A method to divide any given surface into two regions with two properties:
(1) they are visually interlocked since the boundary curve covers the whole surface by meandering over it and
(2) the areas of these two regions are approximately the same. We obtain the duotone surfaces by coloring these regions with two different colors.We show that it is always possible to obtain two such regions for any given mesh surface. Our approach is based on a useful property of vertex insertion schemes such as CatmullClark subdivision: If such a vertex insertion scheme is applied to a mesh, the vertices of resulting quadrilateral mesh are always two colorable. Using this property, we can always classify vertices of meshes that are obtained by a vertex insertion scheme into two groups. We show that it is always possible to create a single curve that covers the whole surface such that all vertices in the first group are on one side of the curve while the other group of vertices are on the other side of the same curve. This single curve serves as a boundary that defines two regions in the surface. If the initial distribution of the vertices on the surface is uniform, the areas of the two regions are approximately the same. We have implemented this approach by appropriately mapping textures on each quadrilateral. The resulting textured surfaces look aesthetically pleasing since they closely resemble planar TSP (traveling salesmen problem) art and Truchetlike curves.

