The Racing Line - 4 Elements of a Perfect Corner (Physics Explained + Sim Training Tutorial & Tips)

This video was created as an introduction to the physics of racing. If you wish to continue your education and are looking for real answers to your driving questions, start with:
The Science of Speed Book 1: The Perfect Corner
www.paradigmshiftracing.com/s...
Topics covered in this video include:
0:00 - Intro
3:28 - The Geometric Line
6:07 - The (super) Late Apex
9:08 - The Perfect Corner
11:51 - The Corner Entry Spiral
14:09 - Maximum Acceleration
19:05 - Use the Whole Track
22:29 - Find the Limit
24:40 - The Universal Cue
27:10 - Driver Training
Science of Speed Bookstore - www.paradigmshiftracing.com/s...
Driver Training Program - www.paradigmshiftracing.com/a...
Track Notes - www.paradigmshiftracing.com/s...
Racing Basics Articles - www.paradigmshiftracing.com/r...

[@bintang7560]

After sim-racing for 4 years, I'm stunned to discover how little I actually knew about the correct racing line. I can't wait to try out the training exercises. Thanks so much!

[@Racc1-1]

Can’t believe this channel doesn’t have way more attention. This was the best visual representation and easy grasp of the apex, 10/10.

[@ParadigmShiftDriverDevelopment]

Thanks, glad you liked it.

[@smoothacceleration437]

Wow, the best video on driving theory I have ever seen.

[@gregoryj99]

This is by far the most thorough and logical breakdown of what is needed to optimize time through a corner. It’s also the precise and explicit articulation of concepts I was dimly and incompletely intuiting. Thank you!

[@gregoryj99]

I was curious, if we maximize forces in the ideal direction on exit, wouldn't that also be a euler spiral in reverse?

[@lorenzodisaro2588]

​@@gregoryj99 From my understanding yes, but only if you have a car powerful enough to reach the grip limit (from the apex). Otherwise keeping higher speed from the apex of the corner on a rounder trajectory is better

[@alecmillea4539]

Man I was going to write a comment but you took the words right out of my mouth! This truly is an invaluable video!

[@bertschmitz]

The video at the 6:02 mark totally blew me away as a frequent user of datalogging. The higher Vmin is slower?! My (previously) favorite driver coach always talks about speeds at those three points. But I totally get it. It explains my lack of success in increasing Vmin at my most troublesome corners and never seeing any gains in time. Excellent video. I'd ask for many more videos, but this one pretty much summarizes every missing piece for me. I have a new datalogging bible, and it is "The Perfect Corner: A Driver's Step-by-Step Guide to Finding Their Own Optimal Line Through the Physics of Racing (The Science of Speed Series Book 1)."

[@yanghu4481]

Best video ever explaining the racing line I've ever seen. The racing line is actually similar to the line introduced in "Drinving on the Edge" book. But this video explained it much better.

[@niklassilen4313]

This is amazingly useful, especially as a reference for new drivers.

[@LightGesture]

Perfect video. I'm at a stage where I'm really going through your points @21:05. Good stuff. I'll keep watching this video plenty of times. It's so packed.

[@joparicutin]

Finally, someone who really knows about it and how to explain it.

[@joshuabenson2568]

Transform my driving it did indeed, thank you so much!

[@zoloatgai]

I'm currently teaching sim racing myself. This is very helpful, thank you very much.

[@hoopla175]

Love the videos! Keep 'em coming.

[@kevinocta9716]

Super interesting! Thanks

[@maciejskiba5363]

Amazing video, thank you for the tips

[@jacohop]

incredible lesson! thank you 🙏

[@_JP18]

Thank you for the great video!!!

[@nidamreps]

wow! Great Video! Thank you very much

[@josemartin3630]

Really helpfull! You should make more of these

[@akioasakura3624]

The moment you started mentioning force I realized that u know ur stuff. Most trash videos only say “this much speed” or “this much steering”, but really everything is contributing to the total force that the tyres can take. That’s what actually matters, everything else follows from that. Good video sir thank u for uploading 🔥🔥🔥

[@ParadigmShiftDriverDevelopment]

Thanks for the kind words

[@alecmillea4539]

Absolutely fantastic video! Thank you so much for all the time and effort put into it!

[@ParadigmShiftDriverDevelopment]

Thanks, glad you liked it!

[@TheIcybus]

The education on your car videos reminds me of engineering explained and savagegeese. Well done 📝

[@sbcrewproject]

I am amazed how good is this video! Bravo. One topic although that I did not understand is: how to visually understand if I am not getting the correct momentum. Difference between geometrical and optimal lines is very very thin visually.

[@rccartips]

Learned something new today :)

[@wadegruber2119]

Wonderful

[@AceKylar]

There isn't any track near where I live, but at least I have a mountain pass. I will definitely implement these drills on my next run!

[@merrick51]

Great video dude

[@ferchuu9]

Thanks!

