# How do I solve an ellipse with three chord lengths and angles?

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.

• Can you clarify if $t, t_1, t_2$ are the parametric angles or the angles of the triangle? – RSS Jan 13 '17 at 12:47
• They are the original theta angles of the circle which created the conic. I didn't want to add them to the conic diagram as I thought it may confuse the issue. – Lee Cook Jan 13 '17 at 12:49
• What are $a$, $b$ and $R$ ? – Yves Daoust Jan 13 '17 at 13:25
• a and b are the major and minor axis, R is the rotation of the circle on the plane which would map the ellipse axis to the circle axis. – Lee Cook Jan 13 '17 at 13:29

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.

• Bah, offline comment creation now... Thank you for your reply, I think I follow what you're saying and I don't think it quite matches the scenario I was (trying) to put forward. The stretch would take place as the plane is tilted but R would effectively rotate the points A,B and C around the ellipse without altering the ellipse itself. – Lee Cook Jan 13 '17 at 13:46
• Any transformation from a circle to an ellipse would be given by the transformation I described, provided we chose the right co-ordinate frame. Unless the points are moving relative the ellipse (under a separate transformation), the angles would not move (except each by a constant amount). If I may ask, is this a problem from computational geometry? – RSS Jan 13 '17 at 14:02
• Sorry, I still think I may not be describing it correctly. The circle is on a 3D plane and only a circle when viewed from a tangential point to that plane. If the plane/circle is tilted about an arbitrary line on the plane through the centre of the circle, the circle would become an ellipse to the viewer, but a point which was at theta 45 in the circle would be theta 45+R (ignoring the apparent rotation of the whole ellipse) for the ellipse. – Lee Cook Jan 13 '17 at 14:20
• Why don't you consider the result as an orthogonal projection say on oxy plane (which amounts to forget $z$ coordinates)? – Jean Marie Jan 13 '17 at 14:33
• I get three sets of exact bearings to the points on the circle from my system but the position and attitude of the circle is unknown. By finding the major & minor axis of the ellipse I can then figure out the amount the plane the circle is on has been rotated - which is the final goal. – Lee Cook Jan 13 '17 at 16:35