An Ellipse

Each point on the ellipse satisfies the condition that the sum of the distances from any point to two fixed points, called foci (singular focus), is constant.

An ellipse with semi-major axis a and semi-minor axis b.

If r and r′ are these two distances then r + r′ is a constant. For a point on the semi-major axis (and the ellipse), this sum is 2a, so it must be 2a everywhere.

\[r+r’=2a\]

Eccentricity:

The eccentricity of an ellipse is the distance between the foci, divided by 2a. A circle is an ellipse of eccentricity zero (both foci are at the same point, the center of the circle). The eccentricity of any ellipse must be less than unity.

From the point where the curve crosses the minor axis,

\[r=r’=a\]

\[b^2=a^2-(ae)^2\]

\[=a^2(1-e^2)\]

at the apastron:

\[r_{apastron}=a(1+e)\]

at the periastron:

\[r_{periastron}=a(1-e)\]

\[\vec r’=\vec {F’F}+\vec r\]

\[r’^2=F’F^2+r^2+2\vec{F’F}\vec r\]

\[r’^2=(2ae)^2+r^2+2r (2ae)\cos (\vec{F’F},\vec r)\]

\[r’^2=(2ae)^2+r^2+4r (ae)\cos \theta\]

but: \(r’+r=2a\)

so: \(r’^2=(2a-r)^2\)

\[(2a-r)^2=(2ae)^2+r^2+4r (ae)\cos \theta\]

\[(2a)^2-4ar+r^2=(2ae)^2+r^2+4aer \cos \theta\]

\[a^2-ar-aer\cos \theta =(ae)^2\]

so:

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