[@007ggman]

Best 3 books I ever bought adam. Thank you

[@loicludwig429]

The 4 mini books on training are a super complement imho.

[@RaceTeq17]

Excellent Adam.

[@ParadigmShiftDriverDevelopment]

Thanks, a lot of work and thought went into this video so it's good to know new people are still finding it.

[@JonesIsHere]

awesome videos. I hope you keep them up. Thank you

[@magnum7068]

Love your video, Please make a video on the double apex corner

[@ParadigmShiftDriverDevelopment]

Thanks, you can find some discussion of the double apex in our Lime Rock Park video.

[@CharlesK441]

Awesome.... wanna fire up the SIM NOW

[@jaymckoskey25]

Extremely interesting, thank you for posting and, BTW, I love your books. Questions: If you replaced the apex points with cones and paved the infield would the ideal line change? If you then removed the walls confining the surface what would the ideal line be?

[@ParadigmShiftDriverDevelopment]

Hey Jay, this is Adam, it’s really good to ask questions like this as it helps to test your knowledge. Most corners are really easy to optimize because of the track limits. Start taking those away and it gets trickier. First off, if you replaced the inside with cones you probably wouldn’t see any difference, especially in the mx-5 as it has a relatively early apex. The inside is already pretty pointy so it’s not getting in the way during your entry or exit. In a later apexing car I do sort of run over the inside curbing a bit. It takes a fairly large 180 with a more squared off inside track to need a double apex. If you start getting rid of the walls, you do start seeing changes though. If you just got rid of one of the walls, the line on the other side would most likely go out a bit, but not very much. That’s because it’s a 180-degree corner and we are already at the 90-degree limit on entry and exit. So for example if we got rid of the wall on the exit we couldn’t really apex much earlier as we would start going over 90 degrees on exit. If we get rid of the entry wall, we are still limited by the exit side. We wouldn’t be able to drive a huge entry spiral and hit the apex at 100mph, as we would never make the exit. Normally a track is the same width throughout so you don’t have this issue, but not always. At COTA some of the entries to the hairpins are very wide compared to the exit. This makes it so there is no reason to start near the corner on very outside of the track. Watch an F1 race there and you’ll see the cars, even when not blocking a pass, not use the entire entry on some of the corners. If we remove both walls, things change even more. A few things to keep in mind. You always want the largest entries and exits you can fit as long as you don’t go past 90 degrees on them. You also only want to double apex if not doing so compromises an entry or exit. For example, here you wouldn’t just drive a big circle around the two apexes. You want the biggest entries and exits you can fit so you would just expand them until they started touching. The exit to one corner would immediately go into the entry of the next. There would be no portion where you are driving straight. I was working with a coaching client recently who needed help with Willow Springs and turn 8 and 9 calls for this. So many people get those corners wrong. How wide you come out of 8 is dictated by where your entry to 9 needs to start, not by the outside of the track. The exit feeds right into the entry.

[@kamz87]

Is this exactly what is covered in Book 1?

[@kamz87]

Hey Adam, Is this video everything that is in Book 1?

[@ParadigmShiftDriverDevelopment]

You would really need to put the knowledge together from all 3 in order to come up with my entire answer. When you start taking away track limits things can become harder to figure out.

[@MattSSquared]

@@ParadigmShiftDriverDevelopment it’s validating to read this as an autocrosser because often the track limits are for you to decide and I wrestle with how wide to go. Thanks for the additional clarification in these comments, as well the very helpful video!

[@Eduardo_Espinoza]

I've thought to myself all of these points, except the cancelling of the tire forces, but I still lack the skills to feel a car, see pixels, react to other cars quickly, predict elevation changes, think, etc.

[@markdoran2543]

It seems like recent changes to the tire modeling in iRacing may have affected what is possible in the Miata -- are the sector times and speeds quoted in the video still current?? Also I'm wondering if you would be willing to share some data (iRacing or MoTeC format, latter first choice ;)) as I'd love to do an overlay to see where the opportunities for improvement lie.

[@ParadigmShiftDriverDevelopment]

Hi Mark, this is Adam. I try to keep an updated current best time for the MX-5 at Charlotte legends on the training page of the website. I think it's around 17.4 last I checked, but I haven't done one in awhile. I try and do a new one every time there's been a change that affects times so let me know if I need to recheck it. I don't think I have the old telemetry files still, but they wouldn't be relevant to the exact current version right now anyway.

[@UltimatumNo5]

In your book, you talk about yaw inertia with regards to a chicane - could you elaborate on this because I don’t understand it at all - inertia can’t add to a polar moment.

[@ParadigmShiftDriverDevelopment]

Sure thing, although it would help if you had a specific question. The basics however is simply that a rotating object wants to continue rotating, but during a chicane you need to stop the rotation of the car and reverse it. The rotating object (car) resists this.

[@Igneous01]

Wow I actually really struggle with the exercise. I keep early apexing the corners. From my point of view it looks like I'm hitting the middle apex or the ideal apex, but when I review from blimp I'm nowhere near it.

