Fixed Point in geometry
A point is called a fixed point (or invariant point) of a geometric transformation if its position remains unchanged after the transformation is applied.
In other words, if a point keeps the same coordinates before and after the transformation, it is considered a fixed point of that transformation.
Formally, given a function \( T: \mathbb{R}^2 \to \mathbb{R}^2 \), a point \( P \) is a fixed point if \( T(P) = P \); that is, the point is mapped to itself.
For example, in a rotation on a plane, the center \( P \) of the rotation is a fixed point because it stays in the same position.
There can be multiple fixed points.
For instance, in an axial symmetry with respect to a line \( r \), every point on the symmetry axis \( r \) remains unchanged.
In some cases, every point in a transformation is fixed.
For example, a full 360° rotation of a plane figure around a central point \( P \) leaves all points exactly where they started.
In this case, every point is fixed, and the geometric figure is said to be pointwise fixed under the transformation.
When all points of a figure are fixed points, the geometric transformation is called the identity.
Note: It may happen that certain points are mapped to other points of the same figure. In such cases, they are not considered fixed points. For example, consider the line \( y = x \) and the geometric transformation $$ T: \begin{cases} x' = x + 1 \\ y' = y + 1 \end{cases} $$ Every point on the transformed figure corresponds to another point on the original line, even though no point remains fixed.
It is worth noting that, in this case, the figure is a invariant figure under an isometry: the distance between any pair of points remains unchanged. However, despite this invariance, the individual points are not fixed points of the geometric transformation.
On the other hand, some transformations have no fixed points at all.
For instance, in a translation, no point remains fixed unless the translation is trivial (i.e., zero displacement).
These are just a few examples of fixed points in geometric transformations.
Note. The concept of a fixed point extends seamlessly from two-dimensional to three-dimensional space. A point is considered fixed if its coordinates remain unchanged after the transformation. The only difference in three-dimensional space is that a point is described by three coordinates, \((x, y, z)\), instead of two, \((x, y)\).
And so on.
