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Interpolation with 2 variables

  1. Jun 5, 2014 #1
    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. jcsd
  3. Jun 6, 2014 #2

    lurflurf

    User Avatar
    Homework Helper

    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
     
  4. Jun 6, 2014 #3
    Yeah! But don't exist the equation of the coefficients in function of the points' coordinates in the wikipage...
     
    Last edited: Jun 6, 2014
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