Mirrors that are convex in some places and concave in others can generate images of extended objects (such as the viewer’s face) that are curiously distorted and often topologically disrupted. Understanding these images involves the caustics of the family of rays emitted by each point of the object, and the totality of all such families constituting the rays from all points of the object. The general theory is illustrated by the simplest mirror with an inflection, whose profile is a cubic function. Simulations, and observations with a flexible plastic mirror, show how the image changes as the viewer moves relative to the mirror.

中文翻译：

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屈折反射：镜子中的图像，其曲率改变符号

在某些地方凸出而在其他地方凹入的镜子可以生成扩展对象（例如观看者的脸）的图像，这些图像奇怪地扭曲并且经常被拓扑破坏。理解这些图像涉及物体每个点发出的光线系列的焦散，以及构成来自物体所有点的光线的所有这些系列的总和。一般理论是用最简单的反射镜来说明的，反射镜的轮廓是三次函数。使用柔性塑料镜进行模拟和观察，可以显示当观察者相对于镜子移动时图像是如何变化的。