Ogami, I think that Conceptual is asking for two parallel racing lines. I posted something about how this could be accomplished, but I've not produced anything you can test.
This the problem to solve:
- Draw two parallel racing lines that do not intersect (allowing for the width of the cars). Both racing lines can be traveled in the same time.
Or this more interesting variation:
- Draw two parallel racing lines that do not intersect. The outer curve can be traveled in less time than the inner curve.
You can use changes in the radius of the curve and in the sideslope to achieve that.
We talked once about designing something like Brooklands, whose curves have a parabollic sideslope (the outer racing line has larger banking).
Brooklands parabollic banking. Flynfrog pointed out that the ride height varies and thus it's not a solution
Of course I'm not proposing a banking as extreme as this. The banking has to be slight to avoid safety issues.
Perhaps, maybe, it's possible to design a curve with enough length and slight "differential banking". Let me show some numbers:
The problem here is that, with a slight increase in grip provided through banking, you need a long curve if you wish for the overtaker to catch the overtaken before the exit of the curve, where the overtaken can block.
For example, I take the Renault curve, which is the longest curve at Catalunya track (the one in red thick lines, lower left:
Length of the curve: 198 m
Radius: 111 m
Deflection (the angle between entrance and exit straights): 102 degrees
Maximum speed: 195 kmh (from FIA site)
The centripetal acceleration is 2.7 Gs, or, what is the same, the lateral friction coefficient is 2.7
The time it takes to travel through the curve is 3.7 seconds.
Now, if you increase the lateral grip to 2.85 Gs (by increasing banking 15%) you can get an increase in speed to 200 kmh.
This means that by increasing grip 15% you gain only 5 km/h. In the 3.7 seconds it takes to move through the curve you gain only 5 meters.
Let's say, for the purpose of this post, that an advantage of 15 meters would do the trick and allow you to overtake.
To gain 15 meters through the curve you need to go at 210 kmh while the overtaken goes at the same 195 kmh we assumed initially as the "regular speed" for the curve.
This means that, at the Renault curve in Catalunya, the overtaker has to produce 3.2 Gs laterally, while the overtaken uses only 2.7 Gs. This will give you a difference in time to travel the curve of a mere 0.3 seconds. The difference in grip is 3.2 - 2.7 = 0.5 Gs.
Conclussion: you need 0.5 Gs of advantage in grip if you want to overtake at Renault.
Now, if the Renault curve were longer, what would happened? I'll add the length of the transition that exists after the circular part of Renault (the curve with the thick orange lines in the previous image). It measures 193 meters more, for a total length of 393 m.
Now the overtaken uses 7.2 seconds to travel the curve. I'll spare you the numbers, but the overtaker now only needs a grip of 2.9 Gs!
As the overtaken car is using a grip of 2.7, you need a banking differential of only 20%. This is how it would look in real life a first drawing showing a section of the curve:
Sorry for the lengthy explanation and the simplifications, but I've been thinking on this for a while and I've had the help of the forum.
I know I performed some "magic tricks" here (who can find the worst? ), but if someone reads this I hope he'll get the idea: in principle, it seems possible to produce a racing line that is faster and allows you to catch and overtake a car along the curve. You can do it only by geometry.