The paper develops a method to design nonlinear splines on a plane via curve evolutions driven by curvature. We consider a curve passing through two given end-points and satisfying prescribed boundary conditions at them (for example, curvature values or tangent directions are specified at the end-points). Each point of the curve moves in the normal direction with speed equal to a function of the curvature and curvature derivatives at the point. Chosen the speed function properly, the evolving curve converges to a desired nonlinear spline. We also consider evolutions of closed curves for purposes of multiscale shape analysis. Smooth curve evolutions are approximated by evolutions of polygonal curves. Discrete analogs of the curvature and its derivatives are considered.