Instantaneous radius & evolute of clubhead trajectory

TGMers are very keen on the geometrical aspects of the golf swing. Points, lines, planes, circles are just some examples of the concepts frequently discussed. I am hence assured in advance of their undivided and enthusiastic attention when looking into the concept of instantaneous centers of rotation and the associated concept of evolute of the curve.

Fig1 illustrates the concept of instantaneous radius and instantaneous center. A curve can be characterized by applying a circle at various  points such that it is tangent to the curve, has the same curvature as the curve and is applied at the concave side of the curve.

The centers of the circles of curvature are defining both the instantaneous centers and the instantaneous radii of curvature. The collection, or locus of the instantaneous centers of curvature is called the evolute of the curve. Figs 2a,b,c illustrate these concepts for a circle, ellipse and parabola.

For a circle, Fig2a, there is a constant instantaneous radius and the evolute is simply a point, the center of the circle. For the ellipse, Fig2b, close family to the circle, the locus of instantaneous centers of rotation is however quite different, indicated by the red curve. Similarly in Fg2c,  the red curve shows the locus of instantaneous centers for a parabola.

Fig3 shows the evolute of the clubhead trajectory (red curve). It can vary greatly for only small changes in the clubhead trajectory.

The same information used in Fig3 is shown again in Fig4 in a different way. Here the clubhead trajectory and some instantaneous radii and corresponding  evolute (red) points are shown.

The evolute is a very ‘sensitive’ characterization of a given curve. Clubhead  trajectories, at first glance almost identical, could have very different loci. If a cluhhead trajectory could be measured with sufficient accuracy its evolute could be used as an unique signature for a particular swing. Any small change, almost undetectable by eye, would show up quite clearly in its evolute. Rather useful for pros verifying if their swing has been changing unnoticed.

mandrin