2 Curvature for parametrically defined curves

An expression for the curvature is also available if the curve is described parametrically:

$x=g\left(t\right)y=h\left(t\right)t_0\le t\le t_1$

We remember the basic formulae connecting derivatives

$\frac{dy}{dx}=\frac{ẏ}{ẋ}\frac{d^{2}y}{d{x}^2}=\frac{ẋÿ-ẏẍ}{{ẋ}^3}$

where, as usual $ẋ\equiv \frac{dx}{dt},ẍ\equiv \frac{d^{2}x}{d{t}^2}\text{etc.}$

Then

$\kappa =\left|\frac{f^{″}\left(x\right)}{{\left\{1+{\left[f^{\prime }\left(x\right)\right]}^2\right\}}^{3∕2}}\right|=\left|\frac{ẋÿ-ẏẍ}{{ẋ}^3{\left[1+{\left(\frac{ẏ}{ẋ}\right)}^2\right]}^{3∕2}}\right|$

$=\left|\frac{ẋÿ-ẏẍ}{{\left[{ẋ}^2+{ẏ}^2\right]}^{3∕2}}\right|$

Task!

An ellipse is described parametrically by the equations

$x=2costy=sint0\le t\le 2\pi$

Obtain an expression for the curvature $\kappa$ and find where the curvature is a maximum or a minimum.

First find $ẋ,ẏ,ẍ,ÿ$ :

Answer

Now find $\kappa$ :

Answer

Find maximum and minimum values of $\kappa$ by inspection of the expression for $\kappa$ :

Answer