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I am in the process of designing a board game involving car chases, and I am stumped by the following problem:

A car will have a maximum speed through a constant radius speed turn, giving a maximum safe cornering speed for a given turn radius. But, if the car follows the racing line (an apex turn?), the radius of the turn is greater, and the car can take the turn faster

A simplified situation would be sufficient to cater to my needs (no need for curves of changing radius, etc), see i.stack.imgur.com/odIyF.jpg for an illustration

enter image description here

The radius of the racing line R(race) must be dependent on 3 factors: Outer radius: R(outer) Inner radius: R(inner) And the length of the turn in degrees.

From this I should get R(race)

R(inner) and R(outer) have the same center, and the center of R(race) must be somewhere on the line that bisects the turn (midpoint of the curve)

I would love to have a formula for the 90 degree turn, but preferably I would like a general solution, where the turn can be of any angle of turn (up to 180 degrees or more). Looking at my sketches, at 180 degrees of turn, the radius of the racing line will equal R(outer), while the radius will approach infinity, as the angle of turn becomes smaller and smaller

I have tried searching online for answers, but the formula I have dug up have given results I haven't been able to reproduce when mocking up on graph paper

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Is the width of the road on the straight parts equal to the difference between the outer radius and the inner radius on the curve? –  Henry Jan 29 '13 at 7:21
Yes, it is - at least it is supposed to be :) - and thanks for embedding the picture, I don't have the privileges to do so :) –  LeZerp Jan 29 '13 at 7:33
If you ever add any more realism, my vote would be to include acceleration vs. deceleration. If a car can break quicker than accelerate (as most cars do) then the apex taken earlier gives more overall speed (more length for regaining speed from the turn gives better top speed in the straightaway.) –  adam W Jan 29 '13 at 15:56
Sounds very interesting, Adam - if it is simple to implement, I'd love to see something on the subject - but for now it would be a lot more detailed than the model for the game :) –  LeZerp Jan 29 '13 at 19:42
Yeah, it would be about as complex as the curves with changing radius...BTW You need the '@' symbol for me to know you addressed me, as in '@Adam'. I only just happened to look here again and saw your comment. –  adam W Feb 1 '13 at 19:27

2 Answers 2

up vote 5 down vote accepted

Let the outer, inner, and race radii be $r_o$, $r_i$, and $r_r$, and the angle of the turn be $\theta$.

$\hskip2in$enter image description here

The arc is tangent to the outside and inside of the road at the points marked $P_1$ and $P_2$, so $$\begin{align} x+r_o&=r_r,\\ x+r_i\cos\frac\theta2&=r_r\cos\frac\theta2. \end{align}$$ The solution is $$r_r=r_i+\frac{r_o-r_i}{1-\cos\frac\theta2}.$$

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In retrospect, the solution can be found without any algebra by observing that adding or subtracting the same value to all radii leaves the tangencies unchanged. So subtract $r_i$ from everything and the inner arc shrinks to a point, and then we can practically read off the solution. –  Rahul Jan 29 '13 at 11:14
Thanks for your answer, Rn However I have some problems putting it to use; I can't fault your diagram, that seems good to me. I got some strange results. I expected 'smooth' Rr, increasing as the curve of the track decreased, but the results show several 'bumps' at discrete angles, and the wrong trend. see: [1]: imgur.com/s1fxvpg What am I missing here? –  LeZerp Jan 29 '13 at 12:28
I am an idiot! Thanks again! :) –  LeZerp Jan 29 '13 at 13:09

The interesting part: the optimal trajectory has no fixed radius. The racer can brake, start with small radius, accellerate during the turn, end it with a larger radius and speed!

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