Linear Interpolation Between Two Known Points
If the two known points are given by the coordinates and, the linear interpolant is the straight line between these points. For a value x in the interval, the value y along the straight line is given from the equation
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 . 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 and, which are normalized distances between the unknown point and each of the end points.
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