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The point of this question is to compile a list of applications of hyperbola because a lot of people are unknown to it and asks it frequently.

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closed as not a real question by Andres Caicedo, Davide Giraudo, muzzlator, vonbrand, Shaktal Apr 11 '13 at 21:37

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Should I upvote the question because it will certainly bring some interesting answers, or should I downvote it since any basic research regarding the word "hyperbola" on the web already gives a lot of answers? –  Djaian Apr 11 '13 at 9:26
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@Djaian: That neutralizes and becomes $0$ vote indeed. ;) –  Inceptio Apr 11 '13 at 10:07
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...not to be confused with "hyperbole", which is a bajillion times more awesome than any hyperbola. –  zzzzBov Apr 11 '13 at 14:51
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8 Answers 8

Applications of hyperbola

Dulles Airport, designed by Eero Saarinen, is in the shape of a hyperbolic paraboloid. The hyperbolic paraboloid is a three-dimensional curve that is a hyperbola in one cross-section, and a parabola in another cross section.

This is a Gear Transmission. Greatest application of hyperbola:

enter image description here

And hyperbolic structers are used in Cooling Towers of Nuclear Reactors.. Doesn't it make hyperbola, a great deal on earth? :)

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+1 for the gear illustration. –  Alex Becker Apr 11 '13 at 8:59
    
@AlexBecker: ..:) –  Inceptio Apr 11 '13 at 9:01
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@MattPressland: hyperboloids are quadric surfaces and contain infinitely many lines, as shown in the picture. –  Matthew Leingang Apr 11 '13 at 17:39
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@MatthewLeingang Hmm, of course - as you say, I was looking at a picture of this fact when I wrote my comment. I always associate the cooling tower picture with Miles Reid's book Undergraduate Algebraic Geometry (where it appears when talking about the infinitely many lines on a quadric surface), and thus with the 27 lines, which is one of Reid's favourite examples and also appears prominently in the book, although of course the two have little to do with each other. Anyway, my previous comment stands if you replace "cubic" by "quadric" and "27" by "infinitely many". –  Matt Pressland Apr 11 '13 at 17:47
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@MatthewLeingang Ha, don't worry! I make silly mistakes often enough that I don't really have time to be too embarrassed about them! Better to correct it. –  Matt Pressland Apr 12 '13 at 10:31
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Of course it does. Among other things, this is the function that describes the trajectory of comets and other bodies with open orbits. Another astronomy related use is Cassegrain telescopes, where hyperbolic mirrors are used (

enter image description here

Image by Szőcs Tamás.

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And similarly, radio antennas (which are a bit more practical). –  Alex Becker Apr 11 '13 at 9:00
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Extreme-telephoto mirror lenses for cameras are also built on this principle. It's the only practical way I know of to get a 1000mm+ focal length on a lens that isn't actually a meter long. –  fluffy Apr 11 '13 at 20:54
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Did you ever take a look at the light projected onto a wall by a nearby lamp with a standard lampshade? That's right: the light on the wall due to the lamp has a hyperbola for a bounday. The reason for this is clear once you think about it for a second: the light out of the lampshade forms a vertical cone, and the intersection of a vertical cone and a vertical wall makes a hyperbola.

Also, consider a pair of sources of ripples in water that produce concentric waves. The intersections of those concentric waves - surfaces of constant phase, are hyperbolae. Why? Because a hyperbola is the locus of points having a constant distance difference from two points (i.e., a phase difference is is constant on the hyperbola).

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+1: Nice examples, and clear explanations to help the "light to go on". –  LarsH Apr 11 '13 at 14:07
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@LarsH: thanks. When my son was in kindergarten, he actually asked me what the shape of the light was on the wall. I told him and had him repeat it to his utterly baffled teacher. –  Ron Gordon Apr 11 '13 at 14:35
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In addition to the awesome answers, here is something mundane: a hyperbola occurs whenever you have a formula of the form $$xy = c$$ Two hyperbolas, if you consider negative values. Equations of this form crop up all over the place, in natural sciences, economics, you name it.

Then, in space, when a small mass passes by a large one (say, comet around a planet), and it is moving faster then escape velocity with respect to the large one, its path is hyperbolic.

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"Two hyperbolas, if you consider negative values." Not to be overly pedantic, but I think that's still one hyperbola (but with both its branches). –  Jesse Madnick Apr 11 '13 at 20:42
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Fair enough, indeed. –  Kaz Apr 11 '13 at 21:38
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Hyperbolas are used extensively in economics and finance (specifically portfolio theory), where they can represent the various combinations of securities, funds, etc. that yield similar risk-return ratios. This is why you often see efficient portfolio frontiers represented as partial hyperbolas. For similar reasons, production frontiers, which represent various combinations of capital and labor that produce a given output, as hyperbolas.

I don't know if that's entirely a "real-world" example because it's not a tangible object, but the mathematics of hyperbolas are still very important.

For example, the upper edge of this hyperbola (the part of the curve above the inflection point) in this plot:

enter image description here

represents the optimal combination of two risky assets, assuming the portfolio doesn't contain any risk free assets like Treasury bills. In this case, an optimal allocation is one that provides the highest ratio of expected return to risk, i.e. standard deviation. This is also known as the Sharpe Ratio.

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Why the downvote? I'd like to improve my answer if necessary. –  John Bensin Apr 18 '13 at 3:13
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Most receptors are made in the shape. e.g. RADARs, television reception dishes, etc. because they need to reflect off the signal and focus it on a single "point".

When two stones are thrown in a pool of water, the concentric circles of ripples intersect in hyperbolas. This property of the hyperbola is used in radar tracking stations: an object is located by sending out sound waves from two point sources: the concentric circles of these sound waves intersect in hyperbolas.

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I was thinking TV dishes etc. used a parabolic shape (Parabola is even used as a brand name) when they're designed to focus on a single point. But I could be wrong ... I don't know why a telescope could have a hyperbolic mirror as well as a parabolic one. –  LarsH Apr 11 '13 at 14:45
    
You are correct of course. Practically, there is no difference between parabola and hyperbola - hyperbola is just a parabola with a mirror image ;-) –  OC2PS Apr 12 '13 at 2:22
    
I thought there was a more significant qualitative difference between the two. For example, in the illustration on this page of a telescope containing a hyperbolic mirror and a parabolic one, the hyperbolic mirror doesn't have a mirror image. I realize that the "conic section" definition hinges on whether a plane intersects both halves or just one half of a double cone. Yet there seems to be more to it than whether the curve has one branch or two. –  LarsH Apr 12 '13 at 14:45
    
I don't believe there's a qualitative difference between the two. –  OC2PS Apr 12 '13 at 23:43
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In computer science, it's the shape of the response-time curve for request-reply pairs. It starts off parallel to the x-axis at low loads, curves upwards and ends up approaching parallel to the line y = (Dmax * x) - Z, where Dmax is the service demand of the slowest part of the system and Z is the user think time between requests.

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Hyperbolas are used extensively in Time Difference of Arrival (TDoA) analysis, which has many applications. For example, it is used for geolocation to determine the location of a vehicle relative to several radar emitters (e.g. passive geolocation of UAVs), localizing cellular phones without requiring a GPS fix (e.g. U-TDOA), or making "tapscreens" that can sense the precise location of a tap on a large display without expensive touchscreens (e.g. MIT's Tapper).

In TDoA, multiple sensors each detect the arrival time of a particular signal. The time differences between any two sensor measurements define a hyperbola of possible origin locations (since those are the points with a constant difference in distance to each sensor). Intersecting the hyperbolas gives you the position of the signal's source very quickly and precisely.

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