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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

Daniyal
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1 Answers1

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I would begin with finding the equation of a 3D paraboloid with central axis the $x$-axis then translate it to get

$$(y-y_0)^2+(z-z_0)^2=4p(x-x_0)$$

3D paraboloid about x-axis

John Wayland Bales
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  • Thank you for your response! I've edited my initial post and derived the equation. Is the way I've taken to translate to your provided equation correct? – Daniyal Oct 06 '16 at 18:13
  • Your use of $p$ is confusing. You write $(x-p)$ where you intend $(x-x_p)$ and you should really use $x_0,y_0,z_0$ rather than $x_p,y_p,z_p$ since the translations do not in any way depend upon the parabolic parameter $p$. – John Wayland Bales Oct 06 '16 at 19:25
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    The key point is that the cross-sections of the untranslated paraboloid taken perpendicular to the axis are circles of some radius $r$ and satisfying the equation $r^2=4px$ where $r^2=y^2+z^2$. – John Wayland Bales Oct 06 '16 at 19:30