Ever seen the shape of planets orbiting the sun? Or the shadow of a circle when viewed from the side? Well, shapes like those are called ellipses! An ellipse isn't just a "flattened" circle, but there's a cool mathematical definition behind it.
So here's the thing, an ellipse is a collection of points where the sum of distances to two specific points is always the same. These two specific points are called foci. Just imagine you have two nails and a string. If you tie the string to both nails, then pull a pencil until the string is tight and draw a complete curve, the curve formed is an ellipse!
Basic Ellipse Concept
Ellipse with two foci and several points showing constant sum of distances.
From the visualization above, notice point P. The distance from P to focus F1 (which we call r1) plus the distance from P to focus F2 (which we call r2) will always be the same for all points on the ellipse. This is the fundamental characteristic of an ellipse!
Before we discuss the formulas, let's get familiar with the important parts of an ellipse. Each part has its own role in determining the shape of the ellipse.
Ellipse Components
Important parts of an ellipse with horizontal major axis.
Here are the components you need to know:
Ellipse center is the midpoint of the ellipse, usually written with letter O. All measurements in the ellipse refer to this point.
Foci (F1 and F2) are two fixed points that serve as reference for the ellipse definition. The distance between the two foci is called the focal distance.
Major axis is the longest line that passes through the ellipse center and both foci. Its endpoints are points A1 and A2.
Minor axis is the shortest line that passes through the ellipse center and is perpendicular to the major axis. Its endpoints are points B1 and B2.
Semi-major (a) is half the length of the major axis, which is the distance from center to the major axis endpoint.
Semi-minor (b) is half the length of the minor axis, which is the distance from center to the minor axis endpoint.
Remember, in an ellipse we always have a>b. If a=b, the shape becomes a circle!
Now, let's get into the fun part: how to write an ellipse in mathematical equation form. There are several forms depending on position and orientation.
There's a formula that always applies to every ellipse:
c2=a2−b2
where c is the distance from center to focus.
Eccentricity of an ellipse is defined as:
e=ac
The eccentricity value of an ellipse is always 0<e<1. The closer to 0, the more circular the ellipse becomes. The closer to 1, the more elongated the ellipse becomes.
