Abstract Drawing movements have been shown to comply with a power law constraining local curvature and instantaneous speed. In particular, ellipses have been extensively studied, enjoying a 2/3 exponent. While the origin of such a non-trivial relationship remains debated, it has been proposed to be an outcome of the least action principle whereby mechanical work is minimized along 2/3 power law trajectories. Here we demonstrate that this claim is flawed. We then study a wider range of curves beyond ellipses that can have 2/3 power law scaling. We show that all such geometries are quasi-pure and with the same spectral frequency. We then numerically estimate that their dynamics produce minimum jerk. Finally, using variational calculus and simulations, we discover that equi-affine displacement is invariant across different kinematics, power law or otherwise. In sum, we deepen and clarify the relationship between geometric purity, kinematic scaling and dynamic optimality for trajectories beyond ellipses. It is enticing to realize that we still do not fully understand why we move our pen on a piece of paper the way we do.

中文翻译：

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绘制椭圆以外的运动时的几何纯度、运动学缩放和动态优化

摘要 绘图运动已被证明符合约束局部曲率和瞬时速度的幂律。特别是，椭圆已被广泛研究，其指数为 2/3。虽然这种非平凡关系的起源仍有争议，但有人提出它是最小作用量原理的结果，即机械功沿 2/3 幂律轨迹最小化。在这里，我们证明这种说法是有缺陷的。然后，我们研究了椭圆以外的更广泛的曲线，这些曲线可以具有 2/3 幂律缩放。我们表明所有这些几何形状都是准纯的并且具有相同的光谱频率。然后我们通过数值估计它们的动态产生最小的混蛋。最后，使用变分微积分和模拟，我们发现等仿射位移在不同的运动学中是不变的，幂律或其他。总之，我们深化并阐明了椭圆以外轨迹的几何纯度、运动学标度和动态最优性之间的关系。意识到我们仍然不完全理解为什么我们以这种方式在纸上移动笔是很诱人的。