I wasn't really clear.

I am not talking about orbits or the model of the solar system. I'm talking purely about changing the frame of reference.

If you want to plot the movement of planets, you set the frame of reference at the sun. That way, the orbits are nice and elliptical. If you plot the movement of the moon, it will look something like this, though. To get a nice ellipse, you have to set the frame of reference at the earth.

So now you have the earth at point (0,0,0) and you have the moon revolving around it in an elliptical orbit. What happens if you plot the sun?

Normally, people consider the ground not to move, and use it as a frame of reference. So if you're just standing, you're not moving. If you're driving 30 miles per hour, you're moving at that rate relative to the earth's surface, right? So, if the ground is not moving but the sun moves from one side to the other, then at this frame of reference the sun is moving around the earth.

So to be clear: I'm not making an argument against the model where the earth orbits around the sun - everything behaves the same. The only thing that changes is the frame of reference.