I have a plane on which is a circle, there are three arbitrary points on the circle ($A$, $B$ and $C$) of which the relative angles are known. The plane the circle is on is then rotated arbitrary line on the plane (through the centre of the circle) to create an ellipse with the three points on it.
I have the equation for the length of the chords from a previous question answered by @coffeemath: $$l = \sqrt{a^2(\cos(t)-\cos(t1))^2 + b^2(\sin(t)-\sin(t_1))^2}$$
This is complicated by the fact that the points are not going to be line with the ellipse axis and will be off by an angle $R$: $$l = \sqrt{a^2(\cos(t+R)-\cos(t_1+R))^2 + b^2(\sin(t+R)-\sin(t_1+R))^2}$$
If I have the three original theta angles ($t_A$, $t_B$ and $t_C$) from the circle, and three sets of chord lengths ($d_{AB}$, $d_{AC}$ and $d_{BC}$) how do I go about working out $a$, $b$ and $R$ (where a and b are the ellipse major/minor axis and R is the rotation of the circle on the plane which would map the ellipse axis to the circle axis)?
In theory the system should produce four answers, two for R (within a 2PI rotation) and the ability for the major/minor ellipse axis to be switched. The four solutions would be based on the angle of the line of rotation on the plane and the sign of the rotation itself.
Thank you in advance! Lee
If I understand the question properly,
You have 3 points on a circle given by the angles $(t, t_1, t_2) $ which the points make with (say) the x-axis.
Now the circle has undergone a stretch:
$$ (r\cos t,r\sin t) \mapsto \begin{bmatrix} a/r &&0 \\ 0 && b/r \end{bmatrix} (r \cos t,r \sin t)$$
Which makes it an ellipse. It is also rotated by an angle $R$.
Now you are given the lengths of the 3 chords connecting the original 3 points.
First, since you are only given lengths in the ellipse, I don't think there is a way of finding the angle $R$ since that leaves the lengths unchanged.
Second, note that the transformation above does not change the parametric angle of the point in the ellipse. So your angles remain unchanged (except for a $+R$ shift). These are different from the geometric angles of the points which indeed transform to $ \tan^{-1} (\frac{b}{a} \tan t). $
But you can use the chord lengths and the formula you mentioned to form 3 equations in 3 variables: $a, b, r$ where $r$ is the radius of the circle.
Hope this helps.