I am watching a series of Lectures by N.Arkani-Hamed on the Positive Geometry of the Real World. And in the second lecture he gives a crash course on the projective geometry and considers some examples of advanteges in working in projective way.
Conformal (Moebius) transformations of $\mathbb{C}$ are given by: $$ z \rightarrow\frac{az+b}{cz + d} $$ Which from the point of view of variable $z$ is a nonlinear transformation. But in projective space $\begin{pmatrix}x_0 \\ x_1 \end{pmatrix} \sim \lambda \begin{pmatrix}x_0 \\ x_1 \end{pmatrix}$, the transformations becomes a linear one: $$ \begin{pmatrix}x_0 \\ x_1 \end{pmatrix} \rightarrow M \begin{pmatrix}x_0 \\ x_1 \end{pmatrix} $$ Where $M$ is a $2 \times 2$ matrix. And this approach can be extended to higher dimensions, where are non-linear transformation in inhomogeneous coordinates becomes a linear one in projective space.
Inhomogeneous coordinates are a subspace in the whole projective space, by $$ \begin{pmatrix} 1 \\ z \end{pmatrix} $$ One covers all the projective space, except one point.
My question is - are there other examples or classes of cases, when a non-linear or complicated transformation on a certain subset/subspace becomes a simple one on some extension of it?