We refine a previously introduced Monte Carlo method for simulating random surfaces. This allows us to calculate with high precision critical indices for planar random surfaces without spikes. We assume standard scaling laws. Within errors of only a few percent our results in four dimensions are: ν=1/4, γ=1/4, dH=4, η=1. In contrast to planar random surfaces with spikes the model is non-trivial: The two-point function has an anomalous dimension η≠0.