Wow, this is a fascinating section! I loved it
Yes, you are right, using the parametric expression is only a compromise when the explicit analytic expression cannot be obtained.
If I understand correctly there is no closed explicit analytic expression for $y(x)$ of Brachistochrone -- only parametric. This is probably why it is hard to put it in this illustration. Not sure though.
Well, I only assume the ball have one degree of freedom
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