Sunday, June 12, 2011

Conic sections

Since early spring I've been hunting conic sections in the wild. Let me start with some examples and not bore you with the mathematics:
Hyperbolas are the easiest to find and to notice - at least if you're looking for conic sections of light cones.
Ellipses are more difficult to find. In real life people don't often aim their spotlights at walls. And I've never found a parabola. The mathematics picture explains it in one look:
Conic sections with plane
An infinite number of dissecting angles produce ellipses or hyperbolas. Just one specific angle produces a parabola. So the chance of finding a parabola is infinitely small.

There are many - more complex - situations that produce conic sections. Marcel Minnaert describes one of them - the interference pattern between vertical fence-posts and their shadows. Actually - I seemed to remember that Minnaert had proven these to be hyperbolas - but when I searched his writings I didn't find the proof.

An even more complex situation occurs at a train station I know. A Quonset hut made from corrugated steel lies behind a fence with a rectangular mesh. The interference of these line patterns creates curved lines that might be conic sections - but I'm undecided it they are ellipses or hyperbolas.

How often do these conic sections occur in nature? One simple experiment is to count their occurrence in the works of Marcel Minnaert, on Flickr and on Google picture search. This yields the following results:

Source Minnaert Flickr Google
Circle 81 1.197.208 51,500,000
Ellipse 29 10.875 1,200,000
Hyperbola 16 244 1,010,000
Parabola 5 6.572 95,800

Conic sections - Wikipedia

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