Affine transformation

An affine transformation is a geometric operation that changes the position or shape of an object while following specific rules. In Euclidean geometry, these transformations are often called affinities, a word derived from the Latin "affinis," meaning connected with.

While these movements can change the distance between points or the angles between lines, they must preserve certain fundamental properties. Most importantly, they preserve lines and parallelism. This means that if you start with a straight line, it remains a straight line after the transformation. If two lines are parallel and never meet, they will remain parallel afterward.

To understand the mechanism, it is helpful to view an affine transformation as a combination of two distinct