if you're at the bottom and not moving then technically you never started moving at all therefor you never moved and you don't apply to the curve
>.> You are just curving away from the fact that if you are at the bottom of the curve, THEN IT IS STILL ON THE CURVE.
a ball placed on a tautochrome curve and acted on by uniform gravity will reach the bottom in the same amount of time no matter the starting point
If the starting point was on the end of the curve, it's still on the curve. There is not a statement implying the ball has to move.