Chapter 2: Space Curves

Section 2.7: Frenet-Serret Formalism

Example 2.7.11

$$ Obtain the expansion in the third conclusion of Theorem 2.7.2. $$ Solution $$ Table 2.7.11(a) contains the calculations by which the coefficients in the expansion   $\mathbf{R}\left(s\right)\=\mathbf{R}\left(0\right)\+s\mathbf{T}\left(0\right)\+\frac{s^2}{2\!}\mathrm{\kappa }\left(0\right)\mathbf{N}\left(0\right)\+\frac{s^3}{3\!}\left(-{\mathrm{\kappa }}^2\left(0\right)\mathbf{T}\left(0\right)\+\mathrm{\kappa }\prime \left(0\right)\mathbf{N}\left(0\right)\+\mathrm{\kappa }\left(0\right)\mathrm{\tau }\left(0\right)\mathbf{B}\left(0\right)\right)\+\cdots$ $$ are obtained.   $$ $\mathbf{R}\prime \left(s\right)\=\mathbf{T}\left(s\right)$ $\mathbf{R}\prime \left(0\right)\=\mathbf{T}\left(0\right)$  $$ $\mathbf{R}″\left(s\right)\=\mathbf{T}\prime \left(s\right)\=\mathrm{\κ}\left(s\right)\mathbf{N}\left(s\right)$  $\mathbf{R}″\left(0\right)\=\mathrm{\κ}\left(0\right)\mathbf{N}\left(0\right)$  $\mathbf{R}‴\left(s\right)$ $\=\left(\mathbf{R}″\left(s\right)\right)\prime$ $$ $\=\left(\mathrm{\κ} \mathbf{N}\right)\prime$ $$ $\=\mathrm{\κ}\prime \mathbf{N}\+\mathrm{\κ} \mathbf{N}\prime$ $$ $\=\mathrm{\κ}\prime \mathbf{N}\+\mathrm{\κ} \left(-\mathrm{\κ} \mathbf{T}\+\mathrm{\τ} \mathbf{B}\right)$ $\mathbf{R}‴\left(0\right)$ $\=\mathrm{\κ}\prime \left(0\right)\mathbf{N}\left(0\right)-{\mathrm{\κ}}^2\left(0\right)\mathbf{T}\left(0\right)\+\mathrm{\τ}\left(0\right)\mathrm{\κ}\left(0\right)\mathbf{B}\left(0\right)$ Table 2.7.11(a)   Coefficients for the expansion $$

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