- #1

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I have a curve defined by following parametric equation

\begin{equation}

\gamma(\theta) = 1 + 0.5 \times \cos (N \theta) (\cos(\theta),\sin(\theta)), 0 \leq \theta \leq 2 \pi \

\end{equation}

where N is an integer. x and y coordinate of any point on the curve are simply

\begin{align}

x &= \gamma (1) \nonumber \\

y &= \gamma (2)

\end{align}

Question is how do I compute a unit normal (n) to above curve in eq. (1) and its x (n_x) and y (n_y) components?

One way to find x and y components of normal n would be to find dx / dn and dy / dn respectively. dx / dn and dy / dn can in turn be found using chain rule as follows -

\begin{align}

n_x &= \frac{dx}{dn} \nonumber \\

& = \frac{dx}{dr} \frac{dr}{dn} \nonumber \\

& = \frac{dx}{d\theta}\frac{d\theta}{dr}\frac{dr}{dn}

\end{align}

and similarly for n_y

\begin{align}

n_y &= \frac{dy}{dn} \nonumber \\

& = \frac{dy}{dr} \frac{dr}{dn} \nonumber \\

& = \frac{dy}{d\theta}\frac{d\theta}{dr}\frac{dr}{dn}

\end{align}

While first two terms on rhs of equations (3) and (4) can be found easily, how to find the dr / dn term? Or if any one has an easy way to compute the normal components of the unit normal using the parametric equation of the curve, it'd be a great help.

Many thanks for help.