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Natural neighbour interpolation has a number of strengths over linear barycentric interpolation. Namely, it provides a smooth or continuous slope between samples3, and is always the same for a given point set, unlike the TIN where the quality of the triangulation can produce biases in the outcome, even in the best case.
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That looks something like this for a small grid where our snake is moving down. Notice that we delete and re-print 3 entire lines!