----- > [!proposition] Proposition. ([[straight iff zero curvature]]) > A [[smooth]] [[regular curve]] is straight iff it has zero [[curvature of parameterized curve|curvature]]. ^9339c2 > [!proof]- Proof. ([[straight iff zero curvature]]) > That straight lines have zero curvature is shown in [[curvature of parameterized curve|lines have zero curvature]]. \ Conversely, we want to show that given a [[smooth]] [[regular curve]] ([[smooth regular parameterized curves admit (re)parameterizations by arc length|WLOG]] [[parameterization by arc length|parameterized by arc length]]) $\alpha(s), s \in (a,b),$ with the property $\kappa(s) \equiv 0$, that $\alpha$ is a segment of a straight [[line]]. Since $\|\alpha''(s)\|_{2} \equiv 0$, we know by the positive definiteness of [[norm|norms]] that $\alpha''(s) \equiv 0$. Hence $\alpha'(s) \equiv \b c$ for some constant $\b c$, and thus $\alpha(s)= \b c t + \b b$ for some constant $\b b$. This is the result. ^60a19e ----- #### ---- #### 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