Ellipse Calculator

It seems in geometry that the ellipse is the "forgotten stepbrother" of the circle even though the ellipse is far more interesting. First some definitions.

An ellipse is the locus of points the sum of whose distances from two fixed points, called foci, is a constant.
Line AB is the Major Axis (also called Long Axis or Line of Apsides).
Lines AO and OB are the Semi-Major axes.
Line CD is the Minor Axis and is the perpindicular bisector of the Major Axis.
Points f₁ and f₂ are the foci of the ellipse.
Points A and B are called apses.

The eccentricity of an ellipse equals (f₁ f₂ / AB)

or (c/a)

or (Line f1 B) - (Line A f1) (Line f₁ B) + (Line A f₁)
As the eccentricity value goes from 0 to 1, the ellipse goes from circular to highly elongated.

The Minor Axis to Major Axis Ratio, which we will call Y/X, equals

The perimeter of an ellipse approximately equals

NOTE: Go to Calculator TWO for Perimeter and Area Calculations.
Johannes Kepler's First Law states that the planets move in elliptical orbits with the Sun located at one of the foci.
If we were dealing with planetary orbits and we were to say the Sun were at f₁ then Line A f1 would be the perihelion distance, Line f1 B would be the aphelion distance and the planet's average (or mean) distance would be one half of the major axis.
Now that you have an ellipse glossary, we can move on to calculations.

Calculator ONE - Eccentricity and Y/X Ratio Conversions
The eccentricity of an ellipse is not such a good indicator of its shape. For example, of all the 9 planets the one with the most eccentric orbit is Pluto with an eccentricity value of .245. This might create the impression that the orbit is somewhat flattened. Actually, by utilizing this calculator, we see that the minor to major axis ratio is about .97 which is practically circular.
Also, you can use this calculator for determinig eccentricity for ellipses found in everyday life. For example, if an elliptical coffee table measures 3.5 feet by 2 feet, its minor to major axis ratio is .5714.... making its eccentricity about .821.

Click on whether you are inputting Y/X or eccentricity.
OR

Calculator TWO
This calculator gives a LOT of information in return for just a small amount of input data.
Choice 1 can be used for calculations concerning ellipses found in everyday life. (see coffee table example above).
Choices 2 and 3 are for astronomy calculations.
Choice 4 would be handy for the mathematics of ellipses.

Click on the data you know:
Major Axis and Minor Axis
Average Distance and Eccentricity
Perihelion Distance and Aphelion Distance
Foci Distance and Major Axis

Calculator THREE - Drawing Ellipses
The traditional way to draw an ellipse is to make a loop of string or thread, place two thumbtacks in a sheet of paper, put the loop over the thumbtacks and then with a pen, keeping the loop tight at all times, go completely around the thumbtacks.
Referring to the ellipse at the top of the page, the triangle C f1 f2 would represent the loop of string, the thumbtacks would be at f1 and f2 and the pen would start out at point C.
This works fine, except you are not exactly sure of what the ellipse is going to look like (how eccentric, minor to major axis ratio, etc.). Therefore this calculator will determine how far apart the thumbtacks have to be for a required eccentricity.

Example: If you make a loop of string 10 inches long, and want an ellipse with an eccentricity of .5, enter these figures in the calculator and your answer will be a thumbtack distance of 6.66666 inches.

Other Ellipse Properties
Referring to the ellipse diagram at the top of the page, let us suppose that it is a pool table and imagine that f1 and f2 are spots on the table. If you were to shoot the ball from anyplace over spot f1, the ball would then go over spot f2, then f1 and so on.
If you shot the ball through line Af1, the ball would go through line f2B and no matter how many times it is deflected, the ball will never cross line f1f2.
Conversely, if you were to shoot the ball anywhere through f1f2, it will continue to cross line f1f2 with each deflection and will never travel through Lines Af1 or f2B.
If we were talking about rays of light and the inside of the ellipse were reflective, the same behavior would be exhibited by the light, just as in the pool table example.
You may have heard of "whispering galleries" in which one person stands in a certain spot and whispers while another person stands at another spot and can hear the other person perfectly. How does this work? The gallery walls are elliptical in shape, and so if one person stands at f1, the sound waves will be directed through f2, where the other person will be standing. One of the chambers in the US House of Representatives is shaped this way.
This webpage has only explored the tip of the "ellipse iceberg" so to speak. For example, the ellipse can also be defined as the locus of points whose distance ratio from one focus to a straight line (called a directrix) is equal to the eccentricity of the ellipse. But you probably knew that right?
Also, we barely discussed the mathematical properties of the ellipse such as ellipse equations. Nor did we discuss the fact that the ellipse is one of the four conic sections. (See? the fun never ends.)
To find out more about ellipses, use your favorite search engine. No doubt you will find loads of information. Good luck with your elliptical internet explorations.