Implicitly defined surfaces f(x,y,z)=0 are usually visualized by an approximating mesh of polygons. One approach of calculating an approximating mesh, practiced in the past, is to subdivide the regions of interests into spatial cells from which the surface is extracted by compiling surface patterns of those cells which are traversed by the surface. Our solution basically follows this approach, but improves it with respect to several aspects. The space is partitioned adaptively using a scheme which neither requires multiple passes nor storing neighborhood information. Furthmore, the cells of the approximating surface mesh are part of the resulting spatial cell decomposition. A post-processing step of ''vertex snapping'' in order to improve the surface mesh is not required.