Advanced Cornering Theory: How To Calculate A Trajectory Path
When navigating around a race course, you'll encounter numerous direction changes and will have to adapt to each of them. Car positioning going into turns is by far one of the most important aspect in competition racing.
Most drivers only know about this general rule: "Outside, inside, outside" or "apex the turn". It's a method of using the most road possible to turn as little as possible making the car travel as fast as possible. Here are the actual physics and mathematics involved in calculating a trajectory arc.
There are several types of "arcs" as noted:
- decreasing radius arc is a turn which gets tighter or smaller
- increasing radius arc is a turn that opens up or straightens out
- constant radius arc is a turn that stays the same without deviating
- multiple arc radii are multiple turns that have more than one arc radius
Each arc has to be converted to an angle in degrees then bisected (cut in half) down the middle by dividing the turn degree number in half. Below is a 90 degree angle bisected into 2 halves, each being 45 degrees. Next step is to map a trajectory path that starts on the outside edge of the turn and allows the biggest arc radius possible while passing through the turn apex or inner corner, and leads back out to the opposing edge of the turn.
Most drivers will use the geometric apex as a marker to put the car over. Others use what's called the "real apex" which is slightly after the geometric apex. By shifting the apex and adjusting the arc radius to allow more room on the turn exit, the car will actually gain speed faster and is especially important if there is a long straight after a turn. If your car is able to accelerate even 1 second ahead of your opponent, that become several feet within the next few seconds time.
Going into turns fast is not as important as maintaining control of the car and having a stable yet quick exit out of the turns.
Advanced Cornering Theory: Trajectory Paths For S-Turns With Multiple Arc Radii
Anytime there's an S-turn leading to a long straightaway, exiting the S-turn is actually more important than entering it. Below is an image showing 2 trajectory paths.
The red path shows the most common line used in this corner which allows a large arc radius for both corners. The 2nd corner though requires the driver to take a smaller arc radius and won't allow the car any room to accelerate.
The yellow path shows a different line that sacrifices the first corner to position the car in a way that allows the 2nd corner to be taken with the largest arc radius possible and allows the car room to accelerate much sooner. As a result, it will exit the corner onto the straightaway at a higher exit speed and will make up for any time lost in the first section of the S-turn.
Advanced Cornering Theory: Exiting Fast Wins Races, Entering Fast Ends Races...
When you enter a corner too fast, odds are you're already losing time just trying to keep the car on the road and most drivers will anticipate this then dive to the inside for the pass. The ideal braking point & turn-in point depends on how much fuel you have, how worn your tires are in addition to their temp + track temp, and how hard you're pushing the car. The ideal racing line would be the shortest path from point A to point B and to win the race, you'll need to gain or lose momentum faster than your competition.
Final Turn - WE1 An example scenario of corner exit theory. In this section, the last turn starts very tight and slowly opens onto the main front straightaway. Taking this turn as fast as possible depends more on how long the car takes to change direction than it does on the moving speed the car goes through it.
Red dotted line: The braking and turning markers are too early causing the car to apex early and go wide since it runs out of road to use.
Green dotted line: The braking and turning markers are just right allowing the car to slow down enough to cut inside the racing line. Cutting inside at the exit of the turn is what will allow the car to accelerate sooner and will carry that speed onto the straightaway.
Black solid line: This is the normal racing line where braking points are a bit late or going too fast so it stays on the outside of the turn most of the time.