How to Normalize the Sum of Two Gaussians

Question

I have the following function:

$I(\theta_i) = I_0 + I_1\exp(\mu(\cos(\theta_i - \theta_s) - 1))$.

Suppose I have two implementations of this function, whose parameters match with the exception of $\theta_s$, and I add them together to make a third function, $G$:

$G(\theta_i) = I_a(\theta_i) + I_b(\theta_i)$.

How can I normalize $G$ to force it to have the same maximum and minimum values as its component functions ($I_0 + I_1$ and $I_0$, respectively)?

Answer 1

This works:

$G_{intermediate}(\theta_i) = \frac{G(\theta_i) - 2I_0}{I_1}$

$G_{norm}(\theta_i) = I_0 + \frac{I_1}{\max(G_{intermediate}(\theta_i))}G_{intermediate}(\theta_i)$.