Given a dog (P) chasing a
pedestrian (Q). The
anxious human creature runs in a straight line, with constant speed u. The dog runs also
with a constant speed (v), always in the direction of its prey. The path the dog runs has
been given the name pursuit curve. The ratio u/v of the two velocities is
just the value of the parameter 'a' in the formula 1).
Leonardo da Vinci was the first to consider this problem.
Arthur Bernhart studied the curve.
Realize that this pursuit curve (for a=1) is not the tractrix: equal distance between pursued and pursuer is not equivalent with equal speed between them.
1) the tangent to the curve of pursuit can be expressed
as dy/dx = (y - ut) / (x -1) (i)
2) To understand that holds: