The diffeomorphism groups of smooth surfaces are classical objects in low dimensional topology. In this talk, we will introduce the deformation spaces of geodesic triangulations as natural discrete analogues of these groups. We will talk about a general framework to identify their homotopy types. The key ingredient of this framework is Tutte's embedding theorem for planar graphs and its generalization to negatively curved surfaces. This is joint work with Tianqi Wu and Xiaoping Zhu.