Interpolation and reconstruction of curves and surfaces

Interpolation is an ubiquitous technique arising in Mathematics, specially in Numerical Analysis. The name interpolation itself means to place something between selected sites, in contrast to extrapolation which means place something outside. Interpolation is generally more accurate than extrapolation leading to better stability in computations and lower error bounds. Since most methods can be used for both interpolation and extrapolation, the word interpolation is preferred. Simple interpolation methods can be dated back to babylonians and the greeks. The Mathematical Syntaxis of Ptolemy shows how to construct a table of chords as a func-tion of its corresponding arcs, the first table of a trigonometric function 2 sin(x/2). John Napier and Joost Bürgi introduced in the 17th century tables of logarithms which were improved by Henry Briggs, who introduced new tables of functions. The tables of func-tions contained a lot of decimal places and computing them was a very laborious task. The idea of approximating a curve by its chord or a parabola leads to a first instance of linear or quadratic interpolation. Interpolation formulae were implicit or explicitly used for constructing and filling up the tables. Briggs propose to subtitute the true logarithmi