A connection between mathematical and literary terms.
In old-fashioned high school math classes "back in the day", we learned about the conic sections, the ellipse, the parabola, and the hyperbola, in a very "classical" way. ( A circle is a special case of an ellipse.)
One way to distinguish among the three is a ratio called the eccentricity. Each of these three types of curves can be described as follows:
There is a fixed point F called the focus and a fixed line D called the directrix with the following property: if we take any point P on the curve, then the ratio
e = (distance from P to F) / (distance from P to D)
is a constant. It has the same value for every point P on the curve. This constant "e" is called the eccentricity of the curve.
If e > 1, the curve is a hyperbola. If e = 1, the curve is a parabola If e < 1, the curve is an ellipse.
On the other hand.....
Hyperbole refers to exaggeration, which in a sense is an "excess" ( e > 1)
A parable is a story in which one thing stands for another, just as the numerator and denominator are equal to each other when e = 1. (In fact, in the Latin Bible, the word that appears in the New Testament is "parabola!")
An elliptical sentence is one that is vague or ambiguous because it "lacks" something ( e < 1).
These Greek prefixes cut across different areas of learning!
I once was talking with a friend about exercise equipment. I knew very little about "elliptical machines", and I asked him what that term referred to. I assumed the machine had a belt in the shape of an ellipse, but he said that he was not aware of that being the case. So he joked, "Maybe it's a machine with parts missing!"
I responded, "In which case, all the advertising for them would be hyperbolic claims about elliptical machines!"
Yeah, he is a math nerd like me.
(Maybe I should have posted this under "Greek Language" rather than English! lol)
This page is a permanent link to the reply below and its nested replies. See all post replies »
If you consider the angle of the cut plane with the axis of the cone, with a uniform probability of any angle, you expect to see ellipses and hyperbolæ, because circles and parabolæ are only produced at 1 specific angle.
This exactly maps to orbital solutions for the "two-body problem" in classical celestial mechanics.
Min total energy for an orbit, E = E_{min} is < 0, (bound orbit) => circular orbit
Neg total energy E_{min} < E < 0 (bound orbit) => elliptical orbit
0 total energy E = 0 (not bound) => parabolic orbit (goes to infinity and has no kinetic energy)
positive total energy E > 0 (free) => hyperbolic orbit (goes to infinity and has extra kinetic energy)
In high school, I learned about moderm physics (statistical mechanics, special relativity, basic quantum theory), the simplex method (The Theory of Games and Economic Behavior by Morgenstern & Van Neumann), 2 yrs of college math (3 sem calc + 1 of ord diff equations), 1 yr of engineering physics (Halliday & Resnick).
