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The Perfect Apex

The perfect corner

Is there such a thing?  The perfect corner, the perfect lap? Many drivers don’t realize, but for a given car, setup, and conditions there is in fact a singular optimum way to navigate a race track in the minimum time possible. A set of fundamental physics based rules exist that can guide you in your never-ending pursuit of speed.

This quest to come ever closer to perfection is what drives many of us.  This pursuit is one of the beauties of racing as the stopwatch always provides a measurable goal that you can always improve on. There is always that last second.  That last 10th. That last 100th.

While a driver will never be able to achieve a truly perfect lap in reality, there is one place an actual perfect lap can exist.   It can exist in the mind and give a driver a goal they can always strive to reach.  That is what this lesson will introduce and explain.  A set of rules that take a physics based approach to finding an optimal solution on track and distilling it into an intuitive way of driving that racers at any level can begin to apply.

All this comes from recent advances from the top levels of motorsport as we have seen great strides in using scientific principles to finding speed on track. Not just with the engineering of the cars, but a greater understanding of the way they should optimally be driven. While the top teams and drivers in the world have been able to take advantage, these findings have not made their way to the general public. Unfortunately for those aspiring drivers looking to learn, these top pro teams are in the business of keeping secrets and winning races. Not educating their competition.

Also unfortunately, all the current driving schools and even the most popular driving books are based on a now mostly obsolete motorsport teaching style. With only the currently available resources, even many naturally talented drivers find they generally hit a wall when trying to find that last second. This is because the current information available is not only sometimes incorrect, but is ultimately too vague and doesn’t provide the exact answers that are now available. So unless you’re naturally skilled and lucky enough to be recruited into a top pro team’s development program, this current information is all you will have available.

The good news is that we have now made these more recent advances available to the general public. We will take you through an intuitive approach to the physics of racing and then begin to apply it as we learn to optimize a corner. All with precise instructions and answers, but broken down so it’s easy to understand. We will look at real world example corners and provide an exact method to find the optimum strategy all from the driver’s eye point of view. No advanced vehicle dynamics knowledge or telemetry systems are needed.

Understand though, that while the strategies we will learn to break down and analyze a track are new, the actual driving techniques are not. We don’t offer a secret new weapon that will have you smashing track records by next weekend. Although they may not all have understood exactly why they drove the way they did, the top drivers in the world have been driving by the principles that we teach for decades. You can find old videos of champion drivers like Senna, Schumacher, and others following these methods. We actually recommend reviewing videos of world class drivers as you work through this lesson. Try and identify how what you are learning is reflected in the videos of their top performances.

While we don’t offer an instant path to the top podium step, what we do offer is knowledge. The knowledge to never be confused about what you should be doing on track. The knowledge to know exactly where you are losing time, and what those champion drivers are doing that makes them faster. You will no longer have to rely on trial and error. You will no longer have to try and simply copy the laps of faster drivers. Instead, you will be able to watch their laps and identify where [*they *]are losing time.

par· a· digm shift


A fundamental change in approach or underlying assumptions.

Learning this new method will require a commitment from you however, because it represents a true paradigm shift in the way you will look at a race track from now on. For many novice drivers with little previous knowledge this should be easier, but for veterans, you may have to set aside previously held assumptions.

A Paradigm Shift

To get you into the proper mindset to absorb this new information we are going to first look at some common driving misconceptions. In today’s top ultra-competitive series, the drivers can’t afford to do anything short of whatever produces the ultimate lap times. If this is also your goal, you might need to set aside the idea having a “driving style”.  One of the side effects of there being an optimal way to navigate a race track is that if you do anything other than the optimal, you are losing time.

The media love to label and will talk about this or that person’s different driving style.  This unfortunately has made its way into the common lexicon of motorsports and people like to try and identify their own driving style.  What drivers should actually try and identify is what aspect of their driving they need to work on and improve. So from now on “I’m not a late braker.” becomes “If I improved my braking, I could be faster.” From now on “I like a really understeering setup.” becomes “If I improved my car control, I could be faster.”  From now on “I’m more of a seat of the pants driver and just go by feel.” becomes “Yeah, I really have no idea what I’m doing.”

All joking aside, this is really important to understand.  Every driver in the entire world has something they can improve on to better their lap times.  Don’t ever let having a certain “driving style” hamper your progress.

There is no “Line”

Most traditional driving instruction usually focuses a lot of time on trying to learn and follow the correct racing line.   Some actually go so far as to claim that all cars should drive on the same line.  Not only is this last part completely false, but there is actually no such thing as a correct line. Only the correct line for you at this exact moment and it will change based on the conditions, your car, and how well you can control it at the limit. There is no reason to worry about trying to drive “the line” until you are driving at your limit, because what happens at the limit is actually creating your line.

Trying to drive on “the line” indicates that there is a certain path you should always follow whereas you will actually learn to focus on the limits of the track and then optimize your current path through them. Because of this, you will need to start thinking of the line more as a result than a technique.  

