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Opt. Soc. Am. A"],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"We call a surface that appears undistorted when viewed in a curved\n\t\t\t\t\tmirror an eigensurface<\/jats:italic> and the mirror an\n\t\t\t\t\teigenmirror<\/jats:italic>. Such pairs are described by\n\t\t\t\t\ta first-order nonlinear partial differential equation of the form \n \n\t\n\t \n\t a<\/mml:mi>\n\t 0<\/mml:mn>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t+<\/mml:mo>\n\t\n\t \n\t a<\/mml:mi>\n\t 1<\/mml:mn>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t\n\t \n\t u<\/mml:mi>\n\t x<\/mml:mi>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t\n\t+<\/mml:mo>\n\t\n\t \n\t a<\/mml:mi>\n\t 2<\/mml:mn>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t\n\t \n\t u<\/mml:mi>\n\t y<\/mml:mi>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t+<\/mml:mo>\n\t\n\t \n\t a<\/mml:mi>\n\t 3<\/mml:mn>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t\n\t \n\t u<\/mml:mi>\n\t x<\/mml:mi>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t\n\t \n\t u<\/mml:mi>\n\t y<\/mml:mi>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t+<\/mml:mo>\n\t\n\t \n\t a<\/mml:mi>\n\t 4<\/mml:mn>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t\n\t u<\/mml:mi>\n\t x<\/mml:mi>\n\t 2<\/mml:mn>\n\t<\/mml:msubsup>\n\t+<\/mml:mo>\n\t\n\t \n\t a<\/mml:mi>\n\t 5<\/mml:mn>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t\n\t u<\/mml:mi>\n\t y<\/mml:mi>\n\t 2<\/mml:mn>\n\t<\/mml:msubsup>\n\t=<\/mml:mo>\n\t0<\/mml:mn>\n <\/mml:math>\n <\/jats:inline-formula>, where \n \n\t\n\t \n\t a<\/mml:mi>\n\t i<\/mml:mi>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t=<\/mml:mo>\n\t\n\t \n\t a<\/mml:mi>\n\t i<\/mml:mi>\n\t <\/mml:msub>\n\t<\/mml:mrow>\n\t(<\/mml:mo>\n\tx<\/mml:mi>\n\t,<\/mml:mo>\n\ty<\/mml:mi>\n\t,<\/mml:mo>\n\tu<\/mml:mi>\n\t)<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>, which we call the anti-eikonal equation<\/jats:italic>. We give examples of\n\t\t\t\t\tsymbolic and numerical solutions, including pairs that are\n\t\t\t\t\tgeometrically congruent. Ray tracing simulations are included that\n\t\t\t\t\tvisually confirm the unusual properties of these surfaces.<\/jats:p>","DOI":"10.1364\/josaa.401301","type":"journal-article","created":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T20:00:21Z","timestamp":1597780821000},"page":"1566","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Anti-eikonal equation of an eigenmirror"],"prefix":"10.1364","volume":"37","author":[{"given":"R. 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