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. 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. If the two known points are given by the coordinates ( x 0 , y 0 ) {displaystyle (x_{0},y_{0})} and ( x 1 , y 1 ) {displaystyle (x_{1},y_{1})} , the linear interpolant is the straight line between these points. For a value x in the interval ( x 0 , x 1 ) {displaystyle (x_{0},x_{1})} , the value y along the straight line is given from the equation of slopes which can be derived geometrically from the figure on the right. It is a special case of polynomial interpolation with n = 1. Solving this equation for y, which is the unknown value at x, gives which is the formula for linear interpolation in the interval ( x 0 , x 1 ) {displaystyle (x_{0},x_{1})} . Outside this interval, the formula is identical to linear extrapolation. This formula can also be understood as a weighted average. The weights are inversely related to the distance from the end points to the unknown point; the closer point has more influence than the farther point. Thus, the weights are x − x 0 x 1 − x 0 { extstyle {frac {x-x_{0}}{x_{1}-x_{0}}}} and x 1 − x x 1 − x 0 { extstyle {frac {x_{1}-x}{x_{1}-x_{0}}}} , which are normalized distances between the unknown point and each of the end points. Because these sum to 1,