You will be constantly correcting and updating your path based on where you are and what the car is doing.  You will never drive a preset “line” or “try different lines” to see which is faster as this indicates you are not optimizing for your current situation. It shows a lack of understanding of the fundamental goals that lower lap times. We refer to these fundamental goals as Line Theory.

Line Theory is the term we use for the set of rules you will use to optimize your path. This will all become clearer as you work through this lesson, but let’s look at a quick example now.

If you are driving on “the line” you might aim for the point on track that you have been using as the apex. As you pass the apex you will smoothly apply throttle and unwind the steering wheel until you arrive at your set track out point. You might check your speed or section times at that point to see how good you did.

Every subsequent lap you might try a different apex followed by checking your times to see what affect the change had. Following “the line” indicates you have pre-planned points you will hit and driver inputs you will make, but you don’t know which is fastest without trial and error.

Alternatively, using Line Theory, as you approach the middle of the turn you will begin looking at the trackout edge and predicting how early you can go to full throttle without going off track at corner exit. You think your corner entry path brought you to a good apex so you apply maximum throttle right as you pass it. You quickly notice this current path would carry you a little wide at trackout so you do a slight lift and then reapply throttle so your path brings you right to the edge of the track. You then mentally catalog this so you can try a slightly later apex next time since it would be more optimal.

This is a highly abbreviated explanation of Line Theory and we’ll explain the specifics and all the rules in depth later in this lesson, but we wanted to point out the key differences. While Line Theory rules will never change, the different conditions, and your ability to control the car will cause the resulting line to change. As your car control skills improve and evolve, so will your optimal line. Even if you have excellent car control skills though, there will always be at least some variations that change how you will optimize a corner each time. The resulting line and the driver inputs used will never be exactly the same even if they only change by the smallest of margins.

The Limits of Control

In this last example we mentioned that your optimal resulting line will change as your car control abilities change. Because of this, it’s important to understand that Line Theory and car control need to be considered completely separate.

The primary reason for separating Line Theory and car control is that the ideal path you take will always be limited by your ability to control the vehicle.   Although the rules will be the same, a novice driver will have a different resulting optimal line than an advanced driver.  A quick example is that a more advanced driver will be able to control a car closer to the limit and would have been able to turn more and achieve a higher speed by the point they put the power down for corner exit.  This will require a different optimal apex than a novice driver.  

There are a lot of Line Theory fundamentals packed into that example we haven’t explained yet, but the take home point is that no matter what your skill level, you can still apply the rules of Line Theory to improve your lap times.

It’s also important to understand that unless we specify otherwise, we are always discussing the situation as if the driver is controlling the vehicle at the limit as best they can.  One rule of Line Theory is that under normal circumstances, for ultimate lap time performance, there is never a time that you are not trying to use the vehicle’s potential to its maximum.  As a novice advances in their car control skills, that potential will change and thus their resulting line will change. How they apply Line Theory fundamentals will always however, stay the same.

It’s also vital that you don’t make the mistake of confusing failures in car control for failures in Line Theory. The differences we are talking about are sometimes only fractions of a second and most of the time a non-optimal technique done perfectly is going to be faster than an optimal technique done poorly. This is probably how the idea of having a “driving style” came into being as very talented drivers can make all sorts of non-optimal approaches work if they are only going up against drivers of lesser skill.

What’s most important is that you understand why a technique is faster even if you aren’t able to do it properly yet. Line Theory is not a list of techniques to try out and see if they work for you and to discard if you are not immediately faster. Doing so shows a lack of understanding of what Line Theory represents. Line Theory is just the term we use for the application of the basic principles of physics on a race track. These principles are immutable until the physics of our universe decide to change.

Therefore, the most powerful aspect of Line Theory is not that it’s just a list of techniques that tells you the fastest way around a track, but once you have a complete understanding, you will [know *]the fastest way. Wouldn’t it be great if you could never have to worry about trying other lines, techniques, etc… because you know what you are doing is correct? It’s a very powerful feeling to remove all doubt from your driving and[ *]know exactly why you were slow or why you were fast. Learning Line Theory will remove that doubt forever, and from then on, the only thing holding you back will be how far you can push the limits of your control.

Vehicle Dynamics Simplified

While there is certainly a lot of vehicle dynamics theory going on behind the scenes in this lesson, from the driver’s perspective there is really very little you need to know once you are in the car. The only thing you really need to understand at this point to work through this lesson is that cars have an overall traction circle.

For those who are unfamiliar, a traction circle represents the concept that a car generates grip (or force) in all directions fairly equally.  While each tire also has a traction circle and is generating a force, if you combine all those tire forces you have an overall force acting on the car’s center of gravity.

