# Linear interpolation

## Linear interpolation derivation
> (wiki) 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.

For example, in the image above, the red dots are the the measurements we know, the blue dots are the data we need to interpolate.


So, where does the fomulation at end of slide come from?

Let's reformulate the problem here. Given two known value $y_0, y_1$ are evaluated at $x_0, x_1$, respectively, and a value of $x \in [x_0, x_1]$, we want to estimate the value of $y$ (see graph below).

Start with slope equation:
$$\frac{y-y_0}{x-x_0} = \frac{y_1 - y_0}{x_1 - x_0}$$

Solving for $y$, we have:

\begin{align*}
y &= y_0 + (x-x_0)\frac{y_1 - y_0}{x_1 - x_0} \\
&= \frac{y_0 (x_1- x_0) + (x-x_0)(y_1-y_0)}{x_1 - x_0} \\
&= \frac{(y_0 x_1 - y_0 x_0) + (xy_1 - xy_0 - x_0 y_1 + x_0y_0)}{x_1 - x_0} \\
&= \frac{y_0 (x_1 - x) + y_1 (x-x_0)}{x_1 - x_0} \\
&= y_0 \frac{x_1 - x}{x_1 - x_0} + y_1 \frac{x-x_0}{x_1 - x_0}
\end{align*}

Now, if we set $t := \frac{x-x_0}{x_1 - x_0}$, then $1-t = \frac{x_1 - x}{x_1 - x_0}$. Thus the above equation becomes:

\begin{align*}
y &= y_0 \times (1-t) + y_1 \times t \\
&=(y_1 - y_0) \times t + y_0
\end{align*}

which matches with the equation in the slide, where $y_0 = a, y_1 = b$.

## Reference
https://en.wikipedia.org/wiki/Linear_interpolation