I see. I don't know CA training techniques. One thing that I was taught is that ou want to work for the turns on the lungeline that are like the turns you want to ride in the saddle. So, for me, I want the horse to feel the rein (leadline) asking for a flexion of the jaw, I want him to flex and THINK around the corner, prepare his body and come around the corner, not come to an abrupt halt and wheel around .
My teacher works very much with the lungeline mimicking the rein in it's signals to the horse and working a lot more on leading the horse or steering via the rein, (lungeline) than driving the horse around with your whip or body language. We only use the body or whip to back up the lungeline if necessary.
Not very common way of lunging , I know.