ELLIPSE

Although the circle is a very common shape, most of the time when we see a circle, it doesn't look like a circle! Because we usually see circles from an angle...and the shape then isn't perfectly round, it's another mathematical curve called an ellipse.One description of an ellipse is a squashed, stretched or otherwise distorted circle.

An ellipse can be seen as a slice of a cone, but it can also be seen when you tilt a glass partially filled with water or juice.

Unlike a circle, whose shape never changes, there are many different shapes of ellipses: long narrow ones ... and rounder ones.

Draftsmen have templates that enable them to draw the correct shape of ellipse for any use. We call the vertical and horizontal lines (shown in red below) the axes; the major axis is the longer of the two ... but it isn't necessarily the horizontal one.

The relationship between the major and minor axes determine the shape of the ellipse. The mathematician's word for shape is this case is eccentricity, and refers to the curvature. It is closely related to the ratio between the lengths of the two axes ... If the major and minor axes are equal, the ellipse is a circle. Shall we define an ellipse as a distorted circle, then, or the circle as a special case of the ellipse? Either one would be reasonable and useful.

The shape of the ellipse makes it more useful in some instances than the circle: since it is longer than it is wide it forms a comfortable shape for your hand ... It is a stable shape: a tank truck is in the shape of an ellipse. If it were circular, it would be taller than it is wide and then be top-heavy. A circle would have given it a larger capacity however, for the circle has the greatest area for a given perimeter, and is therefore the most economical container.

Another way of defining an ellipse is as one of four curves called The Conics, that can be cut from a cone. A, cone sliced parallel to the base has a circular cross-section, but if we angle the slice the result is an ellipse. Varying the angle changes the eccentricity of the ellipse. You will also get ellipses if you slice a cylinder.

The conical beam of light from a flashlight casts an elliptical pattern of light on the wall when held at an angle. If we move the point of the cone further away, an angled slice will still give us an ellipse: no matter how far we move the point of the cone. Imagine the point as far away as infinity...the cone becomes a cylinder and we still have an ellipse in cross-section.

Salami is often cut in elliptical slices ... and tilting a glass of soda to drink with our salami would reveal yet another ellipse! Any cylinder sliced on an angle will reveal an ellipse in cross-section, such as seen in the Tycho Brahe Planetarium in Copenhagen:

The mathematical definition of an ellipse is the set of all points such that the sum of the distances from two fixed points is constant. The distance from any point of the ellipse to one focus plus the distance from that same point to the other focus is always the same number for a particular ellipse: seven in the example below. Two plus five equals seven ... three plus four equals seven ... three and one-half plus three and one-half equals seven.

We can use this definition to derive an equation for an ellipse with its center at the origin on a Cartesian coordinate system:

The numbers 9 and 16 determine the length of the major and minor axes, and therefore the shape of the ellipse; the square root of 9 is 3; the square root of 16 is 4, so the ellipse is 3 units to the left and to the right of the origin and 4 units above and below. We can graph an ellipse, using the equation. 3 and 4 are the x- and y-intercepts. The ellipse will fit into a rectangle drawn 3 units to the left of the origin and 3 units to the right of the origin, 4 units above and below, as you can see in the graph of an ellipse:

We can also use this definition to draw an ellipse with two tacks and a piece of string, as shown on the following website:

http://www.maa.org/math land/mathland_92.html

The two tacks are the focus points. The distance from one focus to the ellipse and then to the other focus is determined by the length of the string. If we make the string longer using the same focus points, we will get a larger ellipse ... this also alters the shape. Another way to change the shape of the ellipse is by moving the foci, the two focus points. If the two foci are moved farther apart and the length of the string remains the same, the ellipse that results is longer and narrower ... If we move the foci closer together again without changing the length of the string, the ellipse becomes rounder and more like a circle.

The two foci of the ellipse have other interesting properties. Any line drawn from one focus to the curve of the ellipse will reflect through the other focus. A pool table in the shape of an ellipse with a pocket at one focus would be foolproof: just hit the ball from one focus and you'd sink it every time!

Rotating a curve around an axis produces a surface of revolution, which has rotational symmetry around that axis. When you rotate an ellipse, you get a 3D shape called an ellipsoid; Planet Earth is actually an ellipsoid rather than a perfect sphere.

The ellipse is found in many different places, even in space: the orbits of the planets are elliptical, and the Earth itself is an ellipsoid. An ellipsoid is a three-dimensional ellipse: imagine an ellipse, spinning on its axis; it would look like a stretched or distorted sphere, almost like a watermelon! Our earth is not a perfect sphere; it is slightly elongated at the North and South poles.

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