We use a circle because the maximum force attainable is fairly even in all directions. If you try and push a parked car from various sides it will take roughly the same amount of force to get it sliding. This same concept applies at speed and although there is a vast multitude of variables affecting the exact grip available at any instant, really all you need to know right now is the basic idea that it’s similar in many different directions. For example, a given car might either brake with 1 g of force or turn with 1 g of force or a combination of braking and turning that generates 1 g of force in a diagonal direction. Very few cars will be able to actually accelerate with the same force that they can brake or turn, but we’ll get to that later.

Also realize that the traction circle is not just a pretty representation. You could actually imagine a jet thruster coming from the center of gravity of the car pushing the car in the direction of the arrow. The tire forces will combine to create a net force on the car and this force will modify the car’s path.

Expanding this concept, you should also understand that for any given radius a car travels on, it can only achieve a certain maximum speed.  This basically means if you drive a bigger circle, you can drive faster before the car is no longer able to maintain its arc. This should be fairly intuitive, but a quick example is that if a given car can drive at the limit around a 100 m radius circle at 100 mph, on a 25 m radius circle it would only be able to drive 50 mph before sliding wide. Grip vs speed is an exponential relationship and this example ignores aero effects, but that is not important for this lesson. Just that you understand the general concept that a bigger circle allows a faster speed at the limit.

This is pretty easy to understand when you have big changes in radius and speed, but realize this also holds true for small changes. Racecars will almost never be on a constant radius arc. They will normally either be increasing or decreasing their radius. As this radius changes, the attainable speed will also change. Even if the radius only changes by 1 cm the attainable speed will also change. This is true with all cars including those with high downforce. A smaller radius will always require a lower speed and vice-versa. There is no free speed or grip, and you can’t cheat the laws of physics.

Astronaut racing

Before we get into learning about how to drive the perfect corner on a race track we find it useful to take a step back and look at the basic physics involved from an intuitive standpoint.   A car, it turns out is actually quite complicated as you have to worry about silly things like steering, throttle, brakes, tires, and so on.   Let’s take all of that out of the equation and take a fun little detour… in space!

Imagine we have an unfortunate astronaut working on his ship when an explosion damages his suit and launches him into space at 100 mph directly away from the ship.  Because his suit was damage he needs to get back to the ship as [*quickly as possible, *]but his only source of propulsion is the fire extinguisher that he brought with him on his spacewalk for some reason.  The fire extinguisher will let him generate maximum thrust in any direction virtually instantly.  He just has to point it and pull the trigger.

Our astronaut quickly surmises that to get back to the hatch as quickly as possible he will simply point the extinguisher away from the ship and blast away to bring him back to the ship in the minimum possible time. This will cause his 100 mph initial velocity to slow to 0 and then he will begin to gain speed on the way back to the hatch.  Let’s ignore the hazardous side effect right now that he would enter the ship at 100 mph.

This simple thought experiment tells us two very important things right off the bat. The first is that to minimize his time to get back he wants to use the absolute maximum thrust he possibly can directly away from the ship for the entire flight. Our illustrations for this section will use arrows to depict the direction he is pointing his extinguisher. You’ll see why this is important as we continue.

Secondly, if we plotted his speed vs time on a graph it will be a V shape with his speed starting at 100, then dropping to 0 and going back up to 100. The more power he can generate, the more angled the V shape and the lower his time in space. His minimum speed is also at the furthest point from the ship. Before moving on, make sure you understand the astronaut’s actions because, believe it or not, what he just did is the very core of Line Theory.

Introducing turns

Now let’s start changing the scenario so we can see how this simple concept can be expanded. On his next mission, the very unlucky astronaut is working on the tail of his new ship at the other end from the entry hatch. There is an explosion yet again and he must get back to safety as quickly as possible. To reach his goal he’s going to have to stop his outward motion and reverse it as well as move to the side in the direction of the hatch.  We are going to assume he needs to enter the hatch straight on or he crashes into the side and explodes. This assumption will help to simplify this thought experiment.

After being launched into space, the astronaut, using his advanced physics knowledge, quickly points the extinguisher away from the ship but partially angled downward in the illustration to decelerate his movement away from the ship, but also start moving him laterally toward his goal. His speed will slow and once he’s reached the farthest point away from the ship, he starts to accelerate back toward it.

At his farthest point he needs to change the angle of the extinguisher to the other side so his path will slowly straighten and he’ll arrive at the airlock dead on.

The astronaut’s actions cause his path through space to be in the shape of a parabola. Savvy readers might recognize this as the basic shape of a racing line through a 180 degree corner.  He decelerates to a minimum speed at the farthest point from the ship (the apex in racing terms) and then accelerates back toward it.

If he aimed his extinguisher perfectly in the proper direction he just optimized his path to the hatch and it’s impossible for him to get there any faster than this. The illustration shows his minimum speed as 30 mph, but the actual speed doesn’t matter right now. What matters is that you understand that it’s the minimum speed he will achieve. If you placed a floating cone out at the point he reaches his minimum speed, he would have also optimized his path around this obstacle as if it was a corner on a track.

