Characteristic directions of closed planar motions

We investigate some properties of closed planar motions. These motions appear between two coordinate systems, fixed and moving. We focus on three aspects: The area enclosed by the paths of points belonging to each of these reference systems, the polar moment of inertia, and the natural average of a path. First, we consider a formula proposed by Jakob Steiner. It states that, for a motion with a complete turn, the points enclosing the same area lie on a circle. Moreover, all circles for different areas have a common center. We start by deriving the Steiner formula relative to both frames. We emphasize that, for practical applications such as in biomechanics, the Steiner circle reduces to a straight line. Then, the centrodes are considered in relation to the Steiner point or normal. We next consider the polar moment of inertia. Its coefficients are shown to be related to those of the Steiner point or normal. Thirdly, the natural average of a path is introduced. Its coefficients are shown to be related to those previously discussed. Finally, we show how the results can be applied to experimentally measured motions. As an example, we consider human gait in the sagittal direction. The most important part of this motion is described as a double hinge driven by the hip and knee angle, which is a simple model for a planar motion. We conclude that, for practical purposes, the Steiner line allows us to visualize a selected direction of motion under consideration.