There is an excellent article in this issue that uses William Mitchell's software to show why the classic racing line around a 180 degree turn of starting wide and making a nice smooth arc down to the apex and then a smooth arc back out is not the fastest way through the corner (even if trailbraking heavily down into the apex or late apexing).

The fastest line is a parabolic line; nearly half of an ellipse. Essentially turning into the corner earlier and driving the car in a straighter arc to the apex. As the car gets nearer to the apex it is heavily braked and after slowing the car is cranked hard around the apex before getting hard on the gas and accelerating back out. The parabolic line has a much slower speed at the apex than the classic line, but its pure cornering phase at the apex is tremendously shorter. The parabolic line carries more speed over a longer period of time before the apex and begins accelerating sooner after the apex. To top it off it also covers less distance than the classic line.

Note that the parabolic line is still different than late apexing a corner. The late apex does allow you to get on the throttle earlier, but its entry speed is still not maintained as long as the parabolic line. Essentially in the car you would start wide and turn in early, aiming the front of the car towards the corner. You can carry more throttle longer because you are not doing as much turning and therefore have more grip for acceleration/braking. Once closer to the apex the car is braked very hard and then trail braked over a short distance while cranking the car very hard around the apex (the speed here at the apex is actually considerably slower than the classic line would produce). The amount of turning done is much tighter than the corner may seem (this also means you may not be on the curb at the apex, you may be a couple feet from it to give room to crank the car around) and so the car will stay to the inside of the track longer just after the apex. Since it already got most of its turning done however it can now use more of it grip to immediately accelerate hard out of the corner. The parabolic line has a much shorter pure cornering phase and much more opportunity to carry throttle longer on the way in and earlier on the way out. Less overall distance is just a bonus.

I can't find any diagrams right now but they are all in the article and it's several pages long. Excellent reading if you can find it at your local magazine stand.

This diagram is not an actual racing pic but illustrates the difference in shape. You can see in the parabolic one the majority of the turning is done right at the apex as opposed to turning it smoothly throughout the corner. Since the parabolic line is straighter on the way in and back out the tires have more grip available for braking/acceleration in these areas.
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Here's the given problem: A 180 degree turn with a short straight leading into and away from the corner, no banking, bumps, sand etc on the surface.

If you were a bystander viewing the track perpendicular to the straightaways you would obviously know right away that the car that can come down the straightaway, turn, and come back out at the highest speed will be faster through the segment. The speed that counts isn't around the corner, it's the speed on the way in and on the way out. By purposely apexing the corner at a slower speed you can do all your turning in a very small portion of the racetrack and therefore not spend as much of your traction budget cornering. You'd spend more of your grip budget and time going fast in a straight line.

These digrams are scans from Erik Zapletal's article in the August 2004 issue of Racecar Engineering. Each of these diagrams shows the force vectors present on the car throughout the corner. The longer the vector, the more force it represents. The "V" vector represents straightline velocity of the car (speed) and the "A" vector represents acceleration force (In physics an acceleration is defined as a change in vector quantity. Therefore driving at a constant speed around a circle is in fact an acceleration since direction is constantly changing. Speeding up and slowing down in a straight line are of course accelerations as well). The parabolic line allows the car to generate greater velocity at a greater acceleration for a longer period of time than does the classic line. This therefore means it can cover the given distance more quickly. The parabolic line allows the car to spend more time at a higher velcoity than the classic line does, even though the classic line does carry more speed at the apex. When viewing the diagrams you can see that the parabolic line has greater vector sums than the classic line when the entire corner is analyzed.

These diagrams are just the tip of the iceberg. The entire article goes much greater into the depths of explanation. You do need to understand vector quantities though.