Curvature varies linearly with arc length $s$ along the clothoid:

$$k(s) = \frac{s}{A^2} = \frac{s}{LR}, \qquad A^2 = LR$$

Natural parameterization (entry clothoid):

$$x(s)=\int_0^s \cos\!\left(\frac{\sigma^2}{2A^2}\right)d\sigma, \quad y(s)=\int_0^s \sin\!\left(\frac{\sigma^2}{2A^2}\right)d\sigma$$

Tangent angle: $\theta(s)=s^2/(2A^2)$. At the end $s=L$: $\theta(L)=L/(2R)$.

Let $s_1$ be the start of the first clothoid, $s_2=s_1+L$, $s_3=s_2+L_{\mathrm{arc}}$, $s_4=s_3+L$, and $s_{\mathrm{tot}}$ the total length. Then:

$$k(s)=\begin{cases} 0 & 0 \le s < s_1 \quad\text{(straight 1)}\\[6pt] \dfrac{s-s_1}{LR} & s_1 \le s < s_2 \quad\text{(clothoid 1)}\\[8pt] \dfrac{1}{R} & s_2 \le s < s_3 \quad\text{(circular arc)}\\[8pt] \dfrac{L-(s-s_3)}{LR} & s_3 \le s < s_4 \quad\text{(clothoid 2)}\\[8pt] 0 & s_4 \le s \le s_{\mathrm{tot}} \quad\text{(straight 2)} \end{cases}$$ $$a_y(s) = V^2\,k(s)$$ $$j_y(s) = V^3\,\frac{dk}{ds}$$

With clothoids: $dk/ds = \pm 1/(LR)$ on the ramps, and $0$ on straights and on the arc.

$$R_{\min} = \frac{V^2}{a_{y,\max}} = \frac{V^2}{0.15\,g}$$

Design lateral jerk on a clothoid ramp: $|j_y| = V^3/(LR)$. Requiring $|j_y|\le j_{y,\max}$ gives a minimum clothoid length

$$L_{\min} = \frac{V^3}{j_{y,\max}\,R}$$

(reference value $j_{y,\max}=0.7\ \mathrm{m/s^3}$ is shown as horizontal dashed lines on the jerk plot; sliders are not clamped to this value.)