Ellipse
An ellipse is like a squashed circle.
A circle has one center, but an ellipse has two foci ("F" and "G" below).
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If you go from point "F" to any point on the ellipse and then go on to point "G", you will always travel the same distance.
f+g is always the same
Definition
An ellipse is the set of all points on a plane whose distance
from two fixed points F and G add up to a constant.
The points "F" and "G" are called the foci of the ellipse (F is a focus, G is a focus, and together they are two foci) |
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Draw It
Put two nails in a board, put a loop of string around them, and insert a pencil into the loop. Keep the string stretched so it forms a triangle, and draw a line ... you will draw an ellipse.
A Circle is an Ellipse
In fact a Circle is an Ellipse, where both foci are at the same point (the center). In other words, a circle is a "special case" of an ellipse.
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Section of a Cone
You can also get an ellipse when you slice through a cone (but not too steep a slice, or you get a parabola or hyperbola).
In fact the ellipse is a conic section (a section of a cone) with an eccentricity between 0 and 1. |
Calculations
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Area
The area of an ellipse is π × a × b
(If it is a circle, then a and b are equal to the radius, and you get π × r × r = πr2, which is right!) |
Perimeter Approximation
Rather strangely, the perimeter of an ellipse is very difficult to calculate, so I created a special page for the subject: read Perimeter of an Ellipse for more details.
But a simple approximation that is within about 5% of the true value (so long as r is not more than 3 times longer than s) is as follows
Remember, this is only a rough approximation!
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Equation
By placing an ellipse on an x-y graph (with its major axis on the x-axis and minor axis on the y-axis), the equation of the curve is:
x2/a2 + y2/b2 = 1
(very similar to the equation of the hyperbola: x2/a2 - y2/b2 = 1, except for a "+" instead of a "-") |
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