What fields/literature should I explore to learn how to parametrize surfaces, with constraints, that fit data? This is a somewhat vague question, but I think a good one!
I sometimes find myself in the position of having to fit a surface to a point cloud. Because of the nature of the data, I sometimes want to enforce a structural constraint. And I sometimes want to enforce a complexity constraint, so that the model has only a handful of parameters.
Let me give an example. Suppose I observe widgets along two dimensions, $s$ (strength) and $v$ (speed), and want to predict time-to-failure $T$. I have observations $(s_i,v_i,T_i)$ associated to widgets $i$. I may want to fit a surface to this which has a few parameters, and which has the property that for any fixed strength, $T$ is a convex function of speed.
What tools and/or fields will help me tackle such problems in a practical way? 
Differential geometry? 
Some subfield of statistics?
 A: Well you first have to decide whether you want to try to fit a parametric or non-parametric model. Since you need convexity in $v$ it probably makes sense to use a parametric model here that automatically suffices this criterion.
A very simple ansatz could be to try a best fit parabola: $f(s,v) = \alpha(s) (v-\beta(s))^2$
Here you could slice your data into strips $S_j = \{(s_i,v_i)\in D :    |s_i-t_j|<\epsilon \}$ for grid points $t_1,\ldots, t_m$ with $\Delta t = 2\epsilon$
Then within each strip project the data along the $s$-axis and find the best fit parabola, i.e.
$$ (\alpha_j, \beta_j) = \underset{(\alpha,\beta)\in\mathbb R^2\\  \alpha>0}{\text{argmin}}\sum_{(s_i,v_i)\in S_j} \big|\alpha(v_i -\beta)^2 - T_i\big|^2 $$
And then find $\alpha(s)$ and $\beta(s)$ via regression of the $(t_j,\alpha_j,\beta_j)$ data.
Of course for this ansatz to work you need that the parabola ansatz gives a decent fit for the data. If the behaviour is more complicated you could try more complex models. For example convex functions are closed under addition (with positive scalars), hence you could try
$$ f(s,v) = \sum_j w_j(s) g_j(v) $$
For a set of fixed convex functions $g_j$ and non-negative functions $w_j$ determined via regression.
