Torsion

• The amount of twisting in a 3D curve
• Measured by the rate of rotation of the osculating plane
• Notated as $\tau$

Equation

$$\frac{\text{d}\hat{B}}{\text{d}s}=-\tau (s)\hat{N}$$ \vec{r}(t) \text{ is any parameterization such } \kappa \neq 0 \iff \vec{r}^\prime(t) \times \vec{r}(t)^{\prime\prime} \neq 0$$

\tau(t) = \frac{\vec{r}^\prime \cdot (\vec{r}^{\prime\prime} \times \vec{r}^{\prime\prime\prime})}{|\vec{r}^\prime \times \vec{r}^{\prime\prime} | ^2} = \frac{\vec{r}^{\prime\prime\prime} \cdot (\vec{r}^\prime \times \vec{r}^{\prime\prime})}{|\vec{r}^\prime \times \vec{r}^{\prime\prime} | ^2}

## Special Curves - Circle - $\tau(s) = 0, \kappa(s) = \kappa_0 \neq 0$ - 0 torsion, constant curvature - Circular Helix - $\tau(s) = \tau_0 \neq 0, \kappa(s) = \kappa_0 \neq 0$ - Constant torsion, constant curvature - Planar Curves - $\tau = 0$