 ## Introduction:

If you consider the above scribbles of Évariste Galois, who developed Galois theory, you will note that some of the scribbles appear random. Yet, upon closer inspection none of the scribbles are completely random. Many of the scribbles are rather smooth which would be improbable if the trajectories were generated by some kind of Brownian-type motion.

This isn’t really surprising if you consider the biomechanical constraints on handwritten text. In fact, some scientists have attempted to distill this observation into a physical law known as the two-thirds power law which I analyse here. Briefly speaking, here’s a breakdown of my analysis:

1. I provide a mathematical description of the law and describe how it may be used as a discriminative model.
2. We may also use this equation as a generative model if we consider symmetries of the equation. Here is the code.
3. The limitations of the ‘law’ are considered and arguments are given to shift focus on plausible generative models.

In spite of its limitations I think that the $2/3$ power law is a very good starting point for understanding biomechanical constraints on realistic drawing tasks.

## Description of the law:

### Brief description:

The $2/3$ power law for the motion of the endpoint of the human upper-limb during drawing motion may be formulated as follows:

\begin{equation} v(t) = K \cdot k(t)^\beta \end{equation}

where $k(t)$ is the instantaneous curvature of the path and the $2/3$ law is satisfied when $\beta \approx -\frac{1}{3}$. By taking logarithms of both sides of the equation we have:

\begin{equation} \ln v(t) = K - \frac{1}{3} \ln k(t) \end{equation}

### Frenet-Serret formulas:

To clarify what we mean by instantaneous curvature $k(t)$ in (2) it’s necessary to use a moving reference frame, aka Frenet-Serret frame, where in two dimensions our reference frame is described by the unit vector tangent to the curve and a unit vector normal to the curve.

With this moving frame we may define the curvature of regular curves(i.e. curves whose derivatives never vanish) parametrized by time as follows:

\begin{equation} k(t) = \frac{\lvert \ddot{x}\dot{y} - \ddot{y}\dot{x} \rvert}{(\dot{x}^2 + \dot{y}^2)^{3/2}} = \frac{\lvert \ddot{x}\dot{y} - \ddot{y}\dot{x} \rvert}{v^3(t)} \end{equation}

Now, if we denote:

\begin{equation} \alpha(t) = \lvert \ddot{x}\dot{y} - \ddot{y}\dot{x} \rvert \end{equation}

we have:

\begin{equation} \ln v(t) = \frac{1}{3} \ln \alpha(t) - \frac{1}{3} \ln k(t) \end{equation}

and we note that our law is satisfied when $\forall t, \alpha(t)=K$. Given that this is a linear equation we may use this equation as a discriminative model by performing a linear regression analysis on drawing data.

### Parallelograms:

If we focus on $(4)$ we may note that this value corresponds to the determinant of a particular matrix:

Furthermore, we may note that this determinant may be identified with the area $K$ of a parallegram with the following vertices:

This formulation is useful as invariants of $\lvert \ddot{x}\dot{y} - \ddot{y}\dot{x} \rvert=K$ now correspond to volume-preserving transformations applied to the above parallelogram.

## Generative modelling via Invariants:

### Invariance via volume-preserving transforms:

Let’s first note that if we always have:

\begin{equation} \lvert \ddot{x}\dot{y} - \ddot{y}\dot{x} \rvert=K \end{equation}

for some $K \in \mathbb{R}$ then we must have:

\begin{equation} \lvert \ddot{x}(0)\dot{y}(0) - \ddot{y}(0)\dot{x}(0) \rvert=K \end{equation}

Now, given that

\begin{equation} \mathcal{M} = \{ M \in \mathbb{R}^{2 \times 2}: det(M)=1 \} \end{equation}

are volume-preserving transformations, we may use $M \in \mathcal{M}$ to simulate arbitrary trajectories that satisfy $(2)$. We may think of this as the Jacobian of a linear, hence differentiable, transformation.

### Computer simulation:

In order to simulate these trajectories, we note that:

where the position is updated using:

\begin{equation} x_{n+1} = x_n + \dot{x}_n\cdot \Delta t + \frac{1}{2} \ddot{x}_n \cdot \Delta t^2 \end{equation}

and in order to make sure that $ad-bc=1$ we may use the trigonometric identity:

\begin{equation} cos^2(\theta) + sin^2(\theta) = 1 \end{equation}

so we have:

\begin{equation} bc = -sin^2(\theta) \end{equation}

and as a result we have a generative variant of the 2/3 power law. Ok, but are these ‘scribbles’ ecologically plausible? I don’t think so, which is why I call the main Julia function I used to simulate these trajectories ‘crazy paths’.

## Criticism:

1. The $2/3$ law is a pretty weak discriminative model because as shown by  the exponent varies with the viscosity of the drawing medium and as shown by  the exponent also depends on the complexity of the shape drawn.
2. The $2/3$ law is an even weaker generative model as it completely ignores environmental cues. The output ‘scribbles’ aren’t the result of any plausible interaction of an agent with an ecologically realistic environment.

This point is even more clear when you consider the underlying minimum-jerk theory that is supposed to justify this ‘law’. A verbatim interpretation of jerk minimisation would imply that humans should mainly draw straight lines. However, there’s certainly a tradeoff between energy minimisation and the expressiveness of the figure drawn since drawing is an activity that involves communicating a particular message.

1. D. Huh & T. Sejnowski. Spectrum of power laws for curved hand movements. 2015.
2. M. Zago et al. The speed-curvature power law of movements: a reappraisal. 2017.
3. U. Maoz et al. Noise and the two-thirds power law. 2006.
4. M. Richardson & T. Flash. Comparing Smooth Arm Movements with the Two-Thirds Power Law and the Related Segmented-Control Hypothesis. 2002.