The coordinate functions of the Heighway dragon curve

In this work, we study properties of the coordinate functions $x_\theta$ and $y_\theta$ of the dragon curve associated with the angle $\frac{\pi}{3} < \theta < \frac{5\pi}{3}$, and we prove that the box-counting dimension of its graph is equal to $1 - \frac{\log \cos\alpha}{\log 2}$, where $\alpha = \frac{\pi - \theta}{2}$.