## Parts of an Ellipse

The ellipse has certain notable parts:

- The major axis is a line that passes through \(C\), foci, and the vertices. It has a length of \(2a\).
- The minor axis is a line that passes through \(C\) and is perpendicular to the major axis; It has a length of \(2b\).
- Center \(C\) is the center of the ellipse. It is the intersection point of the major and minor axes with a coordinate \((h, k)\).
- Vertex \(V\) are the endpoints of the ellipse along the major axis.
- Co-Vertex, \(V_c\) are the endpoints of the minor axis.
- Foci \(F\) – two distinct fixed points that serve as the basis for the loci definition.
- The distance \(a\), semi-major distance, is the halfway distance between vertices or the distance between \(C\) and \(V\).
- The distance \(b\), semi-minor distance, is the halfway distance between the endpoints of the minor axis or the distance between \(C\) and the endpoints of the minor axis.
- The distance \(c\), linear eccentricity, is the distance between \(C\) and \(F\).

Distances \(a\), \(b\), and \(c\) define the shape of the ellipse. There is a relationship among these distances:

\(c^2=a^2-b^2\)

To derive this equation, consider a point \(P(x, y)\) on one of the endpoints of the minor axis and draw distances \(d_1\) and \(d_2\) from the foci.

- We'll notice that sides \(d_1\), \(d_2\), and \(F_1-F_2\) form an isosceles triangle with height \(b\) and base \(2c\).
- Since \(d_1\) and \(d_2\) are equal and that \(d_1+d_2 = 2a\), then \(d_1 = d_2 = a\).
- Considering half of this isosceles triangle, you'll get a right triangle. Apply the Pythagorean theorem to get the equation above.

## How Oval Is an Ellipse?

Sometimes we wonder "how oval an ellipse is." We take the ratio of distances \(c\) and \(a\) – known as eccentricity to measure this.

For an ellipse, eccentricity \(e\) must be between 0 and 1. Furthermore:

- If \(e\) is nearly equal to 0, it is almost circular.
- It is more elongated if \(e\) approaches 1.

## Summary

The ellipse is a set of all points in which the sum of its distances from two unique points (foci) is constant.

The ellipse has certain notable parts: major axis, minor axis, center, vertex, co-vertex, focus, distances a, b, and c.

Distances \(a\), \(b\), and \(c\) define the shape of the ellipse. There is a relationship between these distances: \(c^2=a^2-b^2\)

For an ellipse, eccentricity \(e\) must be between 0 and 1. If \(e\) is nearly equal to 0, it is almost circular. It is more elongated if \(e\) approaches 1.