# Affine automorphisms of real affine varieties

Let $$V \subset \mathbb{R}^d$$ be a real affine variety. I'm hoping I will not butcher existing nomenclature too badly if I say that for the purposes of this question an affine automorphism of $$V$$ is an affine map $$T:\ x \mapsto Ax + b$$ ($$A \in \mathbb{R}^{d \times d}$$, $$b \in \mathbb{R}^d$$) such that $$T(V) = V$$. Is there a convenient way to describe the group of affine automorphisms of a given variety? Is it possible to describe all groups that arise this way?

Motivation: If we restrict further to "scaling automorphisms", namely maps of the form $$x \mapsto \lambda x + b$$ ($$\lambda \in \mathbb{R}$$, $$b \in \mathbb{R}$$), then the group of scaling automorphisms of $$V$$ is readily described. Namely, these are precisely the maps of the form $$x \mapsto \lambda (x-x_0) + x_0$$ where $$\lambda \in \mathbb{R}$$ and $$x_0$$ is a point such that any line passing through $$x_0$$ and at least one other point of $$V$$ is entirely contained in $$V$$ (plus, maps of the form $$x \mapsto x+v$$ where $$v$$ is a direction such that any line $$y_0 + \mathbb{R}v$$ is either contained in $$V$$ or disjoint from $$V$$, which morally corresponds to taking $$x_0$$ as a point in infinity). What is more, the set of possible "centres" $$x_0$$ is an affine space. I'm curious if any geometric picture like this emerges also for general affine maps.