trilinear

Trilinear is an adjective used in mathematics and related fields to describe relationships, operations, or objects that involve three linear components or dimensions. It appears in several distinct contexts, including trilinear interpolation, trilinear coordinates, and trilinear forms.

Trilinear interpolation is a method for estimating values inside a three‑dimensional grid cell from the values at the eight corner points. The procedure performs linear interpolation along one axis to create intermediate values, then along a second axis to produce intermediate values on faces, and finally along the third axis to obtain the interior estimate. The result is continuous and efficient and is commonly used in volume rendering, texture filtering, and finite‑element methods. Its accuracy depends on the assumption that the underlying function varies approximately linearly in each dimension.

Trilinear coordinates are a coordinate system for a point relative to a reference triangle. The coordinates are proportional to the point’s perpendicular distances to the three sides of the triangle (often denoted α, β, γ). They are closely related to barycentric coordinates and are used in planar geometry and computer graphics to simplify certain constructions and computations. The concept can be extended in various geometric contexts, though the classical form is tied to triangles.

In linear algebra, a trilinear form is a scalar-valued function T(x, y, z) that is linear in each argument. Such forms are represented by third‑order tensors and play a role in multilinear algebra, differential geometry, and physics. Symmetric trilinear forms correspond to homogeneous cubic polynomials and describe three‑way interactions in mathematical models.