Definition:Normal Vector of Curve Parameterized by Arc Length/3-Dimensional Real Vector Space
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Definition
Let $\alpha : I \to \R^3$ be a (smooth) curve parameterized by arc length.
Let $s \in I$ be such that the curvature $\map \kappa s \ne 0$.
The normal vector $\map n s$ of $\alpha$ at $s$ is defined as:
- $\map {\alpha} s = \map \kappa s \map n s$
where:
- $\alpha$ denotes the second derivative of $\alpha$
That is:
- $\map n s := \dfrac {\map {\alpha} s} {\norm {\map {\alpha} s} }$
Sources
- 2016: Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces (2nd ed.): $1$-$5$: The Local Theory of Curves Parametrized by Arc Length