Archimedes got an ellipse shrinking a circle along one direction. Then Archimedes could deduce the area of an ellipse as a generalization of the area of a circle.

From the implicit equation of a circle

we can deduce one equation of an ellipse.

When we shrink the circle, each point on the circle goes to a point on the ellipse. Then if a point P with coordinates (x,y) is on the ellipse E then its corresponding point is on the circle C.

Using the implicit equation of a circle and the coordinates of these corresponding points on the circle we can write:

Dividing by a square we get the implicit equation of an ellipse:

A circle is a special case of an ellipse (when a = b).

LINKS

	Ellipse and its foci Every ellipse has two foci and if we add the distance between a point on the ellipse and these two foci we get a constant.
	Archimedes and the area of an ellipse: an intuitive approach In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.
	Archimedes and the area of an ellipse: Demonstration In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. It si a good example of a rigorous proof using a double reductio ad absurdum.
	Plane developments of geometric bodies (4): Cylinders cut by an oblique plane We study different cylinders cut by an oblique plane. The section that we get is an ellipse.
	The Complex Cosine Function The Complex Cosine Function extends the Real Cosine Function to the complex plane. It is a periodic function that shares several properties with his real ancestor.