In this work, we return to Descartes’s idea to develop a formalism to construct rigorously stigmatic singlet lenses comprising two Cartesian surfaces. Optical systems are built using a considerable number of spherical surfaces, presenting in most cases spherical aberration. Wasermann and Wolf proposed eliminating spherical aberration and minimizing third-order coma by using two adjacent aspherical surfaces. That is why, using a parametric formulation for Cartesian ovals, we propose the design of singlet lenses where the condition of rigorous stigmatism is guaranteed for each surface, and therefore, strictly speaking, in the pair of stigmatic points, the lens becomes an optical system free of spherical aberration. This formulation is unified to both refractive and reflective optical surfaces. Therefore, within the framework of the theory of rigorously stigmatic optical systems, making use of Cartesian surfaces for the construction of stigmatic ovoid singlet lenses, we achieve the same functionality of optical systems involving a set of spherical lenses. These lenses have the advantage of being formulated according to a generalized shape factor associated with the Coddington shape factor, allowing an easy classification of these stigmatic lenses. The ideal imaging is carried out by applying an exact ray-tracing method through these ovoid singlet lenses.

中文翻译：

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超圆锥形无规卵形单透镜。

在这项工作中，我们回到笛卡尔的想法，发展形式主义，以构造出由两个笛卡尔曲面组成的严格单调单重透镜。光学系统是使用大量球面构建的，大多数情况下会出现球差。Wasermann和Wolf建议通过使用两个相邻的非球面表面消除球差并最小化三阶彗形象差。因此，我们建议使用笛卡尔椭圆形的参数公式来设计单重透镜，在这种情况下，每个表面都必须具有严格的散光条件，因此，严格来说，在成对的散光点中，透镜成为光学系统没有球差。该配方统一用于折射和反射光学表面。因此，在严格的散光光学系统理论的框架内，利用笛卡尔曲面构造散光的卵形单重透镜，我们实现了涉及一组球面透镜的光学系统的相同功能。这些镜片的优点是可以根据与Coddington形状因子相关的通用形状因子进行配制，从而可以轻松地对这些散光镜片进行分类。理想的成像是通过对这些卵形单线透镜应用精确的光线追踪方法来实现的。这些镜片的优点是可以根据与Coddington形状因子相关的通用形状因子进行配制，从而可以轻松地对这些散光镜片进行分类。理想的成像是通过对这些卵形单线透镜应用精确的光线追踪方法来实现的。这些镜片的优点是可以根据与Coddington形状因子相关的通用形状因子进行配制，从而可以轻松地对这些散光镜片进行分类。理想的成像是通过对这些卵形单线透镜应用精确的光线追踪方法来实现的。