Ellipses

Curves

Allen Klinger, © 12/16/2004
The lune in the night sky is made up of two curves. A replica can be seen in the figure that was composed in Microsoft Word from two overlapping circles.

Circles, squares, and triangles are some basic forms. To understand the movement of bodies in the night sky, people gave names like cube to related ideas. Some of those ideas deal with number and others with shape. Here are two:

An ordinary, counting, whole or round (not fraction) number, is said to be an integer.
The amount of space inside a closed line is known as its area.

Here is a link about area for a square and how it connects to counting and to solid space like that in a cube (or a die, one from apair of dice - essentially two cubes with dots on each side, going from one-dot on one of them, to six-dots on the last, one more on each of the others).

Circles are not the only curve that is useful in understanding the night sky. If a circle becomes a flatter and longer curve, it describes how bodies like planets move around, or orbit stars. The word for this curve is ellipse. To see more about ellipses and orbits go to Ellipses and Planetary Motion.

Johan Kepler was born in 1571; he lived until 1630. He established that planets move in an ellipse (orbit the sun in this curve) and published three laws of planetary motion in 1609 and 1619. We know that an ellipse is a kind of circle ( general type of or variation on - two ways to say this).

The shapes below all are ellipses. The outer ones are circles, ellipses whose two focal points overlap at the center. Another word, vertex, describes end or extreme points of these curved shapes.

Ellipses:

Same focal points.

Same vertices.

Total distance between any point on an ellipse curve and the two focal points is a constant: same sum for both lengths regardless of chosen boundary point.

x + y = u + v = a + b = constant

For a circle the two focal points coincide - they are both at the center. The circle center-to-curve distance is the radius. It is the same for all boundary points.

	12/16/04 Version			http://www.cs.ucla.edu/~klinger/nmath/ellipse.html
	©2004 Allen Klinger