I am reading this article in wikipedia and confused by their definition of coordinate transformation.
With every bijection from the space to itself two coordinate transformations can be associated: Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation). And such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation). For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more.
I am getting confused on what is being transformed here, the geometric points of the space or the coordinates of the points. Does a bijection from the space to itself imply the points are being moved, and the coordinate system is the same. I am familiar with points of $\mathbb{R}^n$ being transformed $\vec{ x } \to T(\vec {x})$ in the same coordinate space or same axes. I do notice that moving the center or origin of the unit circle $x^2 + y^2 -1 = 0$ to the point $(2,3)$ results in the new equation $(x-2)^2 + (y-3)^2 - 1 = 0$ , but I don't know what the coordinate system has to do with it. And how to interpret a coordinate transformation. Also wikipedia mentions active and passive transformations. Active transformations will map points to new points in the same coordinate system, while a passive transformation leaves the points alone and transforms the coordinate system or axes (sometimes there isnt a smooth way to transform the axes, such as cartesian to polar). They are also equivalent somehow.
Also I am getting confused about what points of $\mathbb{R}^n$ are. Rigorously they should be considered independent of geometry (such as the points of a plane), and strictly should be treated as an ordered list of numbers, i.e. n-tuple, which can represent a location of a geometric point after a coordinate system has been chosen. But we often associate geometric points in a plane with their coordinates and omit the distinction. For example often I see mathematicians writing "let $\mathbb{R}^2$ represent the euclidean plane" - but it doesn't, $\mathbb{R}^2$ is just a set of ordered pairs $\mathbb{R} \times \mathbb{R}$.
I apologize if this question seems muddled. There also might be a chicken egg problem here too, i.e. what came first, points or coordinates (of course historically geometric points came before coordinate systems). Also I want to be clear on coordinate transformation, is it a change in the coordinates and not a change in the visual presentation of the points. We could graph the new coordinates on a new graph separately, nice orthogonal axes, as is done in some presentations of a change to polar coordinate for integration, e.g. $\iint f(x,y) ~dA= \iint f(x(r,\theta), y(r,\theta)) |J| dr d\theta$
If what follows is too abstract I apologize, but maybe it will spark your interest in a different, more general, point of view.
From a differential geometry perspective: Let $(M,\mathcal{O})$ be a $d-$dimensional manifold. Then, a pair $(U,x)$ where $U \in \mathcal{O}$ and $x : U \to x(U) \subseteq \mathbb{R}^{d}$ is a homeomorphism, is said to be a chart of the manifold.
The component functions (or maps) of $x : U \to x(U) \subseteq \mathbb{R}^{d}$ are the maps
$$x^{i} : U \to \mathbb{R}$$
where $x^{i}(p) := \text{proj}_{i}(x(p))$ for $1 \leq i \leq d$. Since $x(p) \in \mathbb{R}^{d}$, these component functions can also be seen as $x(p) = (x^{1}(p),...,x^{d}(p))$. The $x^{i}(p)$ are called the coordinates of the point $p \in U$ with respect to the chart $(U,x)$. If we look at it this way, the points are considered "first" and the coordinates of such points are assigned via charts. Some also call these charts a coordinate system for that patch of the manifold.
Then, the coordinate transformations you are familiar with are formulated by introducing another chart $(V,y)$ where $p \in U \cap V \neq \emptyset$ such that $y \circ x^{-1} : x(U \cap V) \to y(U \cap V)$.