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.