Mesh segmentation has become an important step in model understanding and can be used as a useful tool for different applications, for instance, modeling, computer aided design (CAD), and reverse engineering. In this paper, we present a novel application of the platonic solids to find direction vectors for grouping the surface mesh elements. Normal vectors of the faces of the selected platonic solid are defined as the direction vectors. Our algorithm divides a polygonal mesh into the color regions (segments) with polygonal elements with normals that correspond to the direction vectors. Results of experiments on real 3D models demonstrate the performance and efficiency of the proposed algorithm. The original contribution in this paper is using normals of the faces of the platonic solids as the direction vectors for grouping mesh elements of the 3D surface meshes.

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  We propose a new approach to obtain the solutions of HJB equations. Firstly, based on a simple but unique idea, we transform the conventional HJB equations into the virtual-time HJB equations (VT-HJB). As the virtual time goes by, the solution of the VT-HJB equations becomes that of the conventional ones. Then, we solve the VT-HJB equations with respect to the virtual time by means of existing differential equation solvers. Lastly, we demonstrate the effectiveness of our method through numeric experiments with linear and nonlinear control problems.

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