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.


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