# Brachistochrone

A brachistochrone is the curve of fastest descent from to . It is described by parametric equations which are simple to derive:

A cycloid is the path traced by a point on a rolling circle: [asy] draw((0,0)--(9,0)); draw(Circle((0,1),1)); dot((0,0)); draw((0,1)--(3,1),Arrow); [/asy]

If the radius of the circle is and the center of the circle is moving at a speed of units per second, then it moves , or one revolution, every seconds (in other words, it revolves 1 radian per 1 second). Then the x-coordinate of the center relative to the ground is: the x-coordinate of the point relative to the center is: so the x-coordinate of the point is:

Similarly, the y-coordinate of the center relative to the ground is: the y-coordinate of the point relative to the center is: so the y-coordinate of the point is:

Since a brachistochrone is an upside-down cycloid, we reverse the sign of y: This is a brachistochrone starting at .

If you want it to start at you just shift it:

If you want it to go through you need to solve for :

I recommend first solving for in terms of using , then substituting into .