Plane Curve Of Genus Research Articles

Search of Fractal Space-Filling Curves with Minimal Dilation

We introduce an algorithm for a search of extremal fractal curves in large curve classes. It heavily uses SAT-solvers—heuristic algorithms that find models for CNF boolean formulas. Our algorithm was implemented and applied to the search of fractal surjective curves \(\gamma :[0,1]\rightarrow [0,1]^d\) with minimal dilation $$\begin{aligned} \sup _{t_1<t_2}\frac{\Vert \gamma (t_2)-\gamma (t_1)\Vert ^d}{t_2-t_1}. \end{aligned}$$We report new results of that search in the case of Euclidean norm. We have found a new curve that we call “YE”, a self-similar (monofractal) plane curve of genus \(5\times 5\) with dilation \(5+{43}/{73}=5.5890\ldots \) In dimension 3 we have found facet-gated bifractals (which we call “Spring”) of genus \(2\times 2\times 2\) with dilation \(<17\). In dimension 4 we obtained that there is a curve with dilation \(<62\). Some lower bounds on the dilation for wider classes of cubically decomposable curves are proven.

Plane curves of genus p and degree 2p as projections of space curves of the same degree

We characterize plane curves Γ of genus p and degree 2p with respect to the possibility of obtaining them as projections of space curves C′ of the same degree. When Γ is hyperelliptic, we link this characterization with the configuration of the singularities of Γ and with the position of C′ on certain scrolls.