I want to compare the time that it takes to slide a particle in a frictionless hypocycloid curve, so time would be given by the arclength divided by the velocity
So I need first compute the arclength of the hypocycloid curve, but in general the arclength is given by
And by conservation of energy, velocity is given by
Substituting in the integral results
Solving the indefinite integral results in
So now I would just substitute the function y corresponding to the hypocycloid curve
Is my reasoning right?
Then finally to compare times I would just make a graph of the time functions corresponding to the brachistochrone and the hypocycloid