One of the strategies to detect the pose and shape of unknown objects is their geometric modeling, consisting on fitting known geometric entities. Classical geometric modeling fits simple shapes such as spheres or cylinders, but often those don't cover the variety of shapes that can be encountered. For those situations, one solution is the use of superquadrics, which can adapt to a wider variety of shapes. One of the limitations of superquadrics is that they cannot model objects with holes, such as those with handles. This work aims to fit supersurfaces of degree four, in particular supertoroids, to objects with a single hole. Following the results of superquadrics, simple expressions for the major and minor radial distances are derived, which lead to the fitting of the intrinsic and extrinsic parameters of the supertoroid. The differential geometry of the surface is also studied as a function of these parameters. The result is a supergeometric modeling that can be used for symmetric objects with and without holes with a simple distance function for the fitting. The proposed algorithm expands considerably the amount of shapes that can be targeted for geometric modeling.

翻译：检测未知物体姿态与形状的策略之一是通过拟合已知几何实体进行几何建模。传统几何建模方法通常拟合球体或圆柱体等简单形状，但这些形状往往无法涵盖实际遇到物体的多样性。针对此类情况，一种解决方案是采用超二次曲面，其能适应更广泛的形状类别。然而超二次曲面的局限在于无法建模带孔物体（如带把手物体）。本研究旨在将四阶超曲面（特别是超环面）拟合至具有单孔结构的物体。基于超二次曲面的研究成果，推导出主次径向距离的简明表达式，进而实现超环面内外参数的拟合。同时结合参数体系研究了曲面的微分几何特性。最终构建的超几何建模框架可通过简洁的距离函数，实现对称带孔/无孔物体的统一建模。所提算法显著拓展了几何建模可处理的形状类别范围。