Very cool! But I do not think this post shows the actual Brachistochrone curve, right? Also, for the oscillatory curve, should / could not the ball jump a bit? Or the problem somehow is formulated not for a ball, but for a bid on a wire?
Well, I only assume the ball have one degree of freedom
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.
Yes, you are right, using the parametric expression is only a compromise when the explicit analytic expression cannot be obtained.
Wow, this is a fascinating section! I loved it
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