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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?

Jack D'Aurizio
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Art Solver
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  • I am sure that architects, engineers, construction professionals and college professors will disagree with your friend – Vasili Oct 26 '17 at 17:28
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    It is useful to me to make a living as a teacher. BTW, what exactly is "real life"? It means that if I study the properties of figures I live in "Fantasia" – Raffaele Oct 26 '17 at 17:36
  • I'm not saying about basics, but stuff of olympiad or more advanced level – Art Solver Oct 26 '17 at 17:42
  • Try building pyramid without geometry. – Rene Schipperus Oct 26 '17 at 18:31
  • If one aggregates the techniques and the viewpoints of Euclidean geometry and analytical geometry, your friend is completely wrong. For example, I am working on image processing ; there is a huge number of questions that rely on geometry . A good practice of geometry is a blind stick in this domain, especialy when you deal with 3D. – Jean Marie Oct 26 '17 at 20:13

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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.

Jack D'Aurizio
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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.

bubba
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