Real life application of advanced Euclidean geometry

Question

My friend said me that there is no real life application (e.g. in CS, engineering...) of Euclidean geometry of Olympiad difficulty and higher. Is that really true or is there some "practical" use of it?

Answer 1

Not at all. Wave diffusion over smooth domains can be described in terms of the Laplace operator $\Delta$, and such theory leads to interesting results like the following one:

In a elliptic pool, any closed triangle trajectory has the same length.

On the other hand, Poncelet's porism, the optical properties of the ellipse and the relations between the symmedian point of a triangle and its orthic inconic lead to a elementary, Olympiad-style proof of the same claim.

And this is just an instance, of course.

Answer 2

Quite a lot of CAD (computer-aided design) and CAM (computer-aided manufacturing) is based on Euclidean geometry. Design geometry typically consists of shapes bounded by planes, cylinders, cones, tori, etc. CAD/CAM is essential in the design of almost everything, nowadays, including cars, airplanes, ships, and your iphone. A few decades ago, sophisticated draftsmen learned some fairly advanced Euclidean geometry, including things like Pascal's theorem and Brianchon's theorem. But now they don't have to, because the geometric constructions are all done by CAD programs.

There's a good example in this question. It's not an academic problem, it's related to a real situation in engineering design. It's solution appears in 19th century textbooks on Euclidean geometry.