Ellipse

What
An ellipse is mathematically defined as the set of points that satisfies the equation

$$r + r' = 2a$$

Semi Major Axis
The semi major axis $$a$$ is half the length of the long, or major axis of the ellipse

Semi Minor Axis
The semi minor axis $$b$$ is half the length of the short, or minor axis of the ellipse

Focal Point
The focal points are denoted by $$F_1$$ and $$F_2$$ and the distance to from the focal points to the point $$X$$ on the ellipse is denoted as $$r$$ and $$r^'$$

Eccentricity
The eccentricity $$e$$ is defined as the distance between the two foci divided by the major axis of the ellipse. The distance from the center to either foci can be expressed as $$ae$$

For a circle, $$e=0$$

Perihelion
The point on the ellipse that is closest to the principal focus is called perihelion

The distance from the principal focus at perihelion is:

$$d = a - ea = a(1 - e)$$

Aphelion
The point on the ellipse that is farthest from the principal focus is called aphelion

The distance from the principal focus at aphelion is:

$$d = a + ea = a(1 + e)$$

Radius
The radius can be defined as:

$$r = \frac {a(1-e^2)}{1 + e \textrm{cos}\theta}$$

Relationship Between Major and Semi Major Axis
$$b^2 = a^2(1-e^2)$$

Total Energy of a Binary Orbit
$$E = -G \frac {m_1 m_2}{2a}$$