Let $P=(x_1,y_1,0)$ and $Q=(x_2,y_2,0)$ be two points in the $xy$ plane.
Find an equation for a curve (could be a half-ellipse) passing through $P$, $Q$, and a third point $R=(\frac{x_1+x_2}{2},\frac{y_1+y2}{2},h)$ for some fixed height $h$. It is important that the curve does not extend beyond the $xy$ plane.
It's for an art project. I'm rusty on my 3d curve sketching/describing/parameterization.
I have tried to figure it out with the equations for an ellipse $(x/a)^2+(y/b)^2+(z/c)^2=1$ where one of the minor axes $c$ would be represented by height in the problem and solving for $z$ , taking the positive root to ensure we don't go past the $xy$ plane.
Since the three points in question lie in the vertical plane through $(x_{1}, y_{1}, 0)$ and $(x_{2}, y_{2}, 0)$, one approach is to pick your favorite smooth, non-negative function $\phi$ defined on $[0, 1]$ and satisfying $$ \phi(0) = \phi(1) = 0,\quad \phi(1/2) = 1, $$ and to take your parametric curve to be $$ t \mapsto \bigl(x_{1} + t(x_{2} - x_{1}), y_{1} + t(y_{2} - y_{1}), h\phi(t)\bigr). $$
A parabola results from taking $\phi(t) = 4t(1 - t)$. A half-ellipse results from $\phi(t) = 2\sqrt{t(1 - t)}$.