Car racing: How to calculate the radius of the racing line through a turn of varying length
Solution 1:
Let the outer, inner, and race radii be $r_o$, $r_i$, and $r_r$, and the angle of the turn be $\theta$.
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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 $$r_r=r_i+\frac{r_o-r_i}{1-\cos\frac\theta2}.$$