Abstract: Freeform surface rendering is traditionally performed by approximating the surface with polygons and then rendering the polygons. This approach is extremely common because of the complexity in accurately rendering the surfaces directly. Recently several papers presented methods that render surfaces as sequences of isocurves. These methods each have deficiencies in their ability to guarantee a complete coverage of the rendered surface, in their ability to prevent processing the same pixel multiple times, or in their ability to produce an optimal surface coverage under some prescribed norm. In this article, an algorithm is introduced that alleviates the difficulties in all these areas. This algorithm can be combined with a fast curve-rendering method to make surface rendering without polygonal approximation practical.