The point set is a flexible surface representation suitable for both geometry processing and real-time rendering. In most applications, the control of the point cloud density is crucial and being able to refine a set of points appears to be essential. In this work, we propose a new interpolatory refinement framework for point-based geometry. First we carefully select an appropriate one-ring neighborhood around the central interpolated point. Then new points are locally inserted where the density is too low using a sqrt(3)-like refinement procedure and they are displaced on the corresponding curved Point Normal triangle. Thus, a smooth surface is reconstructed by combining the smoothing property produced by the rotational effect of sqrt(3)-like refinements with the points/normal interpolation of PN triangles. In addition we show how to handle sharp features and how our algorithm naturally fills large holes in the geometry. Finally, we illustrate the robustness of our approach, its real-time capabilities and the smoothness of the reconstructed surface on a large set of input models, including irregular and sparse point clouds.