# Creation of the Graphs

In order to show the user how the budget is going to be allocated over the period of 2014 until 2050, we rely on three different forms of graphical interpolation for our indicative pathways. Keep in mind that all per capita values used in these graphs are based on constant population levels of 2014.

## Constant Rate

We always show a plot that uses a constant percentage rate of emission change, g:

\int_{2014}^{2050} \frac{E_i(2014)}{L_i(2014)}e^{-gt} dt = \text{Budget per capita}

## Spline Interpolation

When the budget is large enough and there are no turning points before the end year of 2050, we use cubic spline interpolation with the additional condition that the area below the curve equals the total budget calculated by the climate calculator. You can find a more detailed technical explanation of the curve and function fitting here.

## Decreasing Rate

As an alternative to the spline interpolation, we allocate the budget using a different functional form, when the ratio of the per capita budget and the per capita emission in 2014 becomes small:

E_i (t) = \frac{E_i(2014)}{L_i(2014)}(t-2009)^a

The projection starts with the per capita emission from 2014, (E2014/L2014), and is allocated over the years from 2014 until 2050. We just have to find the a for which the area, hence the integral, under this function equals the budget per capita from the calculator. This implies a decreasing rate of emission reduction.

When the ratio of budget per capita and emission per capita is smaller than 4 we do now show a graph as the budget is just too small to have a meaningful reduction pathway.