Ellipses

Picture of how Circles and Ellipses are formed

I was fascinated when I learned this in Geometry.

Everyone knows what a circle is- it is a series of points that are the same distance from a center point. Now, what is an ellipse? The definition is a little more complicated: a closed plane curve resulting from the intersection of a circular cone and a plane cutting completely through it; the sums of the distances from the foci to any point on an ellipse is constant.

Drawing an Ellipse

The picture on the right demonstrates the first part of the definition. Here's an example of what the second part means: Suppose you want to draw an ellipse. You take two tacks, push them into a piece of paper, and you tie each end of a string that is longer than the distance between the two points to the tacks (each end of the string is tied to a thumbtack). Finally, you push a pencil against the string so the latter is tight (see picture on left). Now if you move the pencil around the tacks, keeping the string tight, you will end with an ellipse. It is a very neat method of drawing an ellipse, but what does this have to do with "the sums of the distance from the foci to any point on an ellipse is constant"? Well, unlike a circle, an ellipse has two foci (plural for focus). If you draw an ellipse using the method stated above, the place where you had your tacks will always be the foci. Now notice how the lines in the picture have names- F1 and F2. In the end, then, statement "the sums of the distance from the foci to any point on an ellipse is constant" simply means that F1 + F2 will always be the same number.