Standardized interpolation formulas

The formulas for quadratic interpolation imply a passage from a linear form in quadratic variables to a quadratic form in linear variables, whose behavior under a change of basis ought to be analyzed. Although this change of emphasis has its merit, not all of the symmetry inherent in the determinantal version is immediately evident in the new form.

In this one dimensional example, we would write

z = U^TQU

with

$$U^T = \left[ \begin{array}{cc} 1 & x \end{array} \right]$$

and

$$Q=\left[ \begin{array}{cc} C&B\\ B&A \end{array} \right].$$

The translation Z,

$$Z=\left[ \begin{array}{cc} 1&0\\ a&1 \end{array} \right].$$

transforms Q into Z^TQZ, which is namely

$$\left[ \begin{array}{cc} 1&a\\ 0&1 \end{array} \right] \left[ \b... ...B&A \end{array} \right] \left[ \begin{array}{cc} 1&0\\ a&1 \end{array} \right]$$ Z^TQZ = (53)

$$\left[\begin{array}{cc}C+2aB+a^2A&B+aA\\ B+aA&A\end{array}\right].$$ = (54)

The choice of a=-B/A removes the off-diagonal elements from Z^TQZ, providing a matricial version of completing the square.

Since the interpolation equations are the same for all grid sites, and since the scale and origin of the equations can be changed so easily, it is reasonable to tabulate the interpolation coefficients for different coordinate grids, particularly for the commonly used square grid.

• two dimensional linear interpolation
  
  $$V = \left[ \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 1 & 1... ... -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array} \right]$$
  
  
  
  
  z = z_0,0+(z_0,1-z_0,0)x+(z_1,1-z_0,1)y
  
  

• two dimensional quadratic interpolation
  
  $$V = \left[ \begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1... ...z_{0,2} & z_{1,0} & z_{1,1} & z_{1,2} & z \end{array} \right]$$
  
  
  
  
  $$M^{-1} = \left[ \begin{array}{cccccc} 1 & -1 & 0 & 0 & 1 & ... ...0 & 0 & 1 & 0 \\ 1 & -1 & 0 & 0 & 1 & 0 \end{array} \right]$$
  
  

• three dimensional linear interpolation
• three dimensional quadratic interpolation