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对其进行几何建模，即通过拟合已知几何实体来实现。经典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的各种物体形态。针对此类情况，一种解决方案是采用超二次曲面，其能够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（如带把手的物体）。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，本文推导了主径向距离与次径向距离的简明表达式，进而实现了超环面内禀参数与外禀参数的拟合。同时研究了该曲面微分几何特性与这些参数的函数关系。最终构建的超几何建模方法，可通过简单的距离函数拟合对称的带孔与无孔物体。所提算法显著拓展了几何建模方法所能处理的形状类别范围。