page_title: Padua points
section_title: The four families
hierarchical_section_title: Padua points / The four families
caption_attribution_description: English: Padua points of family 1 and degree 6, with their generating curve.
context_page_description: In polynomial interpolation of two variables, the Padua points are the first known example of a unisolvent point set with minimal growth of their Lebesgue constant, proven to be O(log² n)
  .
  Their name is due to the University of Padua, where they were originally discovered.
  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.
context_section_description: 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.
  The four generating curves are closed parametric curves in the interval , and are a special case of Lissajous curves.
caption: Padua points of the first family and of degree 6, plotted with their generating curve.
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