---- > [!definition] Definition. ([[normal and geodesic curvature of a curve in a surface]]) > Let $\alpha(t)$ be a [[parameterization by arc length|unit-speed]] [[parameterized curve]] on a [[differentiable Euclidean submanifold|regular surface]] $S$. The [[orthogonal complement|orthogonal complement]] of $T_{\alpha'(t)}S$ has as [[orthonormal basis]] the set $\{ N(t), N(t) \times \alpha'(t) \}$. Since $\alpha''(t)$ is [[orthogonal]] to $\alpha'(t)$, it lives $T_{\alpha'(t)}^{\perp}S$; therefore it is the [[linear combination]] $\alpha''(t)=\kappa_{n}(t)N(t) + \kappa_{g}(t)N(t) \times \alpha'(t).$ We call $\kappa_{n}(t)$ the **normal curvature** of the curve $\alpha(t)$ and $\kappa_{g}(t)$ the **geodesic curvature** of $\alpha(t)$ on the [[surface]] $S$. \ Note that we can 'extract' $\kappa_{n}(t)$ and $\kappa_{g}(t)$ from the above via $\begin{align} \kappa_{n}(t) = & \alpha''(t) \cdot N(t); \\ \kappa_{g}(t)= & \alpha''(t) \cdot (N(t) \times \alpha'(t)). \end{align}$ And as one might hope, $\kappa_{n}(t)$ equals the [[normal curvature]] of $S$ in the direction $\alpha'(t)$: $\kappa_{n}(t)=\text{II}_{\alpha(t)}(\alpha'(t), \alpha'(t)),$ where $\text{II}$ denotes the [[second fundamental form]]. \ A general fact is that for any curve, $\kappa^{2}=\kappa_{n}^{2}+\kappa_{g}^{2}$. ^903f5b > [!justification] > We should verify $\kappa_{n}(t)=\text{II}_{\alpha(t)}(\alpha'(t), \alpha'(t))$. $N(t)$ denotes the [[unit normal vector to a parameterized curve|unit normal vector to the parameterized curve]] $\alpha$. By assumption, $N(t) \cdot \alpha'(t) \equiv 0$. [[derivative|Differentiate]] both sides: $N'(t)$ > [!basicproperties] > - [[curves with same tangent line have same normal curvature]] ---- #### ---- #### 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 > ```