[@ParadigmShiftDriverDevelopment]

That's a good idea to check your line from an overhead view. This is a large enough corner that it can be difficult to keep track of your location and is good for building up your universal cue skills.

[@fl45hman]

Very interesting stuff! I've just started reading your book "The Perfect Corner 2" and I find a lot of absorbing ideas there. There's one thing that troubles me a bit, though. In the chapter on the Suzuka Hairpin, where you suggest treating it as a double apex corner, you show an illustration of the proposed line. Based on the description and looking at it closely, it seems to me that you suggest creating a spiral-shaped entry line to the first apex, where the line suddenly becomes a circular arc of a radius that's smaller than the final radius of the aforementioned spiral. That step change in radius doesn't seem right to me. Are you sure that chapter is entirely correct? Am I possibly missing something here?

[@ParadigmShiftDriverDevelopment]

HI, the double apex portion begins at the exact same radius you reach at the end of the spiral. It looks right to me in the illustration, but maybe your book copy had a slight printing error or something so it looks odd. Ultimately the lines shown are just symbolic to explain the concepts though. The ideal line will always vary somewhat from car to car and mistakes will cause it to vary from lap to lap. The concepts and rules don't change though. We used the Suzuka hairpin for an example, because during high direction change corners with a rounded inner edge, you should need some portion of double apexing in high acceleration cars. Hopefully this answered your question, but please let us know if it didn't.

[@fl45hman]

Paradigm Shift Driver Development Got it. In that case, I believe the illustration comparing the single apex line with the double apex one is slightly mistaken. It shows them overlapping all the way to the first apex, where they split with the double apex constant radius arc drawn to the inside of the continued spiral. I believe they shouldn't overlap and instead the single apex entry spiral should initially go to the inside of the double apex one, crossing it at the first apex.

[@ParadigmShiftDriverDevelopment]

It's hard to see in the book, but you are absolutely correct. The double apex line would have a slightly later, slower 1st apex than the other line at that point. The differences in actual line on the track is so close however that even in the high res version we have, it just looks like a shadow on the inside of the other. Corner entry lines are always very similar, it's really corner exit where they start to diverge more. If you would like the high res version, I can have one made with brighter colors so you can see the differences up close. Just use the contact form on our website and I can have it emailed to you. Again though, it's still just an illustration explaining the concept. It seems you understand the concept and that's what is important.

[@fl45hman]

As I double check, there is indeed a little shadow as you say. I missed it the first time which led to a little confusion. False alarm, then. And thanks for the offer, but yeah, I feel like I get the idea. Anyway, really glad I've stumbled upon your book as it tackles things I haven't found in other recommended titles.

[@ParadigmShiftDriverDevelopment]

Thanks, if you have any more questions feel free to contact us through the website. Certain aspects of the book we're not quite willing to talk about publicly and can be more specific with explanations if needed.

[@arandomp3rs0n]

100 subs!

[@yellowsurge3309]

How long did it take you to develope a sense for universal cue skills?

[@ParadigmShiftDriverDevelopment]

This is something that will always develop and is primarily related to how well you can see the exact placement of your car on track. Vision provides the upper limit to driving skill. As your vision improves, your car control skill will rise to meet it. Adam likes to say, "The better you see, the better you drive." Check out our Academy lessons for exercises teaching you how to develop this.

[@thomasbonielsen3156]

What kind of telemetry logger and viewer are you using here?

[@ParadigmShiftDriverDevelopment]

That is Motec, there is program that converts iracing data files to motec format.

[@loicludwig429]

Hello Adam, I'm currently working on lesson 2 (which required a lot of faith and patience as I had just shifted my mind from rear to the center of gravity :) ). I just noticed that with the lotus 49, you are still braking past the apex. Is this something that you changed since you made the video?

[@ParadigmShiftDriverDevelopment]

I'm not braking past the apex, the Lotus double apexes a bit here in the middle as it needs a fairly late final apex.

[@loicludwig429]

@@ParadigmShiftDriverDevelopment ok, that was my second guess, but I wanted to be sure because the transition on braking is so smooth. It s been a while since I thought that 180 degrees turns are in fact double apex because of the 90 degree limit and the apex angle needed. Would you say they are necessarily decreasing radius?

[@ParadigmShiftDriverDevelopment]

@@loicludwig429 The ideal radius change would depend on the car and exact shape of the apex area, but a high angle rounded apex corner will typically be very close to constant radius. Plus, even the best drivers have slight line variations that will need to be compensated for during the double apex portion. The double apex portion of a corner like this is very much about car control and maximizing force, not necessarily trying to follow a perfect radius change. I wouldn't call out a driver for a definite mistake here unless they are accelerating before they reach the inside of the track or braking after they have left it.

[@loicludwig429]

@@ParadigmShiftDriverDevelopment i get it. Thanks again Adam!

[@loicludwig429]

I'm puzzled now... It means that favoring exit speed by late apexing before the long straights is actualy plain wrong, isn't it? I was also wondering how usefull is trail braking whith a karting that has only rear brakes, because it means that i only have the outside rear tyre connected to the floor, so i guess there is little grip left for braking. Can you help me with that one please?

[@ParadigmShiftDriverDevelopment]

That's right, the length of a straight should not affect your line. It's not about the speed at any certain position in your corner exit, it's about how quickly in time you can get to that speed from the beginning of the corner. Your apex will be "late" in that it will typically be after the midway point of the corner, but you don't want it to be "super late," where you begin accelerating before you reach your apex. Your point of minimum speed reached should always be touching the inside of the track. For rear brake only karts you will still decelerate to the apex, but your entry line will be more circular in shape than the spiral line that would be ideal for cars and 4 wheel brake karts. Good rear brake karters use a lot of slip angle during corner entry and somewhat pitch the kart a bit sideways to aid in decelerating.

[@loicludwig429]

Paradigm Shift Driver Development thanks a lot

[@loicludwig429]

By the way, if I don't care about current lap time, but i'm trying to break my record on the next lap, then, using "super late" apex on the last corner of the circuit should be a valid strategy. Do you agree with that?

[@ParadigmShiftDriverDevelopment]

That's correct. You'll maximize speed as you cross the start/finish on your quali lap.

[@VitorMach]

Is a standard late apex (not a super late one) acceptable though? Are these really ideal for faster cars? I feel like a late apex will always minimize the use of the whole track, as you don't really need to use all of the width of the track on the exit...

[@VitorMach]

Also, I don't quite understand why the apex momentum is determined by power, and not by grip. Shouldn't you always want to carry as much momentum as possible for a given level of grip, regardless of the potential power you have available? At the end of the day, your turn-in is determined by how much grip your tires can put down, power is only there to make sure the tires are using its peak grip capacity. To put it in a different way: let's first optimize the path to the apex. How fast you can reach the apex is only determined by your grip and braking capacity, not power (as it won't be used). As you reach the apex as fast as possible, you now optimize for the exit. This part depends on the grip again, and on your power. At this point you'll put as much power as possible to maximize use of tire grip (which lands you just on the outer edge). As you begin optimizing from the apex to the exit, however, your momentum is already defined by the previous optimization. In the Lotus example. Why wouldn't you want to carry as much momentum on 2nd gear, as you did on 3rd gear?

[@ParadigmShiftDriverDevelopment]

It's actually determined by the ratio of lateral (sideways grip) vs longitudinal acceleration. Power is just an easy shortcut to go by. If you take a given car and increase power, it will typically increase its ability to accelerate forwards, but its lateral grip won't change. This would make its ideal apex for a given corner become later. Power in itself doesn't necessarily mean a car will alway have a very late apex though. A formula 1 car has tremendous power, but also tremendous sideway grip so its ideal apex would be earlier than something like a Corvette that has half the power, but 1/4 the grip. It's the ratio that matters. It's all about maximizing the total net tire force in the ideal direction during corner exit. The type of car that would have the latest apex for a given corner would be a very powerful 4 wheel drive car because it can accelerate forwards as well or possibly better than it can accelerate sideways.

[@zhutwo]

@@ParadigmShiftDriverDevelopment Hi Adam, I find the concept of maximizing your force vector in the ideal direction fascinating. One thing that I'm at conflict with internally though, is how your ideal line is based on a smooth Euler spiral, yet I've read about the merits of "squaring the corner" or taking a V-shaped line in power-biased cars. If for the sake of thought experiment, we imagine a car with lateral grip approaching 0, that car would maximize its force vector through a 90 degree corner by braking in a straight line to a near stop, cheating the turn using rotation, and then accelerating straight out the corner. Could this strategy not be extrapolated to a car that is overwhelmingly longitudinally grip biased, albeit to a less extreme degree? I understand that a V-shaped line results in some wasted distance compared to a smooth line, but surely there are real situations where the calculus checks out?

[@ParadigmShiftDriverDevelopment]

@@zhutwo Yes, I talk about this in my books. The Euler Spiral line is only a guideline for cars that have about equal braking and turning force. This represents just about every car however. As braking force increases in comparison to turning force you would "stretch out" the spiral.

[@zhutwo]

@@ParadigmShiftDriverDevelopment Hey thanks for the reply! I'll definitely be checking out your books.

[@primehellsing6043]

is the universal que similar to weight transfer?

[@ParadigmShiftDriverDevelopment]

The Universal Cue is a driver’s sense of where the car is and how it is moving through the corner. It could be visualized in a similar way to how you would see an RC car being driven on a miniature racetrack as you tried to drive it through the corner as quickly as possible. This is an external car cue, as opposed to internal car cues such as steering forces or tire noises.

