Here’s the proof for the statement I made in the earlier post: The fastest way through any infinite slalom is by crossing over mid-way between the cones. Any other strategy, i.e. being early or late, at any point, will be slower overall.
Let’s assume we are driving a car with infinite transient response, which means we drive through an infinite slalom on perfectly circular arcs. (It doesn’t really matter, but I drew the figure this way.) We can make our crossover either at the midpoint between cones, or before the midpoint (being early) or after the midpoint (being late.) Figure 1, below, shows the path crossing over at the midpoint.
For purposes of illustration I’ve drawn the cone supersized. The point is the same even if the cone is very small as compared to the track width of the car.
If we try to be early on the cone but use the same path then we get the situation shown below in Figure 2.
Do you see the problem? To be early we can no longer drive a path with the minimum lateral movement of the car. The car has to move farther laterally to miss the cone. This means a longer path with a smaller radius. That’s a slower path.
Being late causes exactly the same problem. Most autocrossers think of getting late in a slalom as the kiss of death. We watch novices do it over and over again and shake our heads at how slow it makes them as the car slews sideways and slows terribly in the attempt to not hit the (usually) last cone in the slalom. But, are we really so smart being early?
Any deviation from crossing at the midpoint will cost some time. So, when we think we have a good reason for not crossing at the midpoint it needs to Save More Time than we lose.