# Interpolation with 2 variables

1. Jun 5, 2014

### Jhenrique

If given three points $P_0 = (x_0, y_0)$, $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$, the polynomial function $f(x)$ that intersect those points is $f(x) = a_2 x^2 + a_1 x^1 + a_0 x^0$.

where:
$\begin{bmatrix} a_0\\ a_1\\ a_2\\ \end{bmatrix} = \begin{bmatrix} x_0^0 & x_1^0 & x_2^0 \\ x_0^1 & x_1^1 & x_2^1 \\ x_0^2 & x_1^2 & x_2^2 \\ \end{bmatrix}^{-T} \begin{bmatrix} y_0\\ y_1\\ y_2\\ \end{bmatrix}$

And this ideia can extended for $P_n$ points... so, analogously, given a set of points, exist a relationship between the coefficients of $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ and the coordinates of the points?

2. Jun 6, 2014

### lurflurf

Yes, that is a standard result. Five points in the plane determine a quadratic in two variables, if no three are collinear the quadratic will be unique and non degenerate. The coefficients can be found by solving a linear system like in the one variable case.

http://en.wikipedia.org/wiki/Five_points_determine_a_conic

3. Jun 6, 2014

### Jhenrique

Yeah! But don't exist the equation of the coefficients in function of the points' coordinates in the wikipage...

Last edited: Jun 6, 2014