[@ZartaxtheWise]

Hi! I'm curious about what is said at 9:40 and a bit forward. This seems weird to me. Example: Maximizing deceleration in the direction of the corner entry edge is slamming on the brakes as much as you can until the vehicle is stopped. Which doesn't take you through the corner. Turning into the corner is actually lowering the maximum deceleration in the entry edge direction. Easy example is the Laguna Seca turn 6 shown at 10:05. Here, you actually accelerate in the corner exit edge direction at the START of the turn and decelerate in the corner entry edge direction at the END of the turn, with a transition between the two as you go through the corner. It seems to me like the U-turn example is a bit misleading since the directions are the same. The order in which you accelerate in the different directions is not as straight forward as decelerate first and accelerate after, but with a U-turn you can't see the difference. In the Laguna Seca corner you can. Seems like what you want to do is maximize acceleration in the exit edge direction with just enough deceleration in the entry edge direction to not go off track. Where exactly these take place in the corner is an optimization problem with time as the parameter.

[@ParadigmShiftDriverDevelopment]

Hi Björn, this is Adam. Sorry if that part was confusing. I go into this in much more detail in the books. I do use a 180 degree corner as an example because the ideal directions are both the same. For a corner like turn 6 at Laguna, however you are essentially cutting out the middle and spreading apart what you do in the 180 degree corner. The key point I'm trying to get across is that you do want to maximize the total force in the ideal directions throughout the corner, not entry and exit separately. This will primarily be set by what you need to do to maximize corner exit. If you completely stopped during corner entry, although you might be maximizing deceleration on entry, the ability to produce acceleration in the ideal direction during exit would be severely reduced. Maximized acceleration during exit is a combination of sideways and forward acceleration that the car can create. This will vary by car and effectively sets what you ideal apex is. A higher acceleration car can use a later, slower apex and vice versa. This part will effectively set your apex. Then during corner entry you want to get to this ideal apex while also maximizing the car's deceleration in the entry ideal direction. This is where you get the Euler spiral shaped entry. You do technically "waste" force by needing to drive across the track, but this is necessary. The idea is to minimize this force needed while maximizing the forces pushing you in the ideal directions. Hope this made more sense. Let me know if it doesn't. You might also want to give this article a read. www.paradigmshiftracing.com/racing-basics/the-corner-exit-drag-race-racing-line-physics-explained#/

[@ZartaxtheWise]

​@@ParadigmShiftDriverDevelopment Hi! Thanks for the very quick reply! Maybe I'm misunderstanding something here, so let me clarify what I mean and ask some questions. "The key point I'm trying to get across is that you do want to maximize the total force in the ideal directions throughout the corner, not entry and exit separately."

[@ParadigmShiftDriverDevelopment]

@@ZartaxtheWise In regarding the 90 degree corner, you need to take into account how the radius of the line will increase with speed and how we are going to be limited by the apex. By accelerating in the corner exit ideal direction starting at turn in we would need to severely compromise our entry speed. You would need a much later, slower turn-in to still make it through the apex and corner exit. You've essentially added a little cone right at your turn in point where you achieved minimum speed further constricting your line. This is the definition of a super late apex. The apex should always be the point of minimum speed in the corner where deceleration changes to acceleration. Consider that an ideal apex speed is 100. Is it faster to get to 100 by decelerating to it, or accelerating to it? Does this make sense now?

[@ZartaxtheWise]

​@@ParadigmShiftDriverDevelopment ​I think you are misunderstanding what I mean by "accelerating". When you turn, you accelerate perpendicular to your current velocity. So in a 90 degree turn, as soon as you turn your steering wheel you are accelerating in the direction of the exit edge, in some part. Regardless of entry speed. Note that accelerating in the direction of the exit edge doesn't mean your speed is increasing. It's probably decreasing since you are probably also accelerating in the direction of the entry edge (that is, braking). Your drag race is the perfect example of this. Both cars are accelerating in the direction of the goal, from start, even though one starts perpendicular to the goal. It even wins, because it's possible acceleration in the direction of the goal (the exit edge) is higher when limited by grip and not engine power. I see when I read by previous comment that I was unclear in my use of terms. "If you look at the traction bubble of the car on your link, the forward max force is lower than any other direction. That means we have spare grip at the end of the corner where we are already at full throttle, which we then can use for braking, releasing some grip at the beginning of the turn to be used for acceleration in the exit edge direction."

[@ParadigmShiftDriverDevelopment]

Okay, I understand what you are saying now. I need to walk you through a few steps, which I’ll try to do briefly. If you haven’t read my books, you really should. It would probably clear everything up for you. They are written for people who really like to understand racing at the level you are trying to. First try to understand why the apex is the point of minimum speed. The apex is your primary restraint. Consider your 90 degree turn with a cone as the apex. The highest minimum speed attainable in the corner would be by driving a perfectly circular line. You could pass the cone at a higher speed than this, but only by severely reducing your speed earlier or later in the corner. Your instant radius and speed is linked so that means that if you achieve a lower speed anywhere other than the apex you will need to tighten up your line there more than had you done this at the apex. You have effectively added another cone at your point of minimum speed. Once we understand the cone/apex is the point of minimum speed/radius we can understand we will pass it at exactly 45 degrees. The highest speed we can do this at would be with a perfectly circular line. This will make the velocities of our corner entry edge direction (X) and our corner exit edge direction (Y) equal. Driving at the limit, we can still pass the cone going slower if we wish by driving a spiral shaped line on entry however. Our x and y velocities would still be equal and it would still be at 45 degrees though. Now the question is how fast do we want to pass the cone? Just consider an idealized car with a perfectly round traction circle for now. Our goal from the point we pass the cone is to maximize acceleration in the Y direction while bringing our X speed to zero. With the apex speed from our circular example, we can’t add any additional acceleration in the Y direction because we are already at the limit. We need to use significant amount of the available grip just to bring our X velocity to zero. Our exit line will remain a circle at best. If we pass the cone going slower however, we can bias toward our Y direction acceleration as we have less X direction velocity to bring to zero. There will be an ideal apex speed that allows us to maximize our Y direction acceleration while only using the absolute minimum force necessary to bring our X speed to zero. This will create a spiral shaped exit line of expanding radius. Once you understand this, realize that entry is simply the mirror image. Your maximizing force in the X direction while only using the minimum force needed in the Y direction. You end up with a spiral shaped line of reducing radius.This gets you to the ideal apex you have determined in the minimum time possible.

[@TimurIskhodzhanov]

Found this video after reading your book. I like the theory overall, but it left some open questions for me. One is what is the actual direction of the forces for an optimal corner? Let's assume an AWD car that has more power than traction, and a 180⁰ corner. Applying the "SCUD missile intercept" control theory in the frame of reference moving at the approach speed of the car, the fastest way to get to the apex will involve applying the forces in the same direction diagonally from the turn in to the apex. Similarly, getting from the apex to the track out in minimal time (same theory but with a different frame of reference) involves generating force diagonally inwards. What's weird is that this involves accelerating before the apex, and slowing down after the apex 🤔 Curious if you've done any AWD experiments and visualization of forces.

[@ParadigmShiftDriverDevelopment]

I'm not sure I completely follow your example, but have you read this article. www.paradigmshiftracing.com/racing-basics/the-corner-exit-drag-race-racing-line-physics-explained#/ It might answer your question. I talk about AWD in this as well as in the later books more if you've only read the first one. Let me know if that article didn't answer your question and we can dig into it more.

[@TimurIskhodzhanov]

@@ParadigmShiftDriverDevelopment thanks for a very quick response! That article is awesome, thank you. I wish I read it before the book. Do you plan to republish the book with corrections, and/or maybe redo this video? I wish I had a "one stop" place I could refer my friend to.

[@TimurIskhodzhanov]

Also curious if you can point me at some explanation why the right shape of the entry is a Euler Spiral. I have a major in math and physics, and I studied control theory, but Euler Spiral isn't obviously an optimal line to me. Something like it - possibly, but curvature as a linear function of distance feel suspicious if the speed changes.

[@ParadigmShiftDriverDevelopment]

@@TimurIskhodzhanov Maybe at some point although I don't feel any of it is wrong, I just think of different ways of explaining things. Some might work better for some than others. I'm glad that one helped you.

[@ParadigmShiftDriverDevelopment]

@@TimurIskhodzhanov I talked about in the books how I primarily use the Euler spiral to illustrate the concept that the ideal entry line is one of continually reducing radius. You would need a lap simulator programed to find the exact radius change rate for a given car and corner if you wanted something exact. This wouldn't have much use though as a driver shouldn't be concentrating on trying to follow a preset line regardless. They are concentrating on other cues, but we've found that the entry line for a top driver in a typical car matches the Euler spiral so closely that it's also useful when charting out a line on a track map.

[@TimurIskhodzhanov]

At 17:10 I think you got a bit confused. The amount of "thrust" force the driven wheels provide is in direct relationship with the current torque, not power. A theoretical car with a flat torque curve, no aero drag and no drivetrain losses will generate the same amount of thrust on the driven wheels at any rpm while in the same gear. Changing gears will affect that amount though. That being said, the general point is true: the important part is utilizing the traction of the rear wheels to 100%, which can mean all kinds of throttle pedal inputs depending on the exact powertrain of the car. ...TURBO LAG!

[@ParadigmShiftDriverDevelopment]

I understand what you are saying, but I try to keep my explanations as simple as possible as I know it can already be a bit overwhelming. I'm not trying to explain horsepower vs torque here, I'm primarily trying to explain that you don't necessarily want a steady increase in throttle. For anyone reading this who is curious about power and torque here is a good video kzbin.info/www/bejne/i3q0m6ykbpqWY9E

[@garyrowe58]

You shoukd have mentioned earlier in the video that at 8.38 in the video (super late apex) both cars are limited to a max of 67 mph. Because if they are not, and both keep accelerating ... are you saying that if you travel down a straight at a higher speed than another car, you dont gain on them?

