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!!
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.
At least in the absence of a definition to the contrary, I'd interpret “radial vector” as a vector pointing from the center of the ellipse to a point on its circumference. So if the ellipse is centered at the origin, then this is indeed simply a point on the circumference, but otherwise it's the difference between such a point and the center.
It could be that the vector is meant to be in the direction described above, but of unit length. This makes particular sense if you have a radial vector associated with a parameter value. For circles it is often useful to establish a local coordinate system, with one unit vector in tangential and the other in radial direction. These two are orthogonal. For the ellipse, the radial direction as described above won't be orthogonal, so I guess that most applications for the ellipse would rather use a tangential and a normal direction, the latter being the direction orthogonal to the tangent at a given point. Therefore I doubt that this is what you need, and I doubt that unit length will make sense, but I wanted to note it anyway.
It could possibly be that the radial vector should have its origin not at the center of symmetry but instead at one of the focal points of the ellipse. This is particularly likely if your text discusses something where foci play a special role, e.g. in physics when describing orbits of planets around the sun, since the sun will be located in one focus. I know that many formulas in this context tend to denote the position of a planet relative to the sun using the symbol $\vec r$, where the letter stands for radius as far as I know.
