Can i find a 3D function given some points?

Question

is it possible to find a 3D function given a set of data points? i tried plane-fitting it did not work, too chaotic for a plane.

I am trying to find a 3D equation that cover most of points, how can i get to it ? Is there some terminolgy for this, maybe something i can study?

points are:

$( 3 , 0.48 , 1.25 )$

$( 15 , 3.77 , 5.5 )$

$( 5.5 , 1.3 , 2.75 )$

$( 12.5 , 3.08 , 5.25 )$

$( 2.2 , 0.1 , 0.2 )$

$( 2.2 , 0.2 , 0.3 )$

$( 2.2 , 0.3 , 0.5 )$

Answer 1

You can fit anything. You have 7 points. Assume your function is of the form $f(x,y)=ax+by+cx^2+dy^2+ex^3+fy^3+gx^4$. Now substitute your given points and solve simultaneous linear systems in variables $a,b,c,d,e,f,g,$ in seven variables. If mine choice of $f$ doesn't work, you can adjust $f$ to make the linear system consistent.

Answer 2

Take a look at something like a "thin-plate spline" (Duchon $D^m$ splines) for a $3D$ case, or at rbf interpolation.