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# Linear interpolation¶

See: wikipedia on linear interpolation.

Let us say that we have two known points \(x_1, y_1\) and \(x_2, y_2\).

Now we want to estimate what \(y\) value we would get for some \(x\) value that is
between \(x_1\) and \(x_2\). Call this \(y\) value estimate — an *interpolated*
value.

Two simple methods for choosing \(y\) come to mind. The first is see whether
\(x\) is closer to \(x_1\) or to \(x_2\). If \(x\) is closer to \(x_1\) then we use
\(y_1\) as the estimate, otherwise we use \(y_2\). This is called *nearest
neighbor* interpolation.

The second is to draw a straight line between \(x_1, y_1\) and \(x_2, y_2\). We
look to see the \(y\) value on the line for our chosen \(x\). This is *linear
interpolation*.

It is possible to show that the formula of the line between \(x_1, y_1\) and \(x_2, y_2\) is: