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.

中文翻译：

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平面平滑手势的形状和手势识别器的收敛

在这项工作中，我们提供了新手势识别算法！FTL的数学框架。这使我们可以代数量化平滑平面曲线的形状概念，这受Lester先前在同一本期刊中给出的三角形的形状概念的启发。特别是，我们通过刻在该曲线上的多边形路径来近似被视为平滑平面曲线的每个手势。然后，我们将该多边形路径上连续点的每个三元组视为具有一定形状的三角形的顶点。我们显示出，随着折线点向收敛到原始手势，相应的形状顺序点向收敛到形状的极限曲线，我们认为这是该手势的形状。我们使用欧几里得度量和黎曼积分来测量两个手势的形状之间的距离。三角形形状的位置，比例和旋转不变性仍然适用于手势形状，这是！FTL的主要成就之一。最后，我们提到，为了进一步研究，可以使用Clifford数将二维欧几里得形状概念扩展到更高维度的设置和更通用的度量。