How to calculate start radius, end radius and angle for clothoid segments? I have a following construction:
Clothoid -- Circle Arc -- Clothoid -- Clothoid
which should form a smooth G2-continuous curve.
Clothoids are parameterized with clothoid parameter $A$ and length $L$. For the circle, its radius and arc length are given.
I should get the clothoids parameterized through start radius, end radius and angle. Can this be done?
I tried with usual formulas that can be found on wikipedia and on this site. For the first clothoid i took a big start radius i.e. $R=\infty$ and went on from there, taking the last end radius as the next starting one and so on. This didn't work as I thought.
 A: You didn't say how exactly it didn't work, but it should have worked.
You are given $Clo_1(A_1,L_1)$, $Arc(R,L)$, $Clo_2(A_2,L_2)$, $Clo_3(A_3,L_3)$.
You want  $Clo_1(R_{11},R_{12},\Delta\Theta_1)$, $Clo_2(R_{21},R_{22},\Delta\Theta_2)$, $Clo_3(R_{31},R_{32},\Delta\Theta_3)$ where $R_{12} = R = R_{21}$, $Clo_1$ goes from $s_{11}$ to $s_{12}$, $L_1 = s_{12} - s_{11}$, $R_{Clo_1}(s_{11}) = R_{11}$, $R_{Clo_1}(s_{12}) = R_{12}$, etc.
Then we must have $\{ R_{11}s_{11} = (A_1)^2 = R_{12}s_{12} = Rs_{12}, L_1 = s_{12} - s_{11} \}$ (and similarly for $\__2$ and $\__3$).
The above system of two equations is trivially solvable, for each of the clothoids, giving us their edge radii $R_{ij}$, the $s_{ij}$ values, and hence their edge $\Theta$s, and hence their $\Delta\Theta$s:
$$ \Theta = s^2/2A^2 $$
$$ \Theta_1 = s_1^2/(2R_1s_1) = s_1/(2R_1)$$
$$\Theta_2 = s_2/(2R_2)$$
$$ \Delta\Theta_1 = \Theta_2 - \Theta_1$$
Your error seems to be that
you started from setting $R_{11}=\infty$. Instead, the arc's radius determines the rest.
