A brachistochrone is the curve of fastest descent from to . It is described by parametric equations which are simple to derive.
From Circle to Cycloid
A cycloid is the path traced by a point on a rolling circle:
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 units, or one revolution, every seconds (in other words, it revolves 1 radian per 1 second).
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:
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:
From Cycloid to Brachistochrone
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 .