The tool
MajorAxis[O, A, B, prv]
prv, a free point
Gives points C, D.
Objects which transform the ellipse (here, points A, B, O) should update
SetValue[prv, C]

Much better.
The output is always defined (hoho! I missed lengths of zero. Fix on the way!), and depends upon previous values a, b.
By holding a perpendicular to b and changing which is longer, I can still force a discontinuity: u will flip 90°.
This does not affect the immediate use I have in mind, and I have left it. For my purposes, the major axis has a durable meaning under transformation, and I would like to track it.
I will solve the problem later. Plan of attack: use this particular example to generalize the mapping problem
where (a,b), (u,v) define the same ellipse, and satisfying the well-posed condition: small changes of input (a, b, t) lead to small changes in transformed values.
This will inform me about how to proceed to more general figures.
Note that the continuity condition is on x(a, b, t), and not x(t): the user is free to introduce and manipulate objects. It is incompatible with algebraic resolution.
Now consider a simple figure which can also be translated, scaled and rotated.
How can we establish correct behavior? And how can we, if desired, restrict our figures so that they meet algebraic conditions?
(object space/global space...)
But I have a different purpose for the moment. Mr. the Land is determined to have measured figures in space.
Onward.