We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite compactly supported measure µ, with µ (ℝd) > 0 for p ≥ 1 and λ > 0 we consider the functional E(γ) = ∫ℝd d(x, Γγ)pdµ(x) + λ Length(γ) where γ : I → ℝd, I is an interval in ℝ, Γγ = γ(I), and d(x,Γγ) is the distance of x to Γγ. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure H¹, and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form of estimates on the total curvature of the minimizers. We prove that for measures µ supported in two dimensions the minimizing curve is injective if p ≥ 2 or if µ has bounded density. This establishes that the minimization over parameterized curves is equivalent to minimizing over embedded curves and thus confirms that the problem has a geometric interpretation. [ABSTRACT FROM AUTHOR]