Chapter 2: Space Curves

Section 2.7: Frenet-Serret Formalism

Example 2.7.1

$$ If $s$ is arc length, establish the Frenet equation $\mathbf{T}\prime \left(s\right)\=\mathrm{\κ} \mathbf{N}$. $$ Solution $$ By definition, the curvature is $\mathrm{\κ}\=∥\mathbf{T}\prime \left(s\right)∥$. Since T is a unit vector, $\mathbf{T}·\mathbf{T}\=1$, and $\left(\mathbf{T}·\mathbf{T}\right)\prime \=2 \mathbf{T}·\mathbf{T}\prime \=0$, so $\mathbf{T}\prime$ is orthogonal to T. The direction of $\mathbf{T}\prime$ defines the direction of the principal normal, the unit vector N. Hence, $\mathbf{T}\prime \/\mathrm{\κ}\=\mathbf{N}$, or $\mathbf{T}\prime \=\mathrm{\κ} \mathbf{N}$.

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