This paper presents a computer-aided geometric design approach to realize a new genre of 3D puzzle, namely the 3D Polyomino puzzle . We base our puzzle pieces on the family of 2D shapes known as polyominoes in recreational mathematics, and construct the 3D puzzle model by covering its geometry with polyominolike shapes. We first apply quad-based surface parametrization to the input solid, and tile the parametrized surface with polyominoes. Then, we construct a nonintersecting offset surface inside the input solid and shape the puzzle pieces to fit inside a thick shell volume. Finally, we develop a family of associated techniques for precisely constructing the geometry of individual puzzle pieces, including the ring-based ordering scheme, the motion space analysis technique, and the tab and blank construction method. The final completed puzzle model is guaranteed to be not only buildable, but also interlocking and maintainable.

This paper introduces a novel surface-modeling method to stochastically distribute features on arbitrary topological surfaces. The generated distribution of features follows the Poisson disk distribution, so we can have a minimum separation guarantee between features and avoid feature overlap. With the proposed method, we not only can interactively adjust and edit features with the help of the proposed Poisson disk map, but can also efficiently re-distribute features on object surfaces. The underlying mechanism is our dual tiling scheme, known as the Dual Poisson-Disk Tiling. First, we compute the dual of a given surface parameterization, and tile the dual surface by our specially-designed dual tiles; during the pre-processing, the Poisson disk distribution has been pre-generated on these tiles. By dual tiling, we can nicely avoid the problem of corner heterogeneity when tiling arbitrary parameterized surfaces, and can also reduce the tile set complexity. Furthermore, the dual tiling scheme is non-periodic, and we can also maintain a manageable tile set. To demonstrate the applicability of this technique, we explore a number of surface-modeling applications: pattern and shape distribution, bump-mapping, illustrative rendering, mold simulation, the modeling of separable features in texture and BTF, and the distribution of geometric textures in shell space.

This paper proposes to perform reaction-diffusion on surface tiles. The square tiles fit nicely and cost-effectively in GPU memory, whereas we also apply distortion minimization on tiles so as to precisely reduce the unbalanced scale and resolution problem of chemicals in the reaction- diffusion. The interconnection nature of tiles accounts for the surface topology, and thus allows the chemicals to flow naturally over surfaces of arbitrary genus. Furthermore, by taking advantage of the tile structure, we can efficiently perform localized reaction-diffusion, and adjust the pattern formulation in an interactive manner on the GPU. To demonstrate its performance, we develop an interactive system that allows texture designers to fine-tune and alter the reaction-diffusion process by directly painting chemicals onto the object surface. Finally, we also develop several non-trivial applications of reaction-diffusion, including the geometry-dependent reaction-diffusion and deformation-aware reaction-diffusion.