This occurred to me this morning, when I saw the following three-fold characterization of the conic sections, which may amuse. (I'm basically copying this from Leila Schneps' lecture notes.)
Conic sections can be defined geometrically, analytically, or algebraically.
- Geometrically: a conic section is the curve obtained when a plane intersects with a cone. The conic section is an ellipse, parabola, or hyperbola according to whether the plane is less steeply slanted, exactly as steeply slanted, or more steeply slanted than the cone's generating line.
- Analytically: given a point F the focus and a line D the directrix, a conic section is the locus of points P such that d(P,F)/d(P,D) is a constant e, the eccentricity. The conic section is an ellipse, parabola, or hyperbola according to whether the eccentricity is less than, equal to, or greater than 1.
- Algebraically: a conic section is the solution set of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. The conic section is an ellipse, parabola, or hyperbola according to whether B2 - 4AC, the discriminant, is negative, zero, or positive.
The three definitions are equivalent.
It's fitting that the quote is in Latin, because I don't know Latin, and I also don't believe that mathematics is divided into only these three parts. In particular, you all know that I like probability and combinatorics, which don't naturally fit into this tripartite scheme. But it's a nice division of what one might call "classical" mathematics (it's roughly the content of the semi-standard first-year graduate mathematics curriculum, for example).
Calling the second definition "analytic" is a bit of a stretch, though.
3 comments:
There is a nice geometric treatment of an ellipse at cut-the-knot, hyperbolas and parabolas are left as an exercise, or you can click on Xah Lee for more beautiful pictures and explanations. Enjoy.
It looks like any interesting mathematics doesn't fit neatly into any bureaucratic scheme, chopping mathematics into pieces and hardly mentioning one part while teaching the other really kills it.
Ellipses ...
It's interesting that you formulate the definitions in terms of conic sections. I've been thinking about this lately. Some of the more interesting things that I have seen lately can't be conic sections.
I make the following wild assed guess:
We like math because we like to measure. When we start to measure we end up trying to optimize. We do optimization on convex sets because they don't suffer what we might think of as pathologies (local, non global extrema). Conic sections are well understood convex sets. Unfortunately, some of the most interesting optimization problems are not on convex sets.
Well, the meaning behind the original statement in Latin was that there was more than one tribe of Gauls, which was surprising to Caesar and the Romans.
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