Given a compact convex set $S \subset R^2$ in the first quadrant.
Adding a constant vector of $\vec{C} \in R^2$ to $S$ won't change the convexity.
What if I have a polynomial function $f(d_x, x, y) : R \times R \times R \to R$,
And define a "stretch" transformation $F(x, y) := (x+d_x, y+d_y)$ where $(x, y) \in S$, $d_y =f(d_x, x, y)$, $d_x = C$, $C$ is just some positive constant.
Would the resulting set $T = \{F(x, y)|(x, y) \in S\}$ still convex and compact?
and the $f$ of my interest is in an implicit form.
$$ \frac{(x-d_x)^2}{(y-d_y)} + \frac{(C_1-(x-d_x))^2}{C_2-(y-d_y)} = C_3 $$
Both $C_1, C_3, C_3$ are positive real constants.
And what's more, would the boundary change, if the resulting set $T$ is still a compact convex set.