experimental data (e.g. pressure vs enthalpy vs temperature)

weather data (e.g. longitude vs latitude vs temperature)

GIS (e.g. a LIDAR point cloud)

or finite element analysis on an irregular mesh

The Interpolation package contains tools for the interpolation of irregularly spaced data using several methods (e.g. Kriging and inverse distance weighting). The package generates what is essentially a normal mathematical function that can be analyzed or plotted.

surfdata will generate a surface constructed from triangles passing precisely through each data point.

ScatterPlot3D with the option lowess will generate a surface using lowess smoothing. The surface will not pass precisely through each data point; however, various options can be used to control the type of surface generated.

$N≔75\:$

$$\begin{gathered} X≔\mathrm{LinearAlgebra}:-\mathrm{RandomVector}\left(N\,\mathrm{generator}\=-2..2.\right)\: \\ Y≔\mathrm{LinearAlgebra}:-\mathrm{RandomVector}\left(N\,\mathrm{generator}\=-2..2.\right)\: \\ Z≔\mathrm{Vector}\left(N\,i\to X\left[i\right]\cdot \exp \left(-X{\left[i\right]}^2-Y{\left[i\right]}^2\right)\+\mathrm{rand}\left(\right)\cdot {10}^{-30}\right)\: \end{gathered}$$

$M≔⟨⟨X\|Y\|Z⟩⟩$

$\mathrm{p1}≔\mathrm{plots}:-\mathrm{pointplot3d}\left(M\,\mathrm{style}\=\mathrm{patchnogrid}\,\mathrm{color}\=\mathrm{black}\,\mathrm{symbol}\=\mathrm{solidsphere}\,\mathrm{symbolsize}\=20\, \mathrm{style}\=\mathrm{point}\right)$

$f ≔ \mathrm{Interpolation}:-\mathrm{Kriging}\left(⟨X\|Y⟩\,Z\right)$

$f≔\left(\begin{array}{c}\mathrm{Kriging\; int\ⅇrpolation\; obȷ\ⅇct\; with\; 75\; sampl\ⅇ\; points}\\ \mathrm{Variogram:\; Sph\ⅇrical(.00224838890116667,.0596838206033333,2.112716683)}\end{array}\right)$

$f\left(M\left[1\,1\right]\,M\left[1\,2\right]\right)$

$0.0305112187500000617$

$$\begin{gathered} \mathrm{p2}≔\mathrm{plot3d}\left(f\, -2 .. 2\, -2 .. 2\, \mathrm{transparency}\=0.5\right)\: \\ \mathrm{plots}:-\mathrm{display}\left(\mathrm{p1}\,\mathrm{p2}\right) \end{gathered}$$

$$\begin{gathered} \mathrm{p3}≔\mathrm{plots}:-\mathrm{surfdata}\left(M\,\mathrm{style}\=\mathrm{patchnogrid}\,\mathrm{color}\=\mathrm{ColorTools}:-\mathrm{Color}\left(''RGB''\,\left[\frac{150}{255}\,\frac{79}{255}\,\frac{121}{255}\right]\right)\right)\: \\ \mathrm{plots}:-\mathrm{display}\left(\mathrm{p1}\,\mathrm{p3}\right) \end{gathered}$$

$$\begin{gathered} \mathrm{p4}≔\mathrm{Statistics}:-\mathrm{ScatterPlot3D}\left(M\, \mathrm{lowess}\, \mathrm{fitorder}\=2\, \mathrm{rule}\=3\, \mathrm{grid}\=\left[25\,25\right]\, \mathrm{style}\=\mathrm{surface}\,\mathrm{color}\=\mathrm{ColorTools}:-\mathrm{Color}\left(''RGB''\,\left[\frac{150}{255}\,\frac{79}{255}\,\frac{121}{255}\right]\right)\right)\: \\ \mathrm{plots}:-\mathrm{display}\left(\mathrm{p1}\,\mathrm{p4}\right) \end{gathered}$$