve whenever we need to define a local reference frame along the curve, but the problem with Frenet Serret Frame is that, it is not defined at the point of inflection. The problem gets serious if the point of inflection drags along for some times. i.e. if the direction of the unit tangent is not changing for some time. In such cases if we want to use the Frenet frame of the curve as the local reference frame of a body (a local frame for some sensor e.g camera reference frame) moving along the same curve, it will suffer from the problem of indeterminacy at the point of inflection. Therefore we have to define a frame which exists at all the points of the curve. This defined frame must retain the intrinsic properties of the curve. The Rotation Minimizing Frame is proposed as a solution to the problem in this paper.