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The four red dots show the data points and the green dot is the point at which we want to interpolate. Suppose that we want to find the value of the unknown function f at the point (x, y). It is assumed that we know the value of f at the four points Q 11 = (x 1, y 1), Q 12 = (x 1, y 2), Q 21 = (x 2, y 1), and Q 22 = (x 2, y 2).
Given the two red points, the blue line is the linear interpolant between the points, and the value y at x may be found by linear interpolation. In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
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.
) and the interpolation problem consists of yielding values at arbitrary points (,,, … ) {\displaystyle (x,y,z,\dots )} . Multivariate interpolation is particularly important in geostatistics , where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or ...
Note that for 1-dimensional cubic convolution interpolation 4 sample points are required. For each inquiry two samples are located on its left and two samples on the right. These points are indexed from −1 to 2 in this text. The distance from the point indexed with 0 to the inquiry point is denoted by here.
Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon; the problem may be eliminated by choosing interpolation points at Chebyshev nodes. [5]
The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions.Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point.
Any series of 4 distinct points can be converted to a cubic Bézier curve that goes through all 4 points in order. Given the starting and ending point of some cubic Bézier curve, and the points along the curve corresponding to t = 1/3 and t = 2/3, the control points for the original Bézier curve can be recovered. [9]