We call a surface that appears undistorted when viewed in a curved mirror an eigensurface and the mirror an eigenmirror. Such pairs are described by a first-order nonlinear partial differential equation of the form ${a_0} + {a_1}{u_x} \,+ {a_2}{u_y} + {a_3}{u_x}{u_y} + {a_4}u_x^2 + {a_5}u_y^2 = 0$, where ${a_i} = {a_i}(x,y,u)$, which we call the anti-eikonal equation. We give examples of symbolic and numerical solutions, including pairs that are geometrically congruent. Ray tracing simulations are included that visually confirm the unusual properties of these surfaces.

中文翻译：

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本征镜的反本征方程

我们将在曲面镜中观察时未变形的表面称为本征面，将镜面称为本征镜。这样的对通过形式为$ {a_0} + {a_1} {u_x} \，+ {a_2} {u_y} + {a_3} {u_x} {u_y} + {a_4}的一阶非线性偏微分方程来描述u_x ^ 2 + {a_5} u_y ^ 2 = 0 $，其中$ {a_i} = {a_i}（x，y，u）$，我们称其为反本征方程。我们给出了符号和数值解的示例，包括几何上一致的对。包括光线追踪模拟，可以从视觉上确认这些表面的异常特性。