It’s not so much the shape of the path, but the direction of force that is important right now. The shape of the path is simply the result of that force. His extinguisher is always basically going to be blasting in the same direction opposite the ship.  He does need to angle the extinguisher sideways to start and stop the sideways movement, but this is a relatively shallow angle and the majority of the force he feels is going to be his deceleration and then acceleration toward his ship. If you plotted his path using a speed vs time graph you would again see a V shape. The bottom of the V would be his minimum speed of 30 mph.

If you are having a hard time grasping why this one singular direction of force is so important to minimize the astronauts travel time, try and imagine what would happen if he pointed his extinguisher in any other direction and what that would do to his travel path and time in space. Some people are better at visualizing this than others, but it’s extremely important to have an intuitive understanding of what we are explaining here so it’s definitely worth the time to do your own thought experiments if necessary.

Before moving on we also wanted to point out that while the astronaut must angle his extinguisher to create the sideways movement, a car will also always have a similar sideways force generated while turning. Not to get ahead of ourselves here, but since a car can basically only generate force with its engine while going forward, it needs to do some rotating to be able to do that at corner exit. This needed rotation will require at least some sideways force.

We don’t want to overcomplicate this example, but did want to point out what that sideways angle of the astronaut’s extinguisher related to in car terms. It’s actually a driver’s goal to minimize this sideways force by only using the bare minimum needed to move the car over to the apex. Any extra speed carried past the apex than is necessary just has to be reversed during corner exit. In a car this will cause a driver to not be able to use as much throttle as would be optimal and this hurts their exit.

Minimizing these sideways forces leaves as much force as possible to be used in the primary direction. Throughout this lesson when we speak about the “ideal direction” we are referring to this primary direction of force the driver is trying to maximize. Understand though that there will always be some sideways force needed as well.


Changes in force Direction

So now we understand that optimizing the direction of force is the key to the astronaut’s salvation. So far these forces have basically always been going in the same direction, but let’s now look at what happens when we change things up a bit to look more like a 90 degree turn on a race track and see how this affects our forces needed.

On his third mission, our astronaut starts to suspect maybe he should seek a different career as he is blasted into space yet again. This time however, his ship is damaged beyond repair and he must make it to the sister ship. Unfortunately there is also a piece of space debris in the way that he must first navigate around.

The astronaut again using his great understanding of physics, looks at the obstruction and quickly calculates that he needs to immediately blast his extinguisher at a slight outward angle to slow him down from 100 mph and reach the corner of the obstruction at the proper speed.  For this example we made that speed 50 mph, but again, the actual speed is not important right now, just that you realize it’s the slowest speed he attains on his trip. Also note that because the starting speed is the same as in the previous example, the needed direction of the extinguisher and the resulting shape of the path up to this apex is exactly the same as the beginning part of the previous path. You could overlay them and they would match up exactly. We will look at this in depth in the corner entry section.

Immediately as he passes the obstruction he turns his extinguisher to start moving him toward his goal in the minimum overall time possible. To keep this example simple the explosion launched the astronaut at the exact speed he would need to optimize this corner, but later we’ll see how this optimal starting speed is determined. An analogous situation in a car for the explosion would be the point a driver decides to start turning their car during straight-line braking.

Again, we can see the shape of the astronaut’s path, but also more importantly it shows the direction of force he is generating with the extinguisher. The key difference from the previous example is that once past the obstacle, the optimum direction of force changes. To travel in the minimum time possible around an obstacle, or in racing terms an apex, you want to generate as much force as possible pushing you backwards as you decelerate and turn toward the obstacle (apex) and then after you pass the obstacle you want to generate as much force as possible pushing you toward your final goal.

The direction of force needing to be generated simply follows the angle of the corner. If you moved the sister ship, the forces needed after the apex would follow it. If you changed the point where the astronaut started, the direction of forces needed prior to the apex would simply follow that as well. On a race track this ideal direction will basically just follow the same angle as the track does at corner entry and exit.

It might be confusing to think if you are trying to get to the other ship in the shortest time possible that you would start out by trying to push yourself in a completely different direction, but remember, the astronaut starts his path going 100 mph already. If he does not immediately start slowing down and turning he will fly past the apex out away from his goal and have to spend extra time coming back toward it.

It’s important that you really understand this section and it might be worth rereading a few times and taking the time to think about it if necessary. While driving a race car, having a constant awareness of where the ideal direction of force should be is very important.  In the real world, the better a driver can generate and direct these forces with their vehicle, the lower their lap times will be and that’s really at the heart of what this lesson is about and ultimately the core goal of Line Theory.

