Geometry of the Ellipse

Parts of an Ellipse

‍

The ellipse has certain notable parts:

1. The major axis is a line that passes through \(C\), foci, and the vertices. It has a length of \(2a\).
2. The minor axis is a line that passes through \(C\) and is perpendicular to the major axis; It has a length of \(2b\).
3. Center \(C\) is the center of the ellipse. It is the intersection point of the major and minor axes with a coordinate \((h, k)\).
4. Vertex \(V\) are the endpoints of the ellipse along the major axis.
5. Co-Vertex, \(V_c\) are the endpoints of the minor axis.
6. Foci \(F\) – two distinct fixed points that serve as the basis for the loci definition.
7. The distance \(a\), semi-major distance, is the halfway distance between vertices or the distance between \(C\) and \(V\).
8. 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.
9. 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.