Trilinear interpolation

Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. It approximates the value of an intermediate point ${\displaystyle (x,y,z)}$ within the local axial rectangular prism linearly, using data on the lattice points. For an arbitrary, unstructured mesh (as used in finite element analysis), other methods of interpolation must be used; if all the mesh elements are tetrahedra (3D simplices), then barycentric coordinates provide a straightforward procedure.

Trilinear interpolation is frequently used in numerical analysis, data analysis, and computer graphics.

Trilinear interpolation is the extension of linear interpolation, which operates in spaces with dimension ${\displaystyle D=1}$, and bilinear interpolation, which operates with dimension ${\displaystyle D=2}$, to dimension ${\displaystyle D=3}$. The order of accuracy is 1 for all these interpolation schemes, and it requires ${\displaystyle (1+n)^{D}=8}$ adjacent pre-defined values surrounding the interpolation point. There are several ways to arrive at trilinear interpolation, it is equivalent to 3-dimensional tensor B-spline interpolation of order 1, and the trilinear interpolation operator is also a tensor product of 3 linear interpolation operators.

...
Wikipedia