I promised to convince you about the parabola.

 

 This is the same figure, but X runs now on the x-axis. You can move the point P, if you like.

We can now compute, how the distance y depends on x. d is the y-coordinate of P. Due to Pythagoras in the marked triangle

x2+(d-y)2=y2

A little computing yields

2dy = x2+d2

Divide by 2d and you get the equation of a parabola.

This ends our tutorial about tracks!