How do the tautochrone and brachistochrone curve differ? They're both cycloids or rather parts of cycloids. So does that mean that the brachistochrone = tautochrone curve?

I saw on Wikipedia that "the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can only use up to the first half rotation, and always ends at the horizontal". Does that mean that you can take any part of a cycloid (eg. maybe the first quarter of a cycloid) and it'll still be a brachistochrone curve? Or do you have to use the whole cycloid for it to be a brachistochrone curve? And for the tautochrone curve, do you have to use the whole first half of the cycloid for it to be a tautochrone curve?

Also, just another side question: is the brachistochrone curve still the fastest when friction is included?



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