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Conic Sections: Ellipse

Ellipses also belong to the group of the four curves (circle, parabola, ellipse, and hyperbola) known as conic sections simply because they were originally obtained by slicing a right circular cone with a plane. If the plane is slightly tilted away from the position parallel to the base of the cone, but before it becomes parallel to one element of the cone, the resulting intersection is an ellipse. An ellipse is commonly called an oval.

In plane geometry, an ellipse is the set of points in a plane such that the sum of the distances from two given points to any point in the set stays constant.

An ellipse can be obtained as follows: First, start with two fixed points P and Q. Then, select any constant quantity greater than the distance PQ. Next, determine all points for each of which the distance from P and the distance from Q add up to the selected constant. This last step can be achieved by tying a string, whose length is equal to the selected constant, to the two points P and Q. Then, use a pencil to push the string away from P and Q and try to draw a closed loop in a "circular" fashion. As long as the string stays tight (the sum of the distances from P and Q to the pencil stays constant), the resulting curve is an ellipse.

Image Since the sum of the distances from P and Q stays constant (c), we have:

[(x+r)2+y2]1/2+[(x-r)2+y2]1/2 = c

Simplifying the expression, we can prove that the equation of an ellipse is

x2/a2 + y2/b2 = 1

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