---- > [!definition] Definition. ([[tangent indicatrix of a unit-speed curve]]) > For $\alpha:(a,b) \to \mathbb{R}^{n}$ a [[parameterization by arc length|unit-speed]] curve in $\mathbb{R}^{n}$ we define the curve $T:(a,b) \to \mathbb{R}^{n}$ by $T(s)=\alpha'(s)$. Its trace lies on the unit sphere in $\mathbb{R}^{n}$. Note that $T(s)$ need not be regular; $T'(s)=0$ wherever the [[curvature of parameterized curve|curvature]] of $\alpha$ is $0$. > ![[CleanShot 2024-02-07 at [email protected]]] ^0428fc > [!proposition] (Polar form of the tanget indicatrix and the **Rotation Index**) > Let $\theta(s)$, $0<\theta(s)<2\pi$, be the angle that $T(s)$ (as a point on $\mathbb{S}^{1}$) makes with the $x$-axis. Picture shows that $x'(s)=\cos \theta(s)$, $y'(s)=\sin \theta(s)$, and $\theta(s)=\tan ^{-1} \frac{y'(s)}{x'(s)}.$ We can then write $\begin{align} \frac{dT}{ds} = & \frac{d}{ds} (x'(s), y'(s)) \\ = & \frac{d}{ds} (\cos \theta(s), \sin \theta(s)) \\ = & \theta'(s) (-\sin \theta(s), \cos \theta(s)). \end{align}$ We know from [[curvature of parameterized curve]] and [[unit normal vector to a parameterized curve]] that $\frac{dT}{ds}=\alpha''(s)=\kappa(s) n(s)$. Since $\theta'(s)$ is a scalar and $(-\sin(\theta(s)), \cos \theta(s))$ is a vector, it must be the case that $\kappa(s)=\theta'(s)$ and $n(s)=(-\sin \theta(s), \cos \theta(s))$, i.e., $\frac{dT}{ds}=\theta'(s)n(s)=\kappa(s)n(s)$. So we define the globally differentiable $\theta(s)=\int _{0}^{s} \kappa(s)\, ds $ which measures the 'total rotation of the tangent vector' in $\mathbb{S}^{1}$. Now by [[the fundamental theorem of calculus]] $\int _{a}^{b} \kappa(s) \, ds = \theta(b)-\theta(a)=2\pi I \text{ for some } I \in \mathbb{Z}.$ (recall that the curve is closed). The integer $I$ is called the **rotation index** of $\alpha$. ^a0a3cc ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```