HYPERBOLA

When two stones are thrown simultaneously into a pool of still water, ripples move outward in concentric circles. These circles intersect in points which form a curve known as the hyperbola. The intersections of the white circles are points on hyperbolas (in blue) and ellipses (in green).

The hyperbola has applications in science and industry, and from our own backyards. The hyperbola is one of the four curves shown previously as sections of a cone.

The hyperbola as the pattern cast on a wall by a lamp with a cylindrical shade.The hyperbola can be constructed using two tacks and a piece of string.

Comparisons can be made to the ellipse, and a method for sketching the curve using a rectangle is given both for the hyperbola and its inverse.

The hyperbola is one of four curves called the Conics. These curves can all be described as sections cut from a cone, or pair of cones. Slicing the pair of cones parallel to the base results in a circular section; slicing at an angle results in an elongated circle: the ellipse. The more we tilt the slice, the longer and narrower is the ellipse until suddenly the curve is no longer closed, and is an open curve called the parabola. The parabola occurs only at the moment the slice is parallel to the side of the cone. There is only one shape of parabola, for if we tilt the slice any more, a new curve results: the hyperbola, and it has two branches as we have now intersected both parts of the cone.

Not all of the curve is shown, as the cone is theoretically infinite, and the curve of the hyperbola is also an infinite set of points. There are many different: shapes of hyperbolas, depending on the angle of the slice. The two branches of a hyperbola are the patterns cast on a wall by a lamp with a circular shade: casting two cones of light onto a vertical plane. The conical beam of light from a flashlight casts a circular pattern ... as we change the angle we see an ellipse, a parabola, and finally a hyperbola. Like the ellipse, the hyperbola has two focus points, and these. . .

The hyperbola is an infinite curve, and as it stretches out to infinity it begins to look more and more like a pair of straight lines. Mathematicians say the two branches of the hyperbola approach a pair of intersecting straight lines. These two lines are called the assymptotes of the hyperbola, and they are very useful in drawing the curve on a coordinate system. There is a very simple way to find the 2 2 equations of the assymptotes for the hyperbola x y = 1 The two a2 b2 lines are given by the equations y = b x and b a y -x You see, a and a b in the general equation of the hyperbola determine the shape of the curve, just as they did with the ellipse. A simple way to draw the assymptotes is to first sketch a rectangle "a" units to the left and to the right of the origin, and "b" units above and below. The diagonals of this rectangle are the assymptotes of the hyperbola. Sketch in the hyperbola so it approaches but does not touch the assymptotes. The ellipse fell within a similar type of rectangle, the hyperbola is reflected outside the rectangle. You can see that as "a" and "b" are different numbers, the shape of the hyperbola will change ... We can see many different shaped hyperbolas by constructing two groups of concentric circles. The concentric circles are the same distance apart with each group tangent. The intersections of success- ively larger circles will be points on a hyperbola. Choosing a different set of intersections will give us a hyperbola with different curvature. Mathematically speaking, this curvature is called eccen- tricity and is determined by the ratio between "a" and "b", those important numbers from the original equation for the hyperbola.

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.

The same concentric circle construction was used to create this mathematical poster.

The orbit of a planet is an ellipse With the sun as one of its focal points. It will remain an ellipse as long as it remains a closed curve, according to the laws of gravitation. If the planet were to speed up, the curve would open up into a parabola, and then a hyperbola. An object that passes through the solar system and does not orbit is on a hyperbolic path. The path of a comet as it passes the earth is a hyperbola.

The situation is similar to what occurs when we roll a steel ball past a strong magnet: the ball is deflected into a hyperbolic path. You can find examples of the hyperbola, and all the conics, as far away as the planets, and as near as your own home ... if you look for them.

A lot of things are inversely related to each other. In physics, for example, when you hold the temperature of a body of gas constant, then the volume and pressure are inversely related. If you plot those against each other, you find a hyperbola. Also, if you want to plot the number of people needed to do a huge job versus the total # of minutes needed to do it (assuming no increase in work capacity because somebody brings over a bulldozer instead of using spoons, etc), you also get an inverse relation, and a hyperbola.

Some types of secondary mirrors on telescopes are hyperboloids.

Loran uses time differences between transmitter pairs to find location. These time differences, when graphed, are hyperbolas. Bodies that orbit a larger body travel in elliptical paths. If the body makes one pass, never to return, it moves in a hyperbolic path. Some very interesting roof structures are in the shape of "hyperbolic parabaloids".

The other thing is to consider 3-D structures. Here the hyperboloid of one sheet comes into its own. A beautiful curved shape build of straight lines. As such, it is the most common curved shape you can get with ordinary timbers, and the forms made from them. It appears in curved roofs, in cooling towers for steam plants (including nuclear plants).

One of the most interesting websites on the internet is http://www.GreatBuildings.com This website has beautiful 3-D drawings and photographs of great architecture throughout the world. The photograph below came from this website; it is the Dulles Airport, designed by Eero Saarinen 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.

The next website will allow you to click on a hyperbolic paraboloid and move it around, to see it from different point of views. The webpage may take a minute or so to dowload.

http://mathworld.wolfram.com/HyperbolicParaboloid.html

In the Chicago series of books (FST, I think) it gives examples of how hyperbolas are used to triangulate a position.

SOME SUGGESTED ACTIVITIES FOR TEACHERS:

1. Give students accurate drawings of various hyperbolas with foci drawn for each. Have them measure the distances from one focus in each hyperbola to various points on the curve to the other focus. Have them do this for various points on each hyperbola to verity the definition in a concrete way.

2. Have students graph various hyperbolas from equations, using point plotting with at least ten points per hyperbola. Derive the method for sketching a hyperbola based on a and b as explained in the filmstrip. (letting x equal zero gives the x-intercepts, letting y equal zero gives the y-intercepts.) Have students verify that both the point plotting and the intercept method yield the same results.

3. After the activity above, have students take equations such as 2X2 - 3y2 = 6, put them in standard form and graph them. For more advanced classes the teacher might want to work with hyperbolas which have their centers anywhere on the coordinate system, rather than just at the origin.

4. Students should have the direct experience of drawing a hyperbola using the tack and string method. They can move the tack and T-square, experiment with different lengths of string and different positions of the tacks

5.Derive methods for locating the foci of herpbolas with their centers on the origin, and elsewnere on the coordinate system.

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