Proving an Ellipse can be Parameterized

I'm going to want to do this with a hyperbola as well. If I begin with the
equation of an ellipse in terms of two foci and the invariance of the sum
of the lengths of the line segments connecting the foci to a point on the
curve, what constitutes a necessary and sufficient proof that the ellipse
defined above is uniquely and completely parameterized by x=a cos t, y=b
sin t, |t|<=pi? The traditional meanings of a and b are assumed.

It seems to me that it is sufficient to show that given any point on the
curve, I can solve for t, and given any t, I can uniquely identify a point
on the curve.

I also wonder if I am justified in calling the ellipse a curve before I
demonstrate that it can be parameterized.

The reason I am in proving that an ellipse is parameterizable in the sense
described above is because it seems to fall into the same class of problems
which the derivation of the Lorentz transformation falls into. In _The
Meaning of Relativity_, Einstein basically says (tc)^2=x^2 for a light
sphere. Throw in ict and treat time like a 4th dimension, and we get the
Lorentz transformation expressed in terms trig functions parameterized the
domain of imaginary numbers. To me, that is intuitively obvious, but I'm
not sure what a rigorous proof would constitute.
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Relevant Pages

  • Re: Proving an Ellipse can be Parameterized
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  • Re: Proving an Ellipse can be Parameterized
    ... equation of an ellipse in terms of two foci and the invariance of the sum ... what constitutes a necessary and sufficient proof that the ellipse ... curve, I can solve for t, and given any t, I can uniquely identify a point ...
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