It is known that, besides being stigmatic, spherical refracting surfaces are aplanatic at their Young points since they satisfy the Abbe sine condition rigorously. The Abbe sine condition is commonly applied to different optical systems using numerical methods or optimization processes, obtaining a design of approximately aplanatic systems. Here, we found several families of Cartesian surfaces, whose sets of each of these families constitute exactly aplanatic systems free of spherical aberration and coma. So, studying the different types of systems, it is found that rigorous aplanatism occurs for objects and images on curved surfaces.

中文翻译：

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严格的消球差笛卡尔卵形

众所周知，球面折射面除了是柱头外，还因为严格满足阿贝正弦条件，所以在其杨点处是消球差的。阿贝正弦条件通常用于使用数值方法或优化过程的不同光学系统，从而获得近似消球差系统的设计。在这里，我们发现了几个笛卡尔表面家族，这些家族中的每一个都构成了完全没有球差和彗差的消球差系统。因此，通过研究不同类型的系统，发现曲面上的物体和图像会发生严格的消球差。