Finding triangulations on 2D space by projecting lower hull of 3D So we know that we can get the Delauney triangulation of a polygon if we map all points to the 3D space such that $p'=(p.x,p.y,p.x^2+p.y^2)$, compute the lower hull of that polyhedron, and then project it back to 2D space.
My question is, can we find other mappings which will give us another triangulation of the polygon if we apply this method? Suppose we already have all possible triangulations and we only want to find that mapping.
 A: I can give you some relevant facts, but I don't understand what the question is. 
If $f$ is any convex function $\mathbb{R}^2 \to \mathbb{R}$, and we have a set of points $(x_i, y_i)$ in $\mathbb{R}^2$, then we can take the lower convex hull of the points $(x_i, y_i, f(x_i, y_i))$ in $\mathbb{R}^3$. Projecting down to $\mathbb{R}^2$ gives a polyhedral subdivision of the convex hull of the original points. If $f$ is chosen generically, it will be a triangulation. See Chapter 2.2 in Triangulations, by de Loera, Rambau and Santos.
I have heard this called a "regular triangulation", "coherent triangulation", "convex triangulation" or "generalized Delauney triangulation".  The above mentioned authors say that "Gale triangulation" or "weighted Delauney triangulation" also are used. 
The set of all possible regular triangulations is organized by a combinatorial object known as the secondary polytope (chapter 5 in Triangulations).
Not all triangulations are regular, here is the standard example of a triangulation which is not induced by any convex function. 

The points can be taken to have coordinates $(0,0)$, $(4,0)$, $(0,4)$, $(1,1)$, $(2,1)$, $(1,2)$. 
Given a triangulation, testing whether it is regular is a matter of testing whether a finite number of linear inequalities are compatible, which can be done by linear programming.
I hope some subset of this answers your question.
