# What is a simple formula to interpolate 2 unknowns between 2 knowns?

What is a simple formula to find 2 intermediate values between 2 known values?

f(1)=a, f(2)=?, f(3)=?, f(4)=b


If there would be only 1 unknown, then it would be mean ((a+b)/2), but what to do when there are 2 intermediate values to calculate?

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Define

$$f (x) := a + \left(\frac{b-a}{3}\right) (x-1)$$

and then evaluate $f (2)$ and $f (3)$. You should obtain

$$f (2) = \frac{2 a + b}{3}, \quad{} f (3) = \frac{a + 2 b}{3}$$

which are weighted averages of $a$ and $b$. If you had $4$ known values, you would be able to use a cubic interpolating polynomial, but since you only have $2$, you must use an affine polynomial.

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The four numbers are:

$a=\frac{3a+0}3, \frac{2a+b}3, \frac{a+2b}3, \frac{0+3b}3=b$

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