Question

Find the curvature on the graph of the elliptical helix defined by$r\left(t\right)=\left(a\cos t,b\sin t,ct\right),$where a, b, and c are positive constants.

Short answer

The curvature on the graph of the elliptical helix defined by r(t)is$\kappa =\frac{{\displaystyle \sqrt{b^2{c}^2{\sin }^{2}t+a^2{c}^2{\cos }^{2}t+a^2{b}^2}}}{{\left(a^2{\sin }^{2}t+b^2{\cos }^{2}t+c^2\right)}^{{\displaystyle \frac{3}{2}}}}.$

Step-by-step solution

Step 1. Given Information.

The given elliptical helix is$r\left(t\right)=\left(a\cos t,b\sin t,ct\right).$

Step 2. Find the curvature.

To find the curvature on the graph of the elliptical helix, we will use the formula for the Curvature of a Space Curve $\kappa =\frac{||r\text{'}\left(t\right)\times r\text{'}\text{'}\left(t\right)||}{{||r\text{'}\left(t\right)||}^3}.$

So,

Step 3. Calculate.

Put the values of (a) and (b) in the formula of Curvature of a space curve,

$$\begin{gathered} \kappa =\frac{{\displaystyle \sqrt{b^2{c}^2{\sin }^{2}t+a^2{c}^2{\cos }^{2}t+a^2{b}^2}}}{{\left(\sqrt{a^2{\sin }^{2}t+b^2{\cos }^{2}t+c^2}\right)}^3} \\ \kappa =\frac{{\displaystyle \sqrt{b^2{c}^2{\sin }^{2}t+a^2{c}^2{\cos }^{2}t+a^2{b}^2}}}{{\left(a^2{\sin }^{2}t+b^2{\cos }^{2}t+c^2\right)}^{{\displaystyle \frac{3}{2}}}} \end{gathered}$$