I have a set of points in $\mathbb{R}^2$ given as $G(t;z_0,\theta_0)$ where $z_0 \in \mathbb{R}^2$ and $\theta_0$ represents a specific orientation. I have proved that,
$G(t;z_0,\theta_0) = z_0 + \Pi_{(\theta_0)}G(t,(0,0),0)$.
where $\Pi_{(\theta_0)}$ is the rotation operation.
This indicates computing this set at a specific point $z_0,\theta_0$ is the same as computing it at the origin (with 0 orientation) and applying a linear transformation.
My Question is: What do I call this property of the set? Can I say "$G$ is closed under linear transformation" or " $G$ is invariant under linear transformation". I am looking for a mathematically formal statement which explains this property.