Q:

What are the radial vectors in an ellipse?

I recently came across a term "radial vectors" in terms of ellipse. They appeared like a set of (x,y) coordinates but I am not sure whether I should just treat them as coordinates on the ellipse' circumference or are they something else.
Can anyone help me understand or guide me to a link where I can get more information about radial vectors?
Note: I am completely out of touch with mathematics and it has been years since I practiced it. So please keep the wordings simple.
Thanx in advance!!

A:

The term radial vector does not seem to be usual in English (see the Wikipedia page on ellipses ), but it is in Spanish (see the same page in Spanish, look out for "radio vector" ). It is defined as "the line segment, in a hyperbola or ellipse, that joins a point with the focal points".
In the following plot, this line segment would amount to $ \overline{F_1 P} + \overline{F_2 P}$

As you can see, the point $P$ can be described with a simple pair of Cartesian coordinates $(x, y)$. If those coordinates really belong to an ellipse, then they should satisfy the relationship:
$$\tag{1} E = \left\{P \mid \overline{F_1 P} + \overline{F_2 P} = 2a\right\}$$
i.e. they are the set of all points $P$ such that the radial vector to both focal points $(F_{1}, F_{2})$ is a constant (namely, twice the semi-major radius $a$ of the ellipse).
The same relationship expressed in Cartesian coordinates gives the following:
$$\tag{2} \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
where $b$ is the semi-minor axis, which is related to $a$ through
$$\tag{3} e = \sqrt{a^2-b^2},$$
$$\tag{4} b = a \sqrt{1-e^2}$$
where $e$ is the ellipse eccentricity and may be another information you have on the ellipse.
If you do not know the axes $a$ and $b$ of the ellipse, or $a$ and $e$, pick two points from the ones you have in your data and plug them into the relationship $(2)$: now you have a set of two equations with two unknowns. Solve for $a$ and $b$, and check that the relationship holds with the same parameters for all other points.

