- #1
Jhenrique
- 685
- 4
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}##
Ok, but if is given points in the space with coordinates ##P_1 = (x_1 ,y_1 ,z_1)##, ##P_2 = (x_2 ,y_2 ,z_2)##, ##P_3 = (x_3 ,y_3 ,z_3)##, ..., ##P_n = (x_n ,y_n ,z_n)##... is possible to determine the coefficients of the polynomial function of 2 variables, ##f(x,y) = Ax^2+Bxy+Cy^2+Dx+Ey+F##, in function of the point's coordinates (like above in the 2D case)?
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}##
Ok, but if is given points in the space with coordinates ##P_1 = (x_1 ,y_1 ,z_1)##, ##P_2 = (x_2 ,y_2 ,z_2)##, ##P_3 = (x_3 ,y_3 ,z_3)##, ..., ##P_n = (x_n ,y_n ,z_n)##... is possible to determine the coefficients of the polynomial function of 2 variables, ##f(x,y) = Ax^2+Bxy+Cy^2+Dx+Ey+F##, in function of the point's coordinates (like above in the 2D case)?