smooth curve fitting by triangularization of derivative

Suppose you have some points, for this example I'll use:
[[1,1],[3,3],[4,2]]
Like so:

First connect them with straight lines:

Now imagine you are considering the derivative:

The next step is to do what I'm calling triangularizing the above step function. This means to find the midpoints of AB and CD, E and F. Then find the point G that preserves the total area under the original step function. 

This makes a new curve when you remove B and C, 

Now this is a new approximation for the derivative, so we integrate and set the constant of integration so A goes to [1,1] where it was originally:

To get smoother and smoother approximation, iterate the derivative and triangularization step and integrate the corresponding number of times.