Equation of a parabola in 3D space

Question

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is vertex(lowest point) of the parabola. I only know z-coordinate of this point. I need to find coordinates of the points that lie on a parabola that passes through these 3 points. I am interested in coordinates that lie in between A and B.

Answer 1

According to your comment added later the parabola lies in a vertical plane through the two points $A=(x_0,y_0,z_0)$ and $B=(x_1,y_1,z_1)$ and has a vertical axis. It therefore has a parametric representation of the form $$s\mapsto\left\{\eqalign{x(s)&=(1-s)x_0+sx_1 \cr y(s)&=(1-s)y_0+sy_1 \cr z(s)&=as^2+bs+c\cr}\right.$$ It remains to determine the three coefficients $a$, $b$, $c$ from the data. We want $z(0)=z_0$ and $z(1)=z_1$. The third equation comes from the fact that we are also given the minimal value $z_*$ of $s\mapsto z(s)=as^2+bs+c$. This leads to the equation $$-{b^2\over 4a}+c=z_*\ ,$$ so that we can now determine $a$, $b$, $c$ in a straightforward way.