[@ParadigmShiftDriverDevelopment]

At the point in time the super late apex car crosses the line, both it and the baseline car are going 67 mph, but the baseline car is 46 feet ahead. From that point on, they would both accelerate at the same rate (from 67 mph) and so the baseline car would maintain that 46 foot advantage. Does that help explain it better?

[@garyrowe58]

@ParadigmShiftDriverDevelopment well, firstly, even if the time gap were to remain the same down the straight, that 46 feet will change as they both accelerate. And, there will be a fixed distance to the next corner that they both need to drive to. 2 cars, each driving down a straight, will NOT do so in the same amount of time if they each start at different velocities. The whole point about leaving a corner with a higher speed is to shorten the time needed to reach the next corner. If that involves losing time in the corner itself, then it becomes a trade off ... time lost in ghe corner vs time gained afterwards. If there is no straight after the corner, then overall you've lost time. If there is a straight, then ghe question is - how long is it, and will you make up the time lost, and more. And, of course, this would also have to take on board the respective power, gearing and drag etc of the two cars involved. I rallied, not raced, and it was never about one corner on its own, but where each corner sat in relation to what was before & after it.

[@ParadigmShiftDriverDevelopment]

@@garyrowe58 It is the time gap that will change as the two different cars continue down the straight. The 46 foot difference will remain the same. Consider two equal cars that take off from zero, but one is 46 feet in front of the other. They will always remain 46 feet apart, but the time difference will shrink as they accelerate. 46 feet takes longer to traverse at 10 mph than at 100 mph. In the video example, the baseline car will always reach the next corner 46 feet in front of the super late apex car no matter how long the straight is. How much time difference this makes will depend on the length of the straight. We have an upcoming article that delves into this topic more that might help you understand it better so keep an eye out.

[@H8MadXero]

​@ParadigmShiftDriverDevelopment while the example video is correct, the car chosen and the extremely large radius is the reason this is true. The smaller the radius and or the more powerful the car the more advantageous a late apex becomes assuming a long straight follows. I'm sure you already know all this but it seems a bit misleading for those that may take this as a general rule.

[@ParadigmShiftDriverDevelopment]

@@H8MadXeroLater in the video we show the ideal line for the Lotus 49 which is a more powerful car. A later apex than the MX-5 is ideal, but not a "super late apex" with acceleration prior to reaching the inside of the track and the straightaway length doesn't factor in.

[@loicludwig429]

Do you always choose afternoon? It looks like i m one second down if i choose it instead of morning...

[@ParadigmShiftDriverDevelopment]

I've been choosing morning lately.

[@loicludwig429]

@@ParadigmShiftDriverDevelopment so the 17'5 from the video has been made during the afternoon, but the 16'9 from the website has been made during the morning?

[@ParadigmShiftDriverDevelopment]

The 17'5 was before you could even pick time of day and a good bit has changed in the sim since then. I've also gotten better so there might be a .1 to .2 or so of improvement for me. I haven't updated the time in a while so I'll try to do that soon and check various times against time of day. Primarily time of day affects track temp. Even if ambient temp remains constant, direct sun can raise track temps and this affects times more or less depending on the car. Plus, with a track like Charlotte, if the sun is behind the stands the track temps drop even more. You can always see time of day in the replays as well.

[@loicludwig429]

@@ParadigmShiftDriverDevelopment yeah, the difference felt massive to me, I did 17'5 in the morning and 18'5 in the afternoon... it was like driving on ice, couldn't get a decent apex... I felt so depressed!

[@ParadigmShiftDriverDevelopment]

@@loicludwig429 I got a chance to try out different weather settings. For late afternoon and morning my time was about the same as my current time on website. Late afternoon is slightly faster, but not a huge difference. The track temps are in the low-mid 80's though. For afternoon, the track temp was 137 and the best I could do was an 18'1. Beyond just having way less grip, it slowed the apex speed enough that I was significantly lower in the power band during exit. I might be able to drop that time with a shift to 2nd, but only if I manually clutched. I use autoclutch so I lose a decent amount of time on shifts. I also tried out night just out of curiosity and was able to get down in to the 16'8s. The track temp was 78 Also, the 1st hot lap is the fastest. These mx-5 tires seem like they are best if you take them out of the fridge right before you drive. They like to be cold!

[@Delvokian]

I couldnt really tell what the difference was between baseline and geometric. The baseline seemed to maybe take a tighter corner? They seem like the same thing but with different steering. I might need a better understanding of the fundamentals of cornering

[@Delvokian]

Wait I think I understand. The baseline is just an arbitrary line that feels about right, the specifics don't really matter because it's just a starting point. It could be any line really. But from that point you then adjust where you want the apex to be and where you want the braking and acceleration areas to be little by little until the line is optimal. Do I have that right?

