Chances are you are quite familiar with what is commonly termed the classic line through a constant-radius corner. Beginning at the outside edge of a turn, you bend in toward the apex and carry that line to the outside of the track in an arc with as large a radius as you can possibly scribe. There are variations and adjustments required for camber, hills and decreasing- or increasing-radius turns, but the basic premise is the same. In this diagram, we've drawn in the classic racing line using the whole width of the track. Keeping the radius of this line as large as possible allows you to carry maximum speed through the turn.
As we know, however, track designers are particularly devious and will position turns such that they are too close to use sweeping arcs over the full width of the track, but not close enough for the straight-line method to be practical. Now there is a choice to make: You can straighten the line between the two turns, which forces you to use smaller arcs and go slower over a shorter distance, or you can sweep in a larger arc from one side of the track to the other to keep your speed up, but over a longer total distance. At a recent series of track days and schools, we saw both extremes (shown here) and everything in between. In this diagram, rider A takes the shortest line but has to make the tightest turns, and hence has the lowest speed. Rider B tries to carry more speed by sweeping wide, but ends up traveling much farther than rider A.
It's when you face a series of turns that things get more complicated. A combination of two opposite turns forces you to cross from one side of the racetrack to the other to set up for the second turn. If the individual corners are far enough apart, there is no problem accomplishing this while under full power (2a). However, the closer the turns are the harder it will be to cross over, and at a certain point you will be unable to hold full power and keep your turn sharp enough to make the full width of the track. Here, the turns are far enough apart that there is no problem lining up correctly for the second corner. But if the turns are very close together, as in a true chicane (2b), you will have no choice but to ride practically straight from apex to apex.
Very rarely is either line correct, and you can also see the potential for conflict when rider A and rider B are dicing with each other. Somewhere in the middle of these two extremes is a happy medium, and you should experiment to find the line that allows you to optimize speed and distance to minimize the time involved. Experimentation will also allow you to use one of the extreme lines to pass other riders through that section of track. Here are two options, both with more speed than rider A and a shorter distance than rider B. You can break down any section of track using the same criteria, keeping in mind that, in general, the closer the two corners are, the straighter your line must be.