How to find a 3d equation from a series of points

Question

I have 6 points and I need to find the equation, or an equation, that will go through all of them. How would I go about doing this? The points are as follows. (3.7, 0.45, 0.7) (5.2, 0.8, 0.96) (6, 1.04, 1.15) (7.5, 1.4, 1.4) (10, 2.1, 1.8) (30, 16, 6.6)

Answer 1

The wording of the question is incomplete because on can find an infinity of different solutions. More specifications or conditions should be added in order to ask for only one solution, or only a few solutions.

A manner to find a solution can be :

Take an equation with three variables $x,y,z$ and including 6 adjustable parameters (noted $A,B,C,D,E,F$).

Put in it the given values $x_1,y_1,z_1$ which gives the first equation.

Put in it the given values $x_2,y_2,z_2$ which gives the second equation.

and so on, up to $x_6,y_6,z_6$ which gives the sixth equation.

You have got a system of 6 equations where the six unknowns are $A,B,C,D,E,F$

Then, solve the system for $A,B,C,D,E,F$. You do obtain a result which agrees with the wording of the problem.

As many different equations taken, as many convenient results.

(except if the system is not solvable which occurs in some cases)

See below two examples. The first is made with a polynomial equation. The second is a polynomial plus hyperbolic equation. In order to have a system easy to solve, the equations were chosen to be linear with respect to the parameters. But non-linear could be chosen as well if we are not afraid to solve non-linear systems of equations.

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