Curvelets are known to provide a near-optimal representation of functions smooth away from a C^2 curve, with applications e.g. in image compression. The key to outperform, say, wavelets is to subdivide phase space according to the so-called parabolic scaling. We will show why this geometric view also proves to be successful in harmonic analysis, for the sparsification of Fourier Integral Operators. We will argue why other classical systems like wavelets or ridgelets cannot be expected to behave equally well. We will discuss the potential significance of this fact for the numerical analysis of wave equations.