Of course, just telling a driver to go “generate some forces” doesn’t really do much good on its own.  As usual, the devil is in the details, but to understand the basic physics at work is helpful when trying to get an intuitive understanding of what you are really trying to accomplish on track.  When trying to work through this lesson or trying to work up a particularly tricky section of track, it might be helpful to think back to this section.  If you ever find yourself confused, just ask yourself, “What would the astronaut do?”

Back on Earth

Okay, now we are well on our way to becoming a professional astronaut racer, but how does this help us down here on Earth where drivers need to worry about silly things like steering wheels and gravity.  As it turns out, a car can actually mimic the actions of our astronaut quite well. While the astronaut just points his extinguisher and blasts away, a racecar driver would need to use his steering, brakes, and throttle to generate these same forces with the tires.

It’s important to realize that the forces we talk about are ultimately generated at the tire/track interaction. For example, during corner exit, it’s not really the engine that’s generating the needed force, but the engine’s ability to rotate the tires in combination with the driver turning the steering wheel that generates the force from the tires. The driver not only has control over the total amount of force, but can also alter which way the forces are directed. The better a driver can maximize and direct these forces to mimic the actions of the astronaut the faster they will be able to complete a corner.

Let’s work through this step by step as we learn what we are trying to accomplish in each section of a corner.

Corner Exit

It’s always a debate whether to teach corner entry or corner exit first.  It’s natural to want to go in order, but we really can’t go any further before first developing an understanding of what we are really trying to achieve at corner exit and how everything revolves around the apex.

For our first example, let’s look at the hairpin at Suzuka. Remembering that for any given radius, there is a maximum speed we can achieve, we’ll put a car through the turn that on the limit can drive in a perfect circle at 50 mph.  The blue line represents the path of the car, but the white shaded circle shows how the path of the car during the corner is completely circular.

Now let’s drive the hairpin with a speed limiter in the car set to 50 mph so you just floor the throttle and drive a perfect circle along this line. We are still at the limit of traction of the tires, we just don’t have any extra power to accelerate. This circular path causes the apex to be pretty close to the middle of the corner.

Take note that in this corner exit section we are going to be driving the beginning half of the corner in a perfect circle so we can more easily visualize the differences in speeds and angles at different apexes. This would require the driver held a basically constant steering wheel position and speed from corner entry to the apex.  The car is still at the limit of traction though. As you’ll see later, this is not the way you should optimally drive corner entry, but it helps with explanation and simplicity for now. A circular entry is also much easier from a car control standpoint so for a more novice driver it’s not a bad approach as you improve your skills.

With the speed limiter on, there is no way to exit this corner any faster as we are generating the absolute maximum forward thrust possible. There is also no way to go faster at the apex by driving a bigger circle and still stay on track at corner exit unless you lifted off the throttle. This line is the optimum corner exit for the car right now.

While this circular corner exit would be optimized for a car driving with a speed limiter on, being able to apply more power at corner exit would get us to our goal faster and lower lap times. That should be fairly intuitive, but to understand exactly why from a physics standpoint let’s look at what is happening from the viewpoint of our astronaut.

To illustrate this we’ve added arrows representing the direction of forces the car is generating with its tires.  Just like in space, the arrows should be pushing the car in basically one ideal direction that follows the angle of the track at corner exit. As you can see at the apex the arrows are going in the proper direction pushing the car toward its goal, but as it progresses through the remainder of the corner exit they progressively start pointing in a non-optimal direction away from the desired direction of travel. Only once the corner is fully complete do they move back in the proper direction.

The final small arrow is shown to indicate the force pushing the car down the straightaway after the cornering is complete. Because of the limited power it is a much smaller arrow than the others. Remember, we were driving the entire corner on the limit of traction, but as soon as the cornering is over the tires are no longer required to generate very much grip in comparison. Again, from a physics standpoint, the forces being generated are always from the tire/track interaction. Only a powerful engine would bring the tires near their limit once on the straightaway.

Now let’s remove the limiter and see if we can do better.  Remember though. We were already on the limit of traction, so even with more power, if we try and drive along the same line as previously, but accelerate at any point we will run wide and start heading off track. In fact, if we even try and use one tiny extra bit of power than what we used earlier we will run off track at the exit.  Therefore, in order to use this extra power we must turn more before we begin accelerating.  This changes our apex in two ways.  It moves it further in the corner (a later apex), and we will also arrive at the apex at a slower speed.

A late® apex

It’s important to understand that there is no such thing as a “late apex” or an “early apex.” There is only a correct apex which can either be later or earlier in the corner than it was before. If we overlay a smaller circle going from the edge of the track at corner entry to the inside you can see why you will arrive at a later apex slower. Remember, the smaller a circle that a car drives on, the slower the speed it will be able to attain. The exact speed is not important right now, just that you realize it’s slower than the earlier apex because the circle is smaller.  The later an apex is, the smaller the circle and the lower the apex speed.  

