Plane curves with curvature depending on distance to a line

Motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves, we study the plane curves whose curvature is expressed in terms of the (signed) distance to a line. In this way, we provide new characterizations of some well known curves, like the catenary or the grim-reaper. We also find out several interesting families of plane curves (including closed and embedded ones) whose intrinsic equations are expressed in terms of elementary functions or Jacobi elliptic functions and we are able to get arc length parametrizations of them and they are depicted graphically.