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You know the coordinates (x0, y0) and (x1, y1). You want to pick points on this line with a given x in the interval [x0, x1]. By inspecting the figure we see that:
By manipulating this algebraically, and writing
for the slope, you get:
or if you prefer:
A similar formula can easily be derived for x when y is known.
In numerical analysis a linear interpolation of certain points that are in reality values of some function f is typically used to approximate the function f. Linear interpolation can be regarded as a trivial example of polynomial interpolation. The error of this approximation is defined as
where p denotes the linear interpolation polynomial defined above
It can be proven using Rolle's theorem that if f has two continuous derivatives, the error is bounded by
As you see, the approximation between two points on a given function gets worse with the second derivative of the function that is approximated. This is intuitively correct as well: the "curvier" the function is, the worse is the approximations made with simple linear interpolation.
Linear interpolation is often used to fill the gaps in a table. Suppose you have a table listing the population of some country in 1970, 1980, 1990 and 2000, and that you want to estimate the population in 1994. Linear interpolation gives you an easy way to do this.
The basic operation of linear interpolation between two values is so commonly used in computer graphics that it is sometimes called a lerp in the jargon of computer graphics. The term can be used as a verb or noun for the operation. e.g. " Bresenham's algorithm lerps incrementally between the two endpoints of the line."
Lerp operations are built into the hardware of all modern computer graphics processors. They are often used as building blocks for more complex operations: for example, a bilinear interpolation can be accomplished in three lerps.
Linear interpolation has been used since antiquity for filling the gaps in tables, often with astronomical data. It is believed that it was used in the Seleucid Empire (last three centuries BC) and by the Greek astronomer and mathematician HipparchusFor the Athenian tyrant, see Hipparchus (son of Pisistratus). Hipparchus ( Greek pi;παρχο&sigmaf (circa 190 BC circa 120 BC) was a Greek astronomer, astrologer, geographer, and mathematician. The ESA's Hipparcos Space Astrometry M (second century BC). A description of linear interpolation can be found in the AlmagestAlmagest is the Arabic name of an astronomical treatise explaining the motions of the stars, originally written in Greek as He Megale Syntaxis by Ptolemy of Alexandria, Egypt, sometime around the 2nd century, and accepted for over a thousand years in Arab (second century AD) of PtolemyThis article is about the geographer and astronomer Ptolemy. For Alexander the Great's general, see Ptolemy I of Egypt. For others, see Ptolemy (disambiguation). Claudius Ptolemaeus (Greek: Klaudios Ptolemaios; A. circa 85 circa 165), known in English as.
In demanding situations, linear interpolation is often not accurate enough. In that case, it can be replaced by polynomial interpolation or spline interpolationIn the mathematical subfield of numerical analysis spline interpolation is a special form of interpolation where the interpolant is a piecewise polynomial called spline. Spline interpolation is preferred over polynomial interpolation because the interpola.
Linear interpolation can also be extended to bilinear interpolation for interpolating functions of two variables. Bilinear interpolation is often used as a crude anti-aliasinga) (b) (c) Figure 1 Anti-aliasing in digital signal processing is the technique of minimizing aliasing when representing a high-resolution signal at a lower resolution. In most cases, anti-aliasing means removing data at too high a frequency to represent. filter. Similarly, trilinear interpolationTrilinear interpolation is the process of taking a three-dimensional set of numbers and interpolating the values linearly, finding a point using a weighted average of eight values. Overview The idea behind three-dimensional interpolation is that having so is used to interpolate functions of three variables. Other extensions of linear interpolation can be applied to other kinds of mesh such as triangular and tetrahedral meshes.
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