How to represent the relative geometry of two ellipses with a common focus in GeoGebra?

Question

I'm studying an astrodynamics problem and to help my study I'd like to represent the geometry I'm dealing with. I also obtained a figure in Matlab but I need to represent many angles and so I'd like to avoid to write too lines of code.

My objective is to represent two ellipses, (ell1 and ell2) with a common focus, in the perifocal reference frame of ell1 and this is the main problem that I have, beacuse in the cartesian coordinates of GeoGebra I am able to write the equation of ell1 as (x/a1)^2 + (y/b1)^2 = 1 but I'm having trouble correctly writing the cartesian equation of ell2; it should be something like (x-x0/a2)^2 + (y/b2)^2 = 1. Of course the focus is the same for both ellipses, let's assume F(c,0). In addition, I want to represent the second ellipse rotated of an angle equal to 120 deg about the focus wrt the first ellipse and their intersection point.

The data that I have are semimajor axes, eccentricities and the difference of the arguments of periapsides: a1 = 154448562.453878 km e1 = 0.0524440272062934 a2 = 143768968.725253 km e2 = 0.027842283175748 delta_omega = 120 deg

The data listed above correspond to a tangency condition between two ellipses shown in Matlab figure. It would be amazing to be able to represent in GeoGebra the ell1 fixed and the ell2 rotating and contemporary showing the intersection points (one or two depending on the relative geometry), hence to start with their apsides lines overlapped and then rotate the second one wrt the first one.

Here is the matlab figure that I would like to get:

Here is the scheme I've obtainend up to now in GeoGebra: GeoGebra

My questions are:

• Is it possible to use a perifocal reference frame in GeoGebra or do I have to use the cartesian one?
• Can you help me to draw the problem described above?

Answer 1

You can see below those two ellipses, drawn with GeoGebra. Here's the construction, step by step.

1. Use the Point tool to set point F (the common focus) at the origin.
2. Input a1 = 1.54449..., a2 = 1.43769..., e1 = 0.05244..., e2 = 0.02784... in the command bar.
3. Input F1=(-2*a1*e1,0) to find the focus of the first ellipse.
4. Input A1=(a1*(1-e1),0) to find the periastron of the first ellipse.
5. Use the Ellipse tool to draw the first ellipse (red).
6. Input u2=Rotate[(1,0),120°] to create a unit vector towards the periastron of the second ellipse.
7. Input F2=-2*a2*e2*u2 to find the focus of the second ellipse.
8. Input A2=a2*(1-e2)*u2 to find the periastron of the second ellipse.
9. Use the Ellipse tool to draw the second ellipse (blue).

I then used the Intersect tool to find the intersections between the ellipses (points P and Q).