Remember also we are illustrating the corner entry path as perfectly circular to more easily visualize the change in speed with varying apexes right now. Again, this is not the ideal way to do corner entry, but the most important point to realize is that the speed at the apex will always be slower with a later apex. Even if a driver does a more optimal technique than the perfectly circular path we’ve shown here, the later an apex is, the slower the speed at the apex will be. As we’ll see later, you can have a higher apex speed and a later apex, but this requires a compromised corner entry because this later, higher speed apex is not the true apex.

This figure shows the two previous examples laid on top of each other. Look closely at how the differing size circles representing different speeds possible will hit the apex at different places and angles. Now try and visualize for yourself how different size circles representing different apex speeds will meet up with the corner at different points and angles. The smaller, slower circles will always meet up with the apex further in the corner. Conversely, a larger, faster circle will always hit it earlier. This will create a steady progression of speed and angle as the apex moves along the inside of the track.

Corner exit Power application

Alright, so to use more power we need to turn more and be at a greater angle and therefore be going slower at the apex.  Well how much power do we want to use?  The basic answer is “all of it”, but technically what we are trying to achieve is the maximum force the tires can provide to accelerate us in the ideal direction.

In many lower powered cars this is basically going straight to full throttle in all but the slowest corners.  In a high powered car or slippery conditions you might be only using partial throttle throughout the entire exit.  In a car with lots of aero downforce you might be increasing power as you gain grip with increased speed.

The goal is always the same though.  Maximum acceleration, which is either full throttle if you are at the edge of understeer or the maximum throttle possible that keeps you from going into excessive wheelspin. It’s not technically the wheelspin that is important, but that is a good shortcut that works for many cars. An example of where this shortcut doesn’t work is with off-road driving where maximum acceleration is achieved with high amounts of wheelspin. The goal is still maximum acceleration, but they achieve this in a different way.

It’s also not just forward acceleration we are talking about here, it’s combined vector acceleration. The total combination of lateral and longitudinal force throughout corner exit. This introduces the concept of acceleration arcs which we’ll look at later in the lesson. For now though, just remember that you are trying to achieve maximum acceleration in the ideal direction just like the astronaut.

Okay, so we know we want to use as much power as possible, but when do we begin using this power? You might have heard before that you shouldn’t get on the throttle until you are sure you won’t have to lift.  This is absolutely true, but it’s even simpler than that.  The ideal acceleration point, every single time, is at the apex. That’s right. On every standard corner, on every track in the entire world your goal is to achieve maximum acceleration as quickly as possible at the apex.  Not before, not after.  AT THE APEX. You can see we think this is important to remember.

You actually use your ability to achieve maximum acceleration from the apex to find where the apex should be.  If maximum acceleration from the apex out does not allow you to just barely stay on track at corner exit it simply means you made a mistake and need to adjust your apex until you find one that lets you do that.

Let’s walk through some examples so we can see how this works.  Going back to our high powered car in the Suzuka Hairpin we just crossed our original centrally located apex and then accelerate as hard as possible. We’ve removed the speed limiter now so wheelspin would be no problem to achieve if we wanted to.  The car starts to rocket toward the track edge and we need to lift so we don’t run wide.  

Okay that was wrong.  Let’s move the apex further around the corner so we can turn more before we begin accelerating.  Now this time, as we have traveled on a much smaller circle we are almost all the way through the corner and have a much later apex before we accelerate.  Because we had to drive a smaller circle we have also arrived at the apex slower. Now we are able to use more power which is a step in the right direction, but it actually felt really easy to stay on track so that’s not right either.  Let’s pick somewhere between our first two tries.  Now the placement of our apex as well as our apex speed and angle is in between the previous two.

We’ll come back to this in a moment to see what happens, but you can probably see a pattern beginning to develop here.  Your proper apex is always going to be determined by the current acceleration potential of your car.  The faster a car can accelerate in a given corner, the later the apex needs to be.    When people talk about “momentum cars” what they really should say is that you are driving a relatively earlier (and higher speed) apex than if the car had more acceleration potential in that corner.  Even a Formula 1 car in a very high speed corner might look like it’s driving like a “momentum car” with a relatively early apex. At the other end of the spectrum, a low powered car in a very tight corner would have good acceleration potential and so would need a relatively later apex.

It’s actually a common misconception that the length of the straightaway following a corner is a deciding factor on your apex. This is actually not true as you will see coming up.  If there is a 1 mile straight or a 1 meter straight use the same rules to optimize your corner exit.

As you pass the apex, the only reason you should not be at full throttle is to avoid excessive wheelspin because this reduces the forces the tires can generate. Wheelspin doesn’t always cause oversteer in every car, but when it does, your throttle use shouldn’t be limited until you reach true power oversteer which we’ll look at later.

Conversely, if you experience understeer and run off track you don’t need less throttle. You need a later apex. To put this another way, if you are ever understeering during corner exit, you had better also be at full throttle. If that carries you off track, you need a later apex, not less throttle because less throttle will reduce the maximum tire force produced. The ideal solution is instead changing the apex to redirect that maximum force.

