In differential geometry, curvature-based functionals, such as the total Gaussian curvature, the Willmore energy, and the total geodesic torsion, play a central role in both theoretical investigations and practical applications. In this paper, we study geometric properties of the extremal curves for the next functional \(\begin{aligned} \mathfrak {F}:=\int {\sqrt{H^2-K}}ds=\int {\frac{(k_1-k_2)}{2}}ds, \end{aligned}\) where ds is the arc element on S and \(k_1,k_2\) are the principal curvatures. First, we establish that a necessary and sufficient condition for a surface to be a Dupin cyclide is that its lines of curvature and the extremal curves of functional \(\mathfrak {F}\) intersect at a constant angle. Secondly, we demonstrate that the extremal curves of the functional \(\mathfrak {F}\) are invariant under inversion. Finally, we show that the determination of functional extremal curves of \(\mathfrak {F}\) for any cone, general cylinder, and surfaces of revolution can be reduced to quadratures.
