The points are defined in the domain. It is possible to use the points with four orientations, obtained with subsequent 90-degree rotations: this way we get four different families of Padua points.
Padua points of the first family and of degree 5, plotted with their generating curve.Padua points of the first family and of degree 6, plotted with their generating curve.
We can see the Padua point as a "sampling" of a parametric curve, called generating curve, which is slightly different for each of the four families, so that the points for interpolation degree and family can be defined as
Actually, the Padua points lie exactly on the self-intersections of the curve, and on the intersections of the curve with the boundaries of the square . The cardinality of the set is . Moreover, for each family of Padua points, two points lie on consecutive vertices of the square , points lie on the edges of the square, and the remaining points lie on the self-intersections of the generating curve inside the square.[3][4]
The four generating curves are closed parametric curves in the interval , and are a special case of Lissajous curves.
with representing the normalized Chebyshev polynomial of degree (that is, and , where is the classical Chebyshev polynomial of first kind of degree ).[3] For the four families of Padua points, which we may denote by , , the interpolation formula of order of the function on the generic target point is then
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Caliari, Marco; Bos, Len; de Marchi, Stefano; Vianello, Marco; Xu, Yuan (2006), "Bivariate Lagrange interpolation at the Padua points: the generating curve approach", J. Approx. Theory, 143 (1): 15–25, arXiv:math/0604604, doi:10.1016/j.jat.2006.03.008
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de Marchi, Stefano; Caliari, Marco; Vianello, Marco (2005), "Bivariate polynomial interpolation at new nodal sets", Appl. Math. Comput., 165 (2): 261–274, doi:10.1016/j.amc.2004.07.001
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Caliari, Marco; de Marchi, Stefano; Vianello, Marco (2008), "Algorithm 886: Padua2D—Lagrange Interpolation at Padua Points on Bivariate Domains", ACM Transactions on Mathematical Software, 35 (3): 1–11, doi:10.1145/1391989.1391994