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Self-contractedness (or self-expandedness, depending on the orientation) ishereby extended in two natural ways giving rise, for any$\lambda\in\lbrack-1,1)$, to the metric notion of $\lambda $-curve and the(weaker) geometric notion of $\lambda$-cone property ($\lambda$-eel). In theEuclidean space $\mathbb{R}^{d}$ it is established that for$\lambda\in\lbrack-1,1/d)$ bounded $\lambda$-curves have finite length. For$\lambda\geq 1/\sqrt{5}$ it is always possible to construct bounded curves ofinfinite length in ${\mathbb{R}}^{3}$ which do satisfy the $\lambda $-coneproperty. This can never happen in ${\mathbb{R}}^{2}$ though: it is shown thatall bounded planar curves with the $\lambda$-cone property have finite length.

Metric and geometric relaxations of self-contracted curves