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Barycentric coordinates are strongly related to Cartesian coordinates and, more generally, affine coordinates.For a space of dimension n, these coordinate systems are defined relative to a point O, the origin, whose coordinates are zero, and n points , …,, whose coordinates are zero except that of index i that equals one.
A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or Newton polynomials. Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function.
The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity.
An example Bézier triangle with control points marked. A cubic Bézier triangle is a surface with the equation (,,) = (+ +) = + + + + + + + + +where α 3, β 3, γ 3, α 2 β, αβ 2, β 2 γ, βγ 2, αγ 2, α 2 γ and αβγ are the control points of the triangle and s, t, u (with 0 ≤ s, t, u ≤ 1 and s + t + u = 1) are the barycentric coordinates inside the triangle.
Barycentric subdivision, a way of dividing a simplicial complex; Barycentric coordinates (mathematics), coordinates defined by the vertices of a simplex; In numerical analysis, Barycentric interpolation formula, a way of interpolating a polynomial through a set of given data points using barycentric weights.
They should all work on a regular grid, typically reducing to another known method. Nearest-neighbor interpolation; Triangulated irregular network-based natural neighbor; Triangulated irregular network-based linear interpolation (a type of piecewise linear function) n-simplex (e.g. tetrahedron) interpolation (see barycentric coordinate system)
This method proposes to optimally stack a dense distribution of constraints of the type P″(x) = 0 on nodes positioned externally near the endpoints of each side of the interpolation interval, where P"(x) is the second derivative of the interpolation polynomial. Those constraints are called External Fake Constraints as they do not belong to ...
Thus, it is appropriate to speak of the "Newton form", or Lagrange form, etc., of the interpolation polynomial. However, different methods of computing this polynomial can have differing computational efficiency. There are several similar methods, such as those of Gauss, Bessel and Stirling.