## Morphing

Computational models are used everywhere, from games to movies—e.g., special effects, animations—and **deformations** are the basic tool for animating objects, so we focus on a particular aspect of the vast morphing research field. The key point is: given two different meshes **X** and **Y** undergoing two deformations (called *diffeomorphisms* in this research area), we would like to answer this question: “when it is possible to say that **X** and **Y** are *undergoing the same deformation*”?

If we can answer this question, at least approximately, we could then **transport deformations from a mesh to a different one**, a typical task of the shape analysis research. In particular, it is necessary to perform such a kind of transport when performing group-wise statistical analyses in shape or size and shape spaces. A naive possible answer could be that *the displacement field is the same*, but this gives different results from the one an animator expects. Take the following example:

Let’s assume we deform an object, constituted by three triangles, as in the first two steps; next we want to **apply the same deformation** to object in the third step. As we’re in the plane, and we’re dealing with triangles, everything should work just fine, except **it does not work at all even in a plane**, as we can see in the fourth step. Copy and pasting animations is *not easy* as it might seem.

We assume a different point of view: two templates undergo the same deformation if, given a bijective map between the two underlying affine spaces (i.e., we can map points on one mesh to points in another), the **local metric** (non linear strain) induced by the two diffeomorphisms is the same for corresponding points.