The definitions we use for understeer and oversteer might not be exactly what you are used to and we’ll talk much more in depth about this later. For now just remember if you ever find yourself going wide or not using the whole track, this doesn’t change how much throttle you use, it only changes your ideal apex.

This is an important Line Theory rule to remember, but why exactly is it true? Let’s look at the situation again from a basic physics standpoint. This figure shows our new later apex, but more importantly the direction of forces that the tires are generating. As you can see this now mimics the forces that our astronaut created and they are pushing the car in the ideal direction.

The greater acceleration potential a car has for a given corner, the better it is able to optimally direct its tire forces during corner exit and the later of an apex it will need to do that. This later apex must be at a higher angle and a lower speed than if the car had less acceleration potential in that corner.

But what happens when a car doesn’t have very much acceleration potential in a corner? This could simply be a car with very little power compared to its level of grip or it could be such a fast corner that the car isn’t able to accelerate very much even at full throttle.

The resulting forces would look like this previous figure showing non optimal force generation. This is simply just a case of “as good as it gets”. While later in the corner the forces generated are not directed optimally, the total net force created through the corner exit is the best the car can do. If you attempted to apex later like a more powerful car, the engine simply wouldn’t have the power to bring the tires near their limit. So although you could aim the forces better, the overall net amount would be lower and the result would be a slower corner. Not just corner exit speed, but time spent in the corner as well.

The Perfect Apex

Just remember, you always want to maximize the acceleration of the car using as much track as possible during corner exit because that will always give you the best net result of tire forces.

To bring this full circle let’s continue with our previous Suzuka hairpin example. Back in our more powerful car we cross our new apex and get hard on the throttle as we drive at the absolute limit with all four tires giving everything they can. The outside of the track comes rushing up and the outside wheels just clip the edge as we rocket down the straight.  “That was good” we say to ourselves.  And it was.  We just optimized this corner exit and found our apex.

Apex Angle

It’s important to understand that it’s not really a certain point in the corner that signifies the apex. It’s actually the angle that the car is facing, or more accurately, the direction it is currently traveling that really matters.  For example, imagine if you have a cone on an autocross course as the center of a hairpin turn and a car needs to go down around the cone and back.  Any car driving this corner will have its apex point be right at the cone, but depending on the apex speed needed for a proper corner exit, the angle their car is traveling while passing the cone will be different.

This figure shows two different apexes. You can see that although both cars would hit the apex at almost exactly the same place the more powerful yellow line car that needs a later apex would be at more of an angle as it passed the cone.

When using Line Theory principles on track to identify your apex on shorter corners it’s normally more important to pay attention to not so much where an apex is in a corner, but what direction the car is traveling as you reach the apex.

At the opposite end of the spectrum, there will be big, long corners where it might be easier to remember the apex as a point on the track you hit rather than the angle. It’s still the angle that actually signifies the apex, but using the point it meets up with the track might be easier to remember. Try and visualize how differing size circles representing different apex speeds laid over various shape corners will change the angle and point where they create an apex.

Another way to look at the apex is that it is the point on the inside of the corner that most compromises your ability to go faster through that corner. Starting from the center of the track, if you continuously drive a faster and faster line you will eventually start to hit some point on the inside of the corner. That is now your apex and everything then becomes about optimizing around that point. As you optimize your line, this point might change some as your new optimized apex angle could change where it hits the inside of the track.

Also realize that while the apex is usually some point on the inside curbing of a road course corner, or a cone on an autocross course, it can be anything. It just has to be a point that most limits your speed. It could be the edge of a slick spot while racing in the rain. It could be a deep rut or big bump when driving in dirt. It could be a small kitchen appliance blocking your way. As we’ll see later, it could even be an imaginary point out in the middle of the track if you’ve made a mistake and missed your braking point.

Just remember your acceleration and deceleration always revolves around this one point and angle that is most limiting you right now.

The Myth of the SUPER Late Apex

Wait, wait, wait.  But if our goal is to maximize our corner exit speed why don’t we just start accelerating before the apex and then we will be going even faster than if we wait till the apex to start accelerating?  Just search for “late apex” on the internet and you will find countless examples of websites and images depicting this idea.  Even many of the most popular racing books and schools advocate accelerating prior to the apex.

While you can actually increase your apex speed and initial straightway speed this way, it will result in an overall greater elapsed time. Not just for the lap, but also for the corner. The reason is that normally you gain only a few mph at corner exit, and as we have learned in order to do this you have to drive a tighter radius at the beginning of the corner (a smaller circle). This means an even slower minimum speed and most importantly more time spent turning the car.

It’s this last bit that is key.  You often hear you want to get on the throttle as soon as possible to minimize lap times.  This is absolutely true, but the key is that it should be based on time, not distance.  From the point that you hit the brakes to enter the corner, you want to minimize the time before you are able to get back to maximum acceleration and you do this by changing the direction the car is traveling as quickly as possible at corner entry.  

