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 $\alpha =\frac{\pi -\theta }{2}$.