Editing Linear interpolation (section)

==Extensions==
{{comparison of 1D and 2D interpolation.svg|250px|linear and bilinear interpolation}}

===Accuracy===

If a [[differentiability class|{{math|''C''<sup>0</sup>}}]] function is insufficient, for example if the process that has produced the data points is known to be smoother than {{math|''C''<sup>0</sup>}}, it is common to replace linear interpolation with [[spline interpolation]] or, in some cases, [[polynomial interpolation]].

===Multivariate===

Linear interpolation as described here is for data points in one spatial dimension. For two spatial dimensions, the extension of linear interpolation is called [[bilinear interpolation]], and in three dimensions, [[trilinear interpolation]]. Notice, though, that these interpolants are no longer [[linear functions]] of the spatial coordinates, rather products of linear functions; this is illustrated by the clearly non-linear example of [[bilinear interpolation]] in the figure below. Other extensions of linear interpolation can be applied to other kinds of [[polygon mesh|mesh]] such as triangular and tetrahedral meshes, including [[Bézier surface]]s. These may be defined as indeed higher-dimensional [[piecewise linear function|piecewise linear functions]] (see second figure below).

[[Image:Bilininterp.png|right|thumb|Example of [[bilinear interpolation]] on the unit square with the {{mvar|z}} values 0, 1, 1, and 0.5 as indicated. Interpolated values in between are represented by colour.]]
[[Image:Piecewise linear function2D.svg|right|thumbnail|A piecewise linear function in two dimensions (top) and the convex polytopes on which it is linear (bottom)]]