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Euler used intrinsic equations expressing the radius of curvature as afunction of the angle of inclination to find curves similar to their evolutes.We interpret the evolute of a plane curve optically, as the caustic (envelope)of light rays normal to it, and study the Euler's problem for general caustics.The resulting curves are characterized when the rays are at a constant angle tothe curve, generalizing the case of evolutes. Aside from analogs of classicalsolutions we encounter some new types of curves. We also consider caustics ofparallel rays reflected by a curved mirror, where Euler's problem leads to anovel pantograph equation, and describe its analytic solutions.

Caustics of light rays and Euler's angle of inclination