Point reflection

In geometry, a point reflection (also called a point inversion or central inversion) is a special way a shape can change. Every point moves to the exact opposite side of a chosen center point. This center point, called the inversion center, stays where it is while everything else moves evenly around it.

Simply put, a point reflection makes a shape look like it has been turned inside out through the center point. For example, if you imagine a point in the middle of a piece of paper and flip the paper so that every point moves directly through that middle point to the other side, you have done a point reflection.

Point reflections are useful in many areas of science and math. They help describe the symmetry found in many crystal structures and molecules. When an object looks the same after a point reflection, it is said to have point symmetry. This means it balances perfectly around the center point. This idea helps scientists understand how tiny parts of nature are arranged and how they act.

Terminology

The word "reflection" is used in a broad way when talking about point reflection, though some people prefer the term "inversion." Point reflections are special because doing them twice brings everything back to where it started. This is called an "involution."

In simple terms, a point reflection is its own opposite — applying it twice is the same as doing nothing at all.

When we talk about reflections more narrowly, they usually happen across a line, a plane, or a space that stays unchanged, called a "mirror." But point reflection is a bit different. It’s a special kind of involution where every point moves directly through a central point, flipping to the opposite side at the same distance. This concept is different from another idea in geometry called "inversive geometry," which uses the word "inversion" in a different way.

Main article: Involutions
Main articles: Reflection, Hyperplane, Affine subspace
Diagonalizable, Eigenvalues, Inversive geometry

Examples

In two dimensions, a point reflection is the same as turning something 180 degrees. In three dimensions, it is like turning something 180 degrees and then flipping it over a flat surface. These reflections keep distances the same but might change how the object looks, depending on how many dimensions there are.

Main article: rotation
Main articles: composed, orientation

Formula

In geometry, a point reflection flips every point around a center point. Picture a point and a center; the new point is the same distance on the other side of the center.

If the center is at the origin, the reflection just changes the sign of the point’s coordinates. This keeps distances the same and has only one fixed point—the center of reflection.

Main article: Euclidean geometry
Further information: line segment, vector, mapping, isometric, involutive, affine transformation, fixed point

Point reflection as a special case of uniform scaling or homothety

When we reflect a point across another point called P, and P is at the start point, a point reflection is the same as a special kind of resizing called uniform scaling. In this case, the resizing factor is −1. This is an example of a linear transformation.

If P is not at the start point, a point reflection works like a special type of change called homothety. Here, the homothetic center is at P, and the resizing factor is also −1. This is an example of a non-linear affine transformation.

Main article: Homothety

Point reflection group

When you do two point reflections in a row, it is the same as moving every point by a certain distance in a straight line. This movement is called a translation.

All point reflections and translations together form a special group of movements in geometry. This group is part of the larger set of movements that keep distances the same in shapes.

Point reflections in mathematics

A point reflection is a way of flipping points around a central spot, like looking at the opposite side of a ball through its center. This is similar to turning something 180 degrees, or half a circle.

Symmetric spaces are special shapes where every point has a matching flip across the center. These spaces help mathematicians study groups and shapes in geometry.

Point reflection in analytic geometry

A point reflection moves every point to its mirror image across a fixed center point. If you have a point P and its reflection P', the center point C is exactly halfway between them. You can find the coordinates of P' using these rules:

[ \begin{cases} x' = 2x_c - x \ y' = 2y_c - y \end{cases} ]

If the center point C is at the origin (0,0), the rules become simpler: the reflection of any point (x, y) is just (-x, -y). This shows that a point reflection in two dimensions acts like a half-turn around the center point.

Properties

In flat space with an even number of dimensions, flipping every point around a center point is like doing a series of 180-degree turns. The space still looks the same after this flip.

In our 3D world, this flip looks like turning around 180 degrees and then flipping over. This changes how the space looks.

Some special groups in 3D include this point flip. You can see this in nature and in crystals. This flip is also related to bouncing off a plane, like a mirror image.

Inversion centers in crystals and molecules

Inversion symmetry helps us understand materials. Some molecules have a special point called an inversion center. This means that if you reflect every atom through this point, the molecule still looks the same. For example, six-coordinate octahedra have an inversion center, but tetrahedra do not.

In crystallography, inversion centers help classify crystals. Crystals without inversion symmetry can show special effects like the piezoelectric effect. Real crystals often have irregularities such as distortions or disorder, which can affect their symmetry. Crystals are grouped into thirty-two crystallographic point groups, some of which are centrosymmetric and some are not.

Main article: Crystallographic point group

Further information: Centrosymmetric molecule

Inversion with respect to the origin

Main article: additive inversion

Inversion with respect to the origin is like flipping every point in space to the opposite side of a central point. This central point stays exactly where it is. In simple terms, it's like turning the whole space around that point by 180 degrees.

In math, this is called reflecting through the origin. For example, in a 3D space, a point at (x, y, z) would move to (-x, -y, -z). This flipping keeps distances the same but can change the direction of the space.