Shading makes faceted objects appear smooth. However; the contour will still appear non smooth. Subdivision schemes can handle this problem by introducing new polygons in the mesh. The disadvantage is that a more complex mesh takes more time to render than a simple one. We propose a new method for constructing a curvilinear mesh using quadratic curves with near least square acceleration. This mesh could be used for subsequent subdivision of the surface. This can be done on the fly; at least in software rendering; depending on the curvature of the contour. The advantage is that new polygons are only inserted where needed. However; in this paper we will focus on how such curvilinear mesh can be constructed using vertex points and vertex normals for each polygon. Thus; information about neighboring polygons are not needed and on the fly subdivision is made easier.