I've been playing around with plenty of variants of paraboloid equations. However I couldn't come up with the equation for a x-axis parallel 3D paraboloid of revolution.
For a 2D parabola the equation $$ (y-y_p)^2 = 4p(x-x_p)$$ is derived originally from $$ \vert x+p\vert = \sqrt{(x-p)^2+y^2} $$ and extended by $x_p$ and $y_p$ which describe the vertex coordinates and $p$ for the opening of the parabola.
Is this principle expandable to 3D paraboloids, such as e.g. beginning $$ \vert x+p \vert = \sqrt{(x-p)^2 + y^2 + z^2} $$
Based on the current answer I would further reformulate the equation which leads to $$ (x+p)^2 = (x-p)^2 + y^2 + z^2 $$.
Which, if expanded, leads to: $$ x^2 + 2px + p^2 = x^2 - 2px + p^2 +y^2 + z^2$$
and further simplifies to:
$$ 4px = y^2 + z^2 $$
If we now extend this equation by the vertex coordinates we get:
$$ 4p(x-x_p) = (y-y_p)^2 + (z-z_p)^2 $$
Thank you in advance for any hints
$\endgroup$1 Answer
$\begingroup$I would begin with finding the equation of a 3D paraboloid with central axis the $x$-axis then translate it to get
$$(y-y_0)^2+(z-z_0)^2=4p(x-x_0)$$