[@ParadigmShiftDriverDevelopment]

The "4 elements" of the video explain the baseline, but you can read more about the rules for a standard corner here. www.paradigmshiftracing.com/racing-basics/the-rules-of-the-racing-line#/ We also have a new Racing Line Fundamentals lesson series that will cover this more in depth www.paradigmshiftracing.com/racing-basics/heres-a-simple-way-to-visualize-why-the-ideal-acceleration-point-is-always-at-the-apex-of-a-corner-and-why-straightaway-length-doesnt-matter-racing-line-fundamentals-1#/

[@H_is_irl]

You are so smarte

[@ParadigmShiftDriverDevelopment]

Thankse

[@dlptube1]

And what about a late apex? Not a super insane late but just a late apex?

[@ParadigmShiftDriverDevelopment]

HI Luis, sorry for the late reply. Could you explain a little bit more about what you are asking? A super late apex is a line where acceleration starts prior to reaching the inside of the corner. A normal apex would be one where acceleration begins along the inside and then there is a spectrum between earlier and later. HIgher acceleration cars will need a relatively later apex than lower acceleration ones.

[@LoveLikeaHurricane]

confused between Geometric line and base line......they appear like the same line

[@ParadigmShiftDriverDevelopment]

Both lines reduce radius to the apex and then expand in radius afterwards, but the geometric line is more circular than the baseline with closer to a constant speed. A true geometric line with a perfectly constant radius and speed is impossible to achieve as it requires instantaneous direction changes.

[@68ray]

And (Adam) your qualifications/accomplishments are?

[@ParadigmShiftDriverDevelopment]

Hey, this is Adam. If you are really curious, I could tell you about some racing achievements in my life that I thought were pretty awesome at the time, but I'm the kind of person who always looks for the bigger pond. I certainly can say I am way better than I ever thought was possible, and I'm still getting better. But I think what you are really asking is, can you trust me and what I have to say. To this I would answer, "absolutely not." Please don't trust me. I want people to think critically about everything. Challenge what you read and hear, and don't accept it if it doesn't make sense. While in certain fields it's impractical to to do anything but just defer to the experts, racing is relatively simple. I think just about anyone can learn the physics behind it. Not only does this give someone the confidence to truly know what they should be doing on track and not just trust someone, but understanding the physics can actually help you drive better.

[@urboigetting1052]

Try rental Karting

[@deezynar]

Not enough foundation information is given before using jargon and talking about variations. For instance, skid pad data is fundamental. Can a car take a larger diameter lap in less time than a smaller diameter lap even though the larger diameter is further? Tires allow higher speed in larger turns than tighter turns, but does the higher velocity equate with less time taken per lap? That's worthwhile information for a driver to know. And never did he mention that a perfectly circular turn forces the car to transition from zero lateral force to maximum lateral force in almost an instant when the steering wheel is turned at the start of the turn. Railroads avoid circular turns and lay their tracks in parabolic curves so trains gradually transition from straights to curves. I think he used the term 'spiral' instead of parabola, and in real world driving, they are about the same. The point is that circular turns are spoken of in classes, but are not used in the real world, nor are they even possible.

[@ParadigmShiftDriverDevelopment]

This video is meant to be a general overview. For more in-depth explanations, please check out the books, training program, and free articles available through our website at www.paradigmshiftracing.com.

[@HexlGaming]

Theres a couple inherent problems, one of which is that if the Red car in the data enters the corner ahead of the black car, then this is not looking at the difference the cornering makes but rather the change in difference between the cars as a result of the cornering, which you then mistaked for the difference the cornering made and where you didnt analyze the distance between the two cars before and after the turn, making the comparison kind of pointless.

[@HexlGaming]

Also the MX5 doesn't have enough power to force its tires above the limit when accelerating out of a long, wide corner like this one, so the finding should vary greatly when looking at really high power cars like F1 cars or Indycars.

[@ParadigmShiftDriverDevelopment]

I'm not sure I follow you. The comparison starts with both cars in the same place and time in the corner.

[@ParadigmShiftDriverDevelopment]

@@HexlGaming Yes, a higher acceleration car would have an even larger differential loss between the geometric and baseline. For the super late apex, the loss is simply based on how far from the ideal apex the false apex is.

[@HexlGaming]

@@ParadigmShiftDriverDevelopment I was referring to data like 12:33. the way the graph is set up there is off, making it much harder to compare, because at the same timestamps the cars are at different stages of the track, even just in the acceleration part of the graph. So it’s not „Zeroed“ if you will

[@ParadigmShiftDriverDevelopment]

@@HexlGaming Yes, sorry if that is confusing. You are right, we didn't start them from the same point for that part of the video. They were just zeroed from the lap start. That part was just showing the shape of the line to look for. If you want to see a direct overlay go to 4:56 for the geometric comparison and 7:54 for the super late apex comparison.