I am trying to develop a nomogram which simultaneously shows the exact Fisher equation $(1+u) = (1+v)(1+w)$ and its linear approximation $u \approx v + w$.
This amounts to finding twelve smooth curves such that the following equations hold:
$$\det \begin{bmatrix}f_1(u) & f_2(u) & f_3(u) \\ g_1(v)& g_2(v) & g_3(v) \\ h_1(w) & h_2(w) & h_3(w)\end{bmatrix} = (1+v)(1+w) - (1+u)$$
$$\det \begin{bmatrix}f_4(u) & f_5(u) & f_6(u) \\ g_1(v)& g_2(v) & g_3(v) \\ h_1(w) & h_2(w) & h_3(w)\end{bmatrix} = v + w - u$$
My question is how to begin thinking about this problem --- how can you decide whether such curves exist, and how might you look for them?