example of curvature (space curve)

Example space curves and calculating their curvatures using the formula

$$\kappa (t)=\frac{\parallel {𝐫}^{\prime }(t)\times {𝐫}^{\prime \prime }(t)\parallel }{{\parallel {𝐫}^{\prime }(t)\parallel }^3}$$

1. $𝐫(t)=3t\widehat{i}+t^2\widehat{j}-4t^2\widehat{k}$

the first derivative

${𝐫}^{\prime }(t)=3\widehat{i}+2t\widehat{j}-8t\widehat{k}$

vector magnitude of the derivative

$\parallel {𝐫}^{\prime }(t)\parallel =\sqrt{3^2+{(2t)}^2+{(-8t)}^2}$

$\parallel {𝐫}^{\prime }(t)\parallel =\sqrt{9+4t^2+16t^2}=\sqrt{9+20t^2}$

the second derivative

${𝐫}^{\prime \prime }(t)=2\widehat{j}-8\widehat{k}$

the cross product

${𝐫}^{\prime }(t)\times {𝐫}^{\prime \prime }(t)=24\widehat{j}+6\widehat{k}$

$\parallel {𝐫}^{\prime }(t)\times {𝐫}^{\prime \prime }(t)\parallel =\sqrt{576+36}=\sqrt{612}=2\sqrt{153}$

${\parallel {𝐫}^{\prime }(t)\parallel }^3={(9+20t^2)}^{3/2}$

$\kappa (t)=\frac{2\sqrt{153}}{{(9+20t^2)}^{3/2}}$

2. Calculate the curvature of the right circular helix as given in the plot below and defined as

$𝐫(t)=\cos t\widehat{i}+\sin t\widehat{j}+t\widehat{k}$

${𝐫}^{\prime }(t)=-\sin t\widehat{i}+\cos t\widehat{j}+\widehat{k}$

$\parallel {𝐫}^{\prime }(t)\parallel =\sqrt{{\sin }^{2}t+{\cos }^{2}t+1^2}=\sqrt{2}$

${𝐫}^{\prime \prime }(t)=-\cos t\widehat{i}-\sin t\widehat{j}$

${𝐫}^{\prime }(t)\times {𝐫}^{\prime \prime }(t)=\sin t\widehat{i}-\cos t\widehat{j}+\widehat{k}$

$\parallel {𝐫}^{\prime }(t)\times {𝐫}^{\prime \prime }(t)\parallel =\sqrt{{\sin }^{2}t+{\cos }^{2}t+1^2}=\sqrt{2}$

${\parallel {𝐫}^{\prime }(t)\parallel }^3=2^{3/2}$

$\kappa (t)=\frac{\sqrt{2}}{2^{3/2}}=\frac{1}{2}$