# Physics of Racing - Part 5: Introduction to the Racing Line

### By: Brian Beckman

This month, we analyze the best way to go through a corner. "Best" means in the least time, at the greatest average speed. We ask "what is the shape of the driving line through the corner that gives the best time?'" and "what are the times for some other lines, say hugging the outside or the inside of the corner?" Given the answers to these questions, we go on to ask "what shape does a corner have to be before the driving line I choose doesn't make any time difference?" The answer is a little surprising.

The analysis presented here is the simplest I could come up with, and yet is still quite complicated. My calculations went through about thirty steps before I got the answer. Don't worry, I won't drag you through the mathematics; I just sketch out the analysis, trying to focus on the basic principles. Anyone who would read through thirty formulas would probably just as soon derive them for him or herself.

There are several simplifying assumptions I make to get through the analysis. First of all, I consider the corner in isolation; as an abstract entity lifted out of the rest of a course. The actual best driving line through a corner depends on what comes before it and after it. You usually want to optimize exit speed if the corner leads onto a straight. You might not apex if another corner is coming up. You may be forced into an unfavorable entrance by a prior curve or slalom.

Speaking of road courses, you will hear drivers say things like "you have to do such-and-such in turn six to be on line for turn ten and the front straight." In other words, actions in any one spot carry consequences pretty much all the way around. The ultimate drivers figure out the line for the entire course and drive it as a unit, taking a Zen-like approach. When learning, it is probably best to start out optimizing each kind of corner in isolation, then work up to combinations of two corners, three corners, and so on. In my own driving, there are certain kinds of three corner combinations I know, but mostly I work in twos. I have a long way to go.

It is not feasible to analyze an actual course in an exact, mathematical way. In other words, although science can provide general principles and hints, finding the line is, in practice, an art. For me, it is one of the most fun parts of racing.

Other simplifying assumptions I make are that the car can either accelerate, brake, or corner at constant speed, with abrupt transitions between behaviors. Thus, the lines I analyze are splices of accelerating, braking, and cornering phases. A real car can, must, and should do these things in combination and with smooth transitions between phases. It is, in fact, possible to do an exact, mathematical analysis with a more realistic car that transitions smoothly, but it is much more difficult than the splice-type analysis and does not provide enough more quantitative insight to justify its extra complexity for this article.

Our corner is the following ninety-degree right-hander:

Figure 1 - Our mythical corner.

This figure actually represents a family of corners with any constant width, any radius, and short straights before and after. First, we go through the entire analysis with a particular corner of 75 foot radius and 30 foot width, then we end up with times for corners of various radii and widths.

Let us define the following parameters:

r = radius of corner center line = 75 feet
W = width of course = 30 feet
ro = radius of outer edge = r + 1/2W = 90 feet
ri = radius of inner edge = r - 1/2W = 60 feet

Now, when we drive this corner, we must keep the tires on the course, otherwise we get a lot of cone penalties (or go into the weeds). It is easiest (though not so realistic) to do the analysis considering the path of the center of gravity of the car rather tha