Trilinear interpolation

Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid.

It approximates the value of a function at an intermediate point

within the local axial rectangular prism linearly, using function data on the lattice points.

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

, and bilinear interpolation, which operates with dimension

These interpolation schemes all use polynomials of order 1, giving an accuracy of order 2, and it requires

adjacent pre-defined values surrounding the interpolation point.

There are several ways to arrive at trilinear interpolation, which 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.

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.

On a periodic and cubic lattice, let

and the smaller coordinate related, that is: where

indicates the lattice point below

indicates the lattice point above

First one interpolates along

(imagine one is "pushing" the face of the cube defined by

to the opposing face, defined by

means the function value of

Then one interpolates these values (along

), giving: Finally one interpolates these values along

(walking through a line): This gives us a predicted value for the point.

The result of trilinear interpolation is independent of the order of the interpolation steps along the three axes: any other order, for instance along

, produces the same value.

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

Next, we perform linear interpolation between

Now we do interpolation between

Finally, we calculate the value

via linear interpolation of

In practice, a trilinear interpolation is identical to two bilinear interpolation combined with a linear interpolation: An alternative way to write the solution to the interpolation problem is where the coefficients are found by solving the linear system yielding the result

Trilinear interpolation as two bilinear interpolations followed by a linear interpolation .
Eight corner points on a cube surrounding the interpolation point C
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.