Parabola

1. Equation: \[ y-y_0=\frac{1}{2}a(x-x_0)^2 \]
2. Parametrization ($x_0=y_0=0$): \[ \underline{r}(t) = \begin{pmatrix}x(t)\\y(t)\end{pmatrix} = \begin{pmatrix} t \\ \frac{1}{2}a t^2 \end{pmatrix} \]
3. Velocity: \[ v(t) = \sqrt{1+a^2t^2} \geq 1 \]
4. Frenet System: \[ \underline{t} = \frac{1}{\sqrt{1+a^2 t^2}} \begin{pmatrix}1\\2at\end{pmatrix}, \quad \underline{n} = \frac{1}{\sqrt{1+a^2 t^2}} \begin{pmatrix}-2at\\1\end{pmatrix} \]
5. Determinant: \[ D(t)= \begin{vmatrix} x'(t) & x''(t) \\ y'(t) & y''(r) \end{vmatrix} = \widehat{ \underline{r}' } \cdot \underline{r}'' \]
6. Properties: