#### 5.5.1.2. Linear Multivariate Interpolation

The current implementation of the linear multivariate algorithm works as follows:

1. Search the n+1 nearest neighboring supporting points to the query point.

2. Span a hyperplane through the n+1 nearest supporting points.

3. Evaluate the dependent value along the hyperplane at the position of the query point.

If the independent variable space is 2D, then this procedure is like finding the triangle made up of the 3 nearest supporting points of the query point, then defining a plane in the third dimension through the dependent values of these 3 supporting points, and eventually reading out the dependent variable on the plane at the position of the query point. Another interpretation of this procedure are barycentric coordinates of the query point with respect to the simplex spanned from n+1 nearest neighbors. This approach is well-known in finite element theory.

Writing the query point as a linear combination of its n+1 nearest neighbors:

and assuming a linear interpolation function :

we find:

where is the dependent value of the supporting point . The are a function of the coordinates of of the query point.

More details can be found in [40].

The following figure is an illustration of the linear multivariate algorithm for two independent fields, and . The dashed line represents the bounding of the supporting points. represent the three nearest neighbors to the query point .

Figure : Linear Multivariate Algorithm

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