The shape of planar smooth gestures and the convergence of a gesture recognizer

In this work we provide the mathematical framework of !FTL, a new gesture recognition algorithm. This allows us to algebraically quantify the notion of shape for a smooth planar curve, inspired by the notion of shape of a triangle given previously by Lester in this same journal. In particular, we approximate every gesture, considered as a smooth planar curve, by a polygonal path inscribed on that curve. Then, we consider each triple of consecutive points on that polygonal path as the vertices of a triangle having a shape. We show that, as the polygonal line pointwise converges to the original gesture, the corresponding sequence of shapes pointwise converges to a limiting curve of shapes, that we consider to be the shape of that gesture. We use the Euclidean metric and the Riemann integral to measure the distance between the shapes of two gestures. The position, scale and rotation invariances of the shape of a triangle still hold for the shape of a gesture, and this provides one of the main achievements of !FTL. Finally, we mention, for further research, that the two dimensional Euclidean notion of shape can be extended to higher dimensional settings and more general metrics using Clifford numbers.