# Check whether a point is inside an ellipsoid-like surface

Question: Given points on a surface, how can well check whether another test point falls inside the surface?

I don't have equation defining the surface. All I have, are sample points lying on the surface. It is no exaggeration to say that the sample points I have on the surface define the surface itself. And because of the nature of my application, I always know that those points will come to define an ellipsoid like surface such as this:

For a similar question, you can refer to here. The reason why I asked that question was because I thought that answering that question, I could have answered this question as well. My initial approach was to write out the equations of the surface, I can then check whether a test point is inside the surface or outside of it. More explicitly, if I found that

$$\sum_{i=0}^n \sum_{j=0}^{n-i} \sum_{k=0}^{n-i-j} a_{ijk}\,x^iy^jz^k > 0$$

Then I can be sure that the test point lies outside the surface.

If I found that $$\sum_{i=0}^n \sum_{j=0}^{n-i} \sum_{k=0}^{n-i-j} a_{ijk}\,x^iy^jz^k = 0$$

Then I can be sure that the test points lies exactly on the surface. And so on.

But upon further thought, I am unable to convince myself of the merit of this approach. That's why I ask a more direct question here.

Any idea?