---- > [!definition] Definition. ([[binormal vector to a parameterized curve]]) > Let $I \subset \mathbb{R}$ be an [[open interval]] and $\alpha:I \to \mathbb{R}^{n}$ a curve [[parameterization by arc length|parameterized by arc length]], $\|\alpha'(s)\| \equiv 1$. As long as $s$ is not a [[osculating plane|singular point]] of order $0$ or $1$, the [[osculating plane]] is not degenerate and we call the unit vector $b(s):=t(s) \times n(s)$ > normal to it the **unit binormal vector of $\alpha$ at $s$.** Here, $t(s)=\alpha'(s)$ represents the [[velocity vector of a parameterized curve|(unit) tangent vector]] at $s$ and $n(s)$ represents the [[unit normal vector to a parameterized curve|unit normal vector]] at $s$. ^19168e > [!justification] > That $b(s)$ has unit-length and is normal to $t(s)$ and $n(s)$ follows from properties of the [[cross product]]. ^0a99d3 ---- #### ---- #### 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