# parallel clothoid parameters

I have a clothoid with linear curvature as follows:

$$c(l) = c_0 + c_1 l$$ with starting position $(x_0,y_0)$ and tangent angle $\phi_0$ where the clothoid is described as follows:

$$x(l) = x_0 + \int_0^l \cos(\phi(r))~dr\\ y(l) = y_0 + \int_0^l \sin(\phi(r))~dr\\ \phi(l) = \phi_0 + c_0l + 0.5 c_1 l^2$$ where $l$ is the length or arc along the clothoid.

I want the parameters of the parallel clothoid on a lateral offset $\delta$ up to a length $L$ on the original clothoid. That is, $\hat{c_0}$, $\hat{c_1}$, $\hat{L}$. Is there an exact solution for that? or some good approximation techniques?

More details:

I tried Taylor expansion (for small length $L$ and tangent angle changes) which yields the following: $$x(l) \approx x_0 + l\\ y(l) \approx y_0 + \phi_0 l + 0.5 c_0 l^2 + \frac{1}{6} c_1 l^3\\ \phi(l) = \phi_0 + c_0l + 0.5 c_1 l^2$$ and the two curves were almost identical.

Then the parallel curve should satisfy the following conditions: $$\hat{c_0} = \frac{1}{\frac{1}{c_0}+\delta}\\ \hat{c_0}+\hat{c_1}\hat{L} = c_0 + \frac{1}{\frac{1}{c_1}+\delta} L\\ \hat{L} = L - \delta \phi(L)$$

However, solving these three equations doesn't give a good approximation for the parallel curve.

Are these three equation sufficient? How would I improve the resulting parameters?

• Two things 1) I "googled" with the "words clothoid intrinsic parameters" and fell on the very recent book (I saw the googlebook version) "Mathematical Progress in Expressive Image Synthesis III": Yoshinori Dobashi,Hiroyuki OchiaiDo Editors, where they speak of 3D applications of clothoids etc... 2) For the layman, you should recall at least that $\ell$ is the curvilinear abscissa on the curve. – Jean Marie Aug 8 '16 at 19:10
• May I ask you why you are interested by such a theme ? You may know that in general, parallel curves of non-polynomial curves have no explicit equation ? If your final objective is image analysis or image synthesis, you can proceed in a different way. – Jean Marie Aug 8 '16 at 22:06