----- > [!proposition] Proposition. ([[parallel curve to a convex closed plane curve]]) > Let $\alpha(s)$, $s \in [0, \ell]$ be a [[closed plane curve]], [[convex plane curve|convex]] and positively oriented. The curve $\beta(s)=\alpha(s) - rn(s)$, where $r>0$ is a constant and $n$ is the [[unit normal vector to a parameterized curve|normal vector]], is called a **parallel curve to $\alpha$.** We have that > 1. $\text{length}(\beta)=\text{length}(\alpha)+2\pi r$; > 2. $\text{area}(\beta)=\text{area}(\alpha)+ r\ell + \pi r^{2}$; > 3. $\kappa_{\beta}(s)=\kappa_{\alpha}(s) / (1+r)$. > > [!proof]- Proof. ([[parallel curve to a convex closed plane curve]]) > Recall that since $\alpha$ is a [[closed plane curve]], the [[tangent indicatrix of a unit-speed curve|total curvature]] $\int _{0}^{\ell} \kappa(s) \, ds$ is an integer multiple of $2\pi$. Also recall from the [[Frenet frame|Frenet formulas]] (specialized to 2D) that $n'(s)=-\kappa \alpha'(s)$. WLOG $\alpha(s)$ is [[parameterization by arc length|unit-speed]]. $\begin{align} > \text{length}(\beta)= & \int _{0}^{\ell} \|\beta'(s)\| \, ds \\ > = & \int _{0}^{\ell} \|\alpha'(s)- rn'(s)\| \, ds \\ > = & \int _{0}^{\ell} \|\alpha'(s) - r \kappa(s)\alpha'(s)\| \, ds \\ > = & \int _{0}^{\ell} \|(1+r \kappa(s)) \alpha'(s)\| \, ds \\ > = & \int _{0}^{\ell} (1+r \kappa(s))\| \alpha'(s)\| \ ds \\ > = & \int _{0}^{\ell} \|\alpha'(s)\| \, ds + r\int_{0}^{\ell} \kappa(s) \, ds \\ > = & \text{length}(\alpha) + 2\pi r I, I \in \mathbb{Z}. > \end{align}$ > If we assume simplicity, then the theorem of turning tangents says $I=1$ (recall that the curve is positively oriented), yielding the desired result. ----- #### ---- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag