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I do not know whether from the rotating formula above, can we obtain the whole 360 degrees orientation.

 

[H]owever, if possible, I would like to obtain this difference as well.

[this Geogebra worksheet][1]

Hope that helps. As always, let me know if anything is confusing so I can make it better (more useful to you). [1]: http://www.geogebratube.org/material/show/id/45026

I do not know whether from the rotating formula above, can we obtain the whole 360 degrees orientation.

[H]owever, if possible, I would like to obtain this difference as well.

this Geogebra worksheet

Hope that helps. As always, let me know if anything is confusing so I can make it better (more useful to you).

First, No, your formula does not distinguish positions 180° apart. This is a property of the formula, not the ellipse. Say I make an ellipse out of rigid wire and rotate it. Since I can tell the difference between the two ends...

Second, yes it is possible. The trick is to measure in a way that preserves the difference. The following steps are fleshed out with working examples in

First- No. Your formula does not distinguish positions 180° apart. This is a property of the formula, not the ellipse.

Second-- Yes it is possible. I see you have marked one end with a red dot. That'll do nicely. The trick is to measure in a way that preserves this difference. The following steps are fleshed out with working examples in

When atan(-) falls out of range, dθ leaps suddenly. Go back a step and use atan(+), which is in range. When atan(+) jumps, go back to (-)... etc. You can track θ over the full 360°. Since you're only using the change in angle, it doesn't matter if atan() is close to θ.

The worksheet follows the same principle, but I only use atan(-).

When atan(-) falls out of range, dθ leaps suddenly. Go back one step and use atan(+) (which will be in range) to measure dθ. When atan(+) jumps, go back to (-)... etc. Always use two consecutive values of the same function to calculate dθ. Now you can track θ over the full 360°. Since you're only using the change in angle, it doesn't matter if atan() is close to θ.

The worksheet follows the same principle, but I just use atan(-).