Given \(F\left ( x,y,c\right ) \) we need to find the orthogonal projections. The first step is to find the slope of the orthogonal projection, which is given by (it is orthogonal to the given curve slope)
\begin{equation} \frac{dy}{dx}=\frac{F_{y}}{F_{x}}\tag{1} \end{equation} Next step, check if \(c\) still shows up in the above (i.e. did not cancel out), then solve for \(c\) from \(F\left ( x,y,c\right ) =0\) and replace it in (1). Now (1) will not have \(c\) in it any more. Next, solve (1) for \(y\). This gives the curve for the orthogonal projection. This solution will have new \(c\) in it (since we need to integrate to find \(y\)). See HW 2 for example problem.