Unit Vectors of Curves

Equations

Tangent:

$$\hat{T}(t)=\frac{\text{d}r}{\text{d}s}=\frac{{\vec{r}}^{\prime }(t)}{\Vert {\vec{r}}^{\prime }(t)\Vert }$$

Normal:

$$\hat{N}(t)=\frac{{\hat{T}}^{\prime }(t)}{\Vert {\hat{T}}^{\prime }(t)\Vert }$$

Basis:

$$\hat{B}(t)=\frac{\vec{r}\prime (t)\times {\vec{r}}^{\prime \prime }(t)}{\Vert {\vec{r}}^{\prime }(t)\times {\vec{r}}^{\prime \prime (t)}\Vert }$$

Cyclic Permutation Equations

$$\begin{gathered} \hat{B}=\hat{T}\times \hat{N} \\ \hat{T}=\hat{N}\times \hat{B} \\ \hat{N}=\hat{B}\times \hat{T} \end{gathered}$$

For Planar Curves

$$\hat{T}(t)=\frac{⟨1,f^{\prime }(t)⟩}{\sqrt{1+f^{\prime }(t{)}^2}}$$ $$\hat{N}(t)=\frac{\pm ⟨-f^{\prime }(t),1⟩}{\sqrt{1+f^{\prime }(t{)}^2}}$$

The $\pm$ is so it can go in the direction of concavity

Extra Notes

• ${\vec{r}}^{\prime \prime }=a\hat{T}+b\hat{N}$

Frenet-Serret Equations

$$\begin{gathered} \frac{\text{d}\hat{T}}{\text{d}s}=\kappa (s)\hat{N}(s) \\ \frac{\text{d}\hat{B}}{\text{d}s}=-\tau (s)\hat{N}(s) \\ \frac{\text{d}\hat{N}}{\text{d}s}=\tau (s)\hat{B}(s)-\kappa (s)\hat{T}(s) \end{gathered}$$

Means that torsion and curvature define a cure uniquely:

$$\vec{r}(s)=\vec{r}+0+{\int }_0^s\hat{T}(s\prime )\text{d}s\prime$$