Corner reflector

Every­one knows that there is no light emit­ter in the cat's eyes attached to the bicy­cle wheels. How­ever, a dri­ver pass­ing by the cyclist sees the cat's eye very well when it's in the light of his car. Have you ever thought that late pedes­tri­ans may not see the light of the reflec­tor? It may seem amaz­ing, but the cat's eye prop­er­ties are based on sim­ple geo­met­ric facts.

As we know from geo­met­ric optics, a ray reflects accord­ing to the law "the angle of reflec­tion equals the angle of inci­dence".

Con­sider the plane case: two mir­rors at right angles. A ray in this plane falling on one of the mir­rors, after reflect­ing from the sec­ond returns in the same direc­tion it came from. Check it com­put­ing the angles or ana­lyz­ing the direc­tion vec­tor of the ray.

To get a sim­i­lar effect in our ordi­nary three-dimen­sional space we need to place the mir­rors in pair­wise orthog­o­nal planes. Take a cor­ner of a cube with a bor­der form­ing a reg­u­lar tri­an­gle.

A ray falling on such a mir­ror sys­tem after reflect­ing from all the planes returns par­al­lel to the incom­ing ray in the oppo­site direc­tion. Check it!

This sim­ple geo­met­ric device with its prop­er­ties is called a cor­ner reflec­tor. To use it one makes a bat­tery of such cor­ners increas­ing the area of reflec­tion. The sim­plest math­e­mat­i­cal ideas are used: we can tile a plane with tri­an­gles, so the cor­ner reflec­tors are easy to attach to each other.

This is how a bicy­cle or a car cat's eye work. How­ever, this geo­met­ric con­sid­er­a­tions are used in much more tech­ni­cal devices.

When one started to build Luno­hod, no one knew what was the sur­face of the Earth satel­lite like. Was it hard or there was so much dust that the landed appa­ra­tus would have to float in it. There were long debates that fin­ished with  a note of Sergei Pavlovich Korolev (1907-1966): "One should expect quite a hard soil, kind of pumice. [...] Korolev". That is how great sci­en­tists were not afraid to solve dif­fi­cult prob­lems and take all the respon­si­bil­ity.

On Novem­ber 17, 1970 a sta­tion, called in the mes­sage of the main infor­ma­tion agency, Tele­graph Agency of the Soviet Union (TASS), "Luna-17", landed near the Sea of Rains on the Moon. A soviet appa­ra­tus first rut. It was dri­ven from the Earth and one could see a small plot in front of the appa­ra­tus. Intended to work dur­ing three months, the appa­ra­tus worked three times longer: 11 full lunar days. The last com­mu­ni­ca­tion ses­sion with the first lunar rover hap­pened on Sep­tem­ber 14, 1971. Dur­ing this period Luno­hod-1 passed 100 km 540 m mak­ing a cir­cle and return­ing to its orig­i­nal point.

Sur­pris­ingly, Luno­hod was equipped cor­ner reflec­tors! First of all, they allowed every coun­try check the pres­ence of a soviet appa­ra­tus on the Moon. And what is more impor­tant, such a sim­ple geo­met­ric device helped sci­ence to mea­sure the dis­tance to the Earth. Sci­en­tists from all coun­tries used cor­ner reflec­tors of Luno­hod-1 even in the XXI cen­tury.

This is how sim­ple geo­met­ric con­sid­er­a­tions help peo­ple start­ing from every­day ques­tions of safety to study­ing the Uni­verse.