It turns out the fastest way to accomplish this puts your acceleration point [*at the apex.  *]Remember how the astronaut always reached their slowest speed at the apex. The force generated by the extinguisher decelerating the astronaut before the apex was the equivalent of a car trailbraking, and the extinguisher force after the slowest point is the equivalent of applying power at corner exit. We are just using the car to mimic what the astronaut was doing.

When you start acceleration before the apex you are actually optimizing the corner for a false apex out in the middle of the track somewhere. Basically driving as if you are going around an obstacle that is not really there. To fully understand this we will need to work through the corner entry section coming up soon, but let’s look at it briefly right now.

This figure shows an optimized apex as well as the super late apex. They both still have a circular entry path though. If you do a super late apex and begin accelerating early you might have a speed of 52 mph at the point you hit the inside of the track, whereas the optimum apex would have an apex speed of only 50 mph. But at the same point in time that the super late apex car hits that 52 mph apex, the optimum apex car is 10 meters further down the track and is already going 53 mph. Their ideal acceleration arcs will match from the point they were both at 52 mph and so the optimum apex wins every time. We’ll talk much more about acceleration arcs later in the lesson.

So the problems with the super late apex is actually related more to corner entry than exit. You’ll see how the problems with the super late apex are even greater once we start driving an optimized corner entry.

Truthfully, you do actually use this early acceleration technique sometimes because it can gain you straightaway speed that can be used in certain circumstances.  This could be on the last corner of a lap to maximize speed as you cross the start line on a qualifying run or to setup a pass during a race. But from a current lap time perspective it is always slower than optimal because you are taking longer than necessary to turn the car and begin accelerating.

It’s this necessity of turning that brings us to corner entry and our overriding goal there which is, in simple terms, to get the car turned as quickly as possible. It’s important to understand that when we say turn, we mean to actually change the car’s direction of travel, not just rotate the car.  After all, you can pull on the handbrake to initiate a quick rotation, but you haven’t actually changed the car’s direction of travel much. It’s very important to understand the difference.  If it’s hard to grasp this, think about the astronaut.  He uses his extinguisher to change his direction in space (or turn), but it actually doesn’t matter which way he is facing (rotation) as he does it.

For the rules of Line Theory, it’s not rotation that matters, but the actual changing of the direction that you are traveling. Later we’ll examine how these are related, but for now we are simply talking about the car’s change in direction of travel, or again simply, turning.  It’s this necessity to turn that brings us to the importance of the Euler spiral.

Spiraling into control

Since we strive for an intuitive approach, this is really as advanced as the physics and math are going to get in this lesson.  Looking at the spiral, really all you need to know is that as the line progresses it continuously bends more and more. Technically speaking, the radius decreases as the line length increases. Engineers that lay out train track designs use portions of Euler spirals to smoothly transition trains along the tracks so there is no sudden increase in g force. The spiral continues into infinity, but we are really only concerned with the first 90 degrees of it. We’ll see later why the 90 degree limit is so important.


To be continued in the perfect corner

We hope you have enjoyed this introductory lesson. In the remainder of The Perfect Corner lesson we will cover such topics as:

  • Using the Euler Spiral on track
  • Spiral variables (starting speed and size)
  • Beginner vs advanced driver spirals
  • Car and track effects on spiral shape
  • The “artist rendition” line compared
  • Circular entry compared
  • Myth of the corner exit speed check
  • How corner entry determines your apex
  • The physics of trail braking
  • Determining brake points
  • The infinite straightaway problem
  • Comparing acceleration arcs
  • Optimizing entry and exit together
  • Introduction to advanced corners
  • Full throttle into the spiral
  • Braking or deceleration?
  • Real world considerations
  • The driver’s eye view
  • Introduction to the Universal Cue
  • Bonus section: Advanced racing physics


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Table of Contents

  • The perfect corner
  • A Paradigm Shift
  • There is no “Line”
  • The Limits of Control
  • Vehicle Dynamics Simplified
  • Astronaut racing
  • Introducing turns
  • Changes in force Direction
  • Back on Earth
  • Corner Exit
  • A late® apex
  • Corner exit Power application
  • The Perfect Apex
  • Apex Angle
  • The Myth of the SUPER Late Apex
  • Spiraling into control
  • (Untitled)
  • To be continued in the perfect corner…

The Perfect Apex

This introductory lesson will take you through the beginning of The Perfect Corner. We dispel some common driving misconceptions and then introduce the physics of racing with a fun and intuitive approach. We will also take you through the all important corner exit. You will learn why the ideal acceleration point is always at the apex in standard corners and how you actually use this to determine your perfect apex placement.

  • Author: Paradigm Shift Driver Development
  • Published: 2015-11-06 15:50:10
  • Words: 9363
The Perfect Apex The Perfect Apex