Editing Trilinear interpolation (section)

===Algorithm visualization===
[[File:Trilinear_interpolation_visualisation.svg|thumb|A geometric visualisation of trilinear interpolation. The product of the value at the desired point and the entire volume is equal to the sum of the products of the value at each corner and the partial volume diagonally opposite the corner.]]

The above operations can be visualized as follows: First we find the eight corners of a cube that surround our point of interest. These corners have the values <math>c_{000}</math>, <math>c_{100}</math>, <math>c_{010}</math>, <math>c_{110}</math>, <math>c_{001}</math>, <math>c_{101}</math>, <math>c_{011}</math>, <math>c_{111}</math>.

Next, we perform linear interpolation between <math>c_{000}</math> and <math>c_{100}</math> to find <math>c_{00}</math>, <math>c_{001}</math> and <math>c_{101}</math> to find <math>c_{01}</math>, <math>c_{011}</math> and <math>c_{111}</math> to find <math>c_{11}</math>, <math>c_{010}</math> and <math>c_{110}</math> to find <math>c_{10}</math>.

Now we do interpolation between <math>c_{00}</math> and <math>c_{10}</math> to find <math>c_{0}</math>, <math>c_{01}</math> and <math>c_{11}</math> to find <math>c_{1}</math>. Finally, we calculate the value <math>c</math> via linear interpolation of <math>c_{0}</math> and <math>c_{1}</math>

In practice, a trilinear interpolation is identical to two [[bilinear interpolation]] combined with a linear interpolation:
:<math>c \approx l\left( b(c_{000}, c_{010}, c_{100}, c_{110}),\, b(c_{001}, c_{011}, c_{101}, c_{111})\right)</math>