The Interpolate command is a general-purpose command used for interpolation. To perform the interpolation, a variety of different methods can be used as specified by the method option. This command will return an interpolation object which can then be used as a procedure, taking the coordinates of a point as parameters and giving the interpolated result as the return value.

The values for the method option correspond to the following interpolation methods, and the supported dimensions for the independent points in points:

The default method is spline if the input points are one-dimensional, naturalneighbor if the input points are two-dimensional, and radialbasisfunction otherwise.

The methods that are listed as supporting only dimension 1, above, do support higher dimensions, but only for independent points that are arranged in a grid. The way such points are then specified is not compatible with the way points are specified for the other commands, so the Interpolate command doesn't support this. However, you can make use of this functionality by calling the corresponding commands (linked in the table above) directly.

$\mathrm{with}\left(\mathrm{Interpolation}\right)\:$

$\mathrm{points}≔\left[\left[0\,0\right]\,\left[1\,0\right]\,\left[2\,0\right]\,\left[0\,1\right]\,\left[1\,1\right]\,\left[2\,1\right]\,\left[0\,2\right]\,\left[1\,2\right]\,\left[2\,2\right]\right]$

$\mathrm{points}≔\left[\left[0\,0\right]\,\left[1\,0\right]\,\left[2\,0\right]\,\left[0\,1\right]\,\left[1\,1\right]\,\left[2\,1\right]\,\left[0\,2\right]\,\left[1\,2\right]\,\left[2\,2\right]\right]$

$\mathrm{data}≔\left[0\,0\,0\,0\,1\,0\,0\,0\,0\right]$

$\mathrm{data}≔\left[0\,0\,0\,0\,1\,0\,0\,0\,0\right]$

$f≔\mathrm{Interpolate}\left(\mathrm{points}\,\mathrm{data}\right)$

$f≔\left(\mathrm{Natural\; Neighbor\; interpolation\; object\; with\; 9\; sample\; points}\right)$

$f\left(0.5\,0.5\right)$

$0.250000000000000$

$g≔\mathrm{Interpolate}\left(\mathrm{points}\,\mathrm{data}\,\mathrm{method}=\mathrm{radialbasisfunction}\,\mathrm{gaussian}\,1.5\right)$

$g≔\left(\begin{array}{c}\mathrm{Ra\ⅆial\; Basis\; Function\; int\ⅇrpolation\; obȷ\ⅇct\; with\; 9\; sampl\ⅇ\; points}\\ \mathrm{Ra\ⅆial\; Basis\; Function:\; gaussian}\end{array}\right)$

$g\left(0.5\,0.5\right)$

$0.271060313298269628$

$\mathrm{plot3d}\left(\left[f\,g\right]\,0..2\,0..2\,\mathrm{transparency}=0.5\,\mathrm{color}=\left[\mathrm{red}\,\mathrm{blue}\right]\right)$

$\mathrm{points}≔\left[0\,1\,3\,4\,6\right]$

$\mathrm{points}≔\left[0\,1\,3\,4\,6\right]$

$\mathrm{values}≔\left[1\,0\,4\,4\,5\right]$

$\mathrm{values}≔\left[1\,0\,4\,4\,5\right]$

$f≔\mathrm{Interpolate}\left(\mathrm{points}\,\mathrm{values}\right)$

$f≔\left[\begin{array}{c}\text{a spline interpolation object}\\ \text{with 5 points in 1-D}\end{array}\right]$

$f\left(2\right)$

$1.84677419354839$

$\mathrm{plot}\left(f\,0..6\right)$

The Interpolation[Interpolate] command was introduced in Maple 2018.

For more information on Maple 2018 changes, see Updates in Maple 2018.