Figure 2.4.9(a)   Graph of the curvature

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$\mathrm{BasisFormat}\left(\mathrm{false}\right)\:$

$\mathbf{R}\=⟨3 p-p^3\,3 p^2\,3 p\+p^3⟩$$\stackrel{\text{assign}}{\to }$

Obtain the curvature

$\mathbf{R}$

$\stackrel{\text{curvature}}{\to }$

$\frac{1}{6}\frac{\sqrt{{\left(-\frac{1}{6}\frac{\sqrt{2}\left(-3p^2\+3\right)\mathrm{csgn}\left(1\,p^2\+1\right)}{{\mathrm{csgn}\left(p^2\+1\right)}^2\left(p^2\+1\right)}-\frac{1}{3}\frac{\sqrt{2}\left(-3p^2\+3\right)p}{\mathrm{csgn}\left(p^2\+1\right){\left(p^2\+1\right)}^2}-\frac{\sqrt{2}p}{\mathrm{csgn}\left(p^2\+1\right)\left(p^2\+1\right)}\right)}^2\+{\left(-\frac{\sqrt{2}p\mathrm{csgn}\left(1\,p^2\+1\right)}{{\mathrm{csgn}\left(p^2\+1\right)}^2\left(p^2\+1\right)}-\frac{2\sqrt{2}{p}^2}{\mathrm{csgn}\left(p^2\+1\right){\left(p^2\+1\right)}^2}\+\frac{\sqrt{2}}{\mathrm{csgn}\left(p^2\+1\right)\left(p^2\+1\right)}\right)}^2\+{\left(-\frac{1}{6}\frac{\sqrt{2}\left(3p^2\+3\right)\mathrm{csgn}\left(1\,p^2\+1\right)}{{\mathrm{csgn}\left(p^2\+1\right)}^2\left(p^2\+1\right)}-\frac{1}{3}\frac{\sqrt{2}\left(3p^2\+3\right)p}{\mathrm{csgn}\left(p^2\+1\right){\left(p^2\+1\right)}^2}\+\frac{\sqrt{2}p}{\mathrm{csgn}\left(p^2\+1\right)\left(p^2\+1\right)}\right)}^2}\sqrt{2}}{\mathrm{csgn}\left(p^2\+1\right)\left(p^2\+1\right)}$

$\stackrel{\text{assuming positive}}{\to }$

$\frac{1}{3{\left(p^2\+1\right)}^2}$

Obtain the curvature as $\mathrm{\κ}\=\parallel \mathbf{R}\prime \times \mathbf{R}″\parallel \/{\mathrm{\ρ}}^3$ 

$\mathbf{R}\prime \=\frac{\ⅆ}{\ⅆ p} \mathbf{R}$$\stackrel{\text{assign}}{\to }$

$\mathbf{R}″\=\frac{{\ⅆ}^2}{{\ⅆ}^{}{p}^2} \mathbf{R}$$\stackrel{\text{assign}}{\to }$

$\mathrm{\ρ}\=\parallel \mathbf{R}\prime \parallel$$\stackrel{\text{assign}}{\to }$

$\frac{∥\mathbf{R}\prime \times \mathbf{R}″∥}{{\mathrm{\ρ}}^3}$ = $\frac{1}{3{\left(p^2\+1\right)}^2}$$$

Obtain the curvature as $\mathrm{\κ}\=∥\mathbf{T}\prime \left(s\right)∥$ then graph $\mathrm{\κ}$

$\mathbf{T}\=\mathbf{R}\prime \/\mathrm{\ρ}$$\stackrel{\text{assign}}{\to }$

$∥\left(\frac{\ⅆ}{\ⅆ p} \mathbf{T}\right)\/\mathrm{\ρ}∥$

$\frac{1}{3}\frac{\mathrm{csgn}\left(\frac{1}{{\left(p^2\+1\right)}^2}\right)}{{\left(p^2\+1\right)}^2}$

$\stackrel{\text{assuming positive}}{\to }$

$\frac{1}{3{\left(p^2\+1\right)}^2}$

$\to$

Display computed quantities

$\mathbf{R}\prime \,\mathbf{R}″\,\mathrm{\ρ}\,\mathbf{T}$

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\:$

$\mathbf{R}≔⟨3 p-p^3\,3 p^2\,3 p\+p^3⟩\:$

$\mathrm{simplify}\left(\mathrm{Curvature}\left(\mathbf{R}\,p\right)\right) assuming \mathrm{positive}$

$\frac{1}{3{\left(p^2\+1\right)}^2}$

$\mathbf{T}≔\mathrm{TangentVector}\left(\mathbf{R}\,p\,\mathrm{normalized}\right)\:$

$\mathrm{\ρ}≔\mathrm{Norm}\left(\mathrm{diff}\left(\mathbf{R}\,p\right)\right)\:$

Obtain the curvature as $\mathrm{\κ}\=\parallel \mathbf{R}\prime \times \mathbf{R}″\parallel \/{\mathrm{\ρ}}^3$ 

$\frac{\mathrm{Norm}\left(\mathrm{CrossProduct}\left(\mathrm{diff}\left(\mathbf{R}\,p\right)\,\mathrm{diff}\left(\mathbf{R}\,p\,p\right)\right)\right)}{{\mathrm{\ρ}}^3}$

$\frac{1}{3{\left(p^2\+1\right)}^2}$

Obtain the curvature as $\mathrm{\κ}\=∥\mathbf{T}\prime \left(s\right)∥$ 

Apply the diff, Norm, and simplify commands.

$\mathrm{\κ}≔\mathrm{simplify}\left(\mathrm{Norm}\left(\frac{\mathrm{diff}\left(\mathbf{T}\,p\right)}{\mathrm{\ρ}}\right)\right) assuming \mathrm{positive}$

$\frac{1}{3{\left(p^2\+1\right)}^2}$

Graph the curvature

$\mathrm{plot}\left(\mathrm{\κ}\,p\=-2..2\,\mathrm{size}\=\left[200\,200\right]\right)$