Definition:Curvature 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 $s \in I$.
The curvature of $\alpha$ at $s$ is defined as:
- $\map \kappa s := \norm {\map {\alpha' '} t}$
where:
- $\alpha ' '$ denotes the second derivative of $\alpha$
- $\norm \cdot$ denotes the Euclidean norm on $\R^3$
Also known as
Some sources use the spelling parametrized.
Also see
- Definition:Curvature: for curves in $\R^2$.
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