### Linear Interpolation

We start with linear interpolation, which fits a polynomial of degree 1 (or linear function) in each interval. By definition, the interpolated values will change linearly within each interval (between two consecutive data points), and at each intermediate \( t_i\) from \(t_1\) to \(t_{n-1}\), the left and right polynomials will meet.

We explain the approach using a generic interval- say, the interval between the data points \( (t_i, f_i) \) and \( (t_{i+1}, f_{i+1}) \). You may remember the linear equation as:

$$ P_{i}(t)= a_i + b_i (t-t_i) $$

Where \(a_i\) and \(b_i\) represent the intercept and the slope, respectively; and \( P_{i}(t) \) represent the interpolating polynomial for the interval \(t_i\) to \(t_{i+1}\) (We will have n such functions to cover the whole range from \(t_0\) to \(t_n\)). We can solve this equation for \(a_i\) and \(b_i\) using the two known values at both ends of the interval. However, we present the interpolating function in an alternative form in the spirit of Langrange polynomial:

$$ P_{i}(t)= a_i (t_{i+1}-t) + a_{i+1} (t-t_i) $$

We require that the value of the interpolating polynomial matches the value of the function at both ends of the interval:

$$ P_{i}(t_i)=f_i$$ $$ P_{i}(t_{i+1})=f_{i+1}$$

Equating the values, we get solution of \( a_i \) and \( a_{i+1} \):

$$ P_{i}(t_i)=f_i \quad \Rightarrow \quad a_i (t_{i+1}-t_i) =f_i \quad \Rightarrow \quad a_i = \frac{1}{(t_{i+1}-t_i)} f_i $$ $$ P_{i}(t_{i+1})=f_{i+1} \quad \Rightarrow \quad a_{i+1} (t_{i+1}-t_i) =f_{i+1} \quad \Rightarrow \quad a_{i+1} = \frac{1}{(t_{i+1}-t_i)} f_{i+1} $$

Substituting for \(a_i\) and \(a_{i+1}\) and rearranging, we get our linear interpolation formula:

$$ P_{i}(t)= a_i (t_{i+1}-t) + a_{i+1} (t-t_i) $$ $$ \quad = \frac{(t_{i+1}-t)}{(t_{i+1}-t_i)} f_i + \frac{(t-t_i)}{(t_{i+1}-t_i)} f_{i+1} $$

This is the linear interpolation formula. Now to find interpolated value for a given t, we just need to find the interval that brackets the given t, and then calculate the interpolated value using the above formula.