Reading for students: patchworking
More systematically about the same technique:
A new point of view on the patchworking relating it to the Maslov dequantiztion of the semiring of positive real numbers. Here it is shown how to degenerate real algebraic geometry to the piecewise linear geometry, and that the patchworking is the opposite deformation, that is a quantization of a piecewise linear variety. This is a talk at the Third European Congress of Mathematicians (Barcelona, 2000).