Euler Spiral Animation [gnuplot]

YouTube

Formulation

Parametric equations ^{[1] [2]}

\begin{align*} x(t) &= \int_0^t \cos\theta^2\,\diff\theta\\ y(t) &= \int_0^t \sin\theta^2\,\diff\theta \end{align*} The limits of $x(t)$, $y(t)$ as $t\rightarrow\pm\infty$ are known: \begin{align*} \lim_{t\rightarrow\pm\infty}x(t) &= \int_0^{\pm\infty} \cos\theta^2\,\diff\theta = \pm\frac{1}{2}\sqrt{\frac{\pi}{2}}\\ \lim_{t\rightarrow\pm\infty}y(t) &= \int_0^{\pm\infty} \sin\theta^2\,\diff\theta = \pm\frac{1}{2}\sqrt{\frac{\pi}{2}} \end{align*}

Curvature and osculating circle

Let \begin{align*} \bm{r} = \left[x(t), y(t)\right]^T \end{align*} Then \begin{align*} \frac{\diff\bm{r}}{\diff t} &= \left[\cos t^2, \sin t^2\right]^T \\ \frac{\diff s}{\diff t} &= \left|\frac{\diff\bm{r}}{\diff t}\right| = 1\ \ (s:\text{Arc length paramenter}) \end{align*}

Unit tangential vector $\bm{t}$

\begin{align*} \bm{t} = \frac{\diff \bm{r}}{\diff s}=\frac{\diff\bm{r}}{\diff t}\frac{\diff t}{\diff s}=\left[\cos t^2, \sin t^2\right]^T \end{align*}

Curvature $\kappa$

\begin{align*} \frac{\diff\bm{t}}{\diff s} &= \frac{\diff\bm{t}}{\diff t}\frac{\diff t}{\diff s} = \left[-2t\sin t^2, 2t\cos t^2\right]^T\\ \kappa &= \left|\frac{\diff\bm{t}}{\diff s}\right| = \sqrt{(-2t\sin t^2)^2+(2t\cos t^2)^2} = \sqrt{(2t)^2}=2|t| \end{align*}

Unit principal normal vector $\bm{n}$

\begin{align*} \bm{n} = \frac{1}{\kappa}\frac{\diff\bm{t}}{\diff s} =\frac{1}{2|t|}\left[-2t\sin t^2, 2t\cos t^2\right]^T =\left[-\frac{t}{|t|}\sin t^2, \frac{t}{|t|}\cos t ^2\right]^T \end{align*}

Radius of curvature $\rho$

\begin{align*} \rho = \frac{1}{\kappa} = \frac{1}{2|t|} \end{align*}

Center of curvature $\bm{c}$

\begin{align*} \bm{c} &= \bm{r}+\rho\bm{n}\\ &= \left[x(t), y(t)\right]^T+\frac{1}{2|t|}\left[-\frac{t}{|t|}\sin t^2, \frac{t}{|t|}\cos t^2\right]^T\\ &= \left[x(t)-\frac{1}{2t}\sin t^2, y(t)+\frac{1}{2t}\cos t^2\right]^T\ \ (\because |t|^2=t^2) \end{align*}

Simulation [gnuplot]

Source code (PLT file)

Output (PNG file → GIF file)

Only curve	Curve with limit points
Curve with osculating circles	Curve with the points and the circles

Graph of Fresnel integrals $C(t)$ and $S(t)$

Source code (PLT file)

Output (PNG file)

Note : Numerical Integration

Composite trapezoidal rule ^[3]

\begin{align*} \int_a^b f(x)\,\diff x \approx \frac{h}{2}\left(f(x_0)+2\sum_{i=1}^{N-1}f(x_i)+f(x_N)\right) \end{align*} where \begin{align*} x_i &= a+ih\ (i=0, 1, \cdots,\ N-1, N)\\ h &= \frac{b-a}{N} \end{align*} This time, \begin{equation*} a=0, b=t\ \rightarrow\ h=\frac{t}{N}, x_i=ih \end{equation*}

Composite Simpson's rule ^[4]

\begin{align*} \int_a^b f(x)\,\diff x \approx \frac{h}{3}\sum_{i=1}^{N/2}\left[f(x_{2i-2})+4f(x_{2i-1})+f(x_{2i})\right] \end{align*} where \begin{align*} x_i &= a+ih\ (i=0,\ 1,\ \cdots,\ N-1,\ N)\\ h &= \frac{b-a}{N} \end{align*} This time, \begin{equation*} a=0, b=t\ \rightarrow\ h=\frac{t}{N}, x_i=ih \end{equation*}

More details

GitHub Repository

https://github.com/hiroloquy/euler-spiral.git

References

1. ^ Euler spiral - Wikipedia
2. ^ Fresnel integral - Wikipedia
3. ^ Trapezoidal rule - Wikipedia
4. ^ Simpson's rule - Wikipedia