Tool/algorithmic library for determining equation of quadric surface through 9 given points

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Is there any software that can calculate(and eventually plot) the quadric surface generated by 9 given points in 3D space? I know I can calculate that by defining 9 equations in Mathematica and solving them using $Solve$, but that is quite a process(inputting the coordinates by hand) and the solution is not always that accurate(from my experience, when plotting in Geogebra the quadric found by Mathematica, the surface usually does not pass through the input points.) Thanks!

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If your goal is that of having a dynamic-geometry tool, graphing a quadric surface in real time while you drag the points around, then you can achieve that with GeoGebra.

GeoGebra has a simple CAS embedded, quite cumbersome but apt to the purpose. I used it to substitute into the general equation of a quadric the coordinates of the nine points, solve the resulting system of equations and again substitute the solution into the equation. I thus obtained the equation of the quadric, which is automatically displayed by GeoGebra.

I uploaded this GeoGebra file to GeoGebraTube, for you to experiment, download and possibly improve it.