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When doing simple linear interpolation on a data set in $\mathbb{R}^2$, I just sort the points by one co-ordinate and then do piecewise interpolation between a point and its successor in the sorted data set.

  1. How would I do this in $\mathbb{R}^3$, and generally in $\mathbb{R}^k$?

  2. How would I know which points to "connect"? (In the two-dimensional case the sorting takes care of that.)

  3. How complex would it be to calculate such simple interpolations?

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You need to triangulate the data set then do linear interpolation within each triangle (simplex, in higer-dimensions).

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    Mostly, yes, although some people say *tetrahedralization* in 3d solid data, in contrast with 3d surface data, which can be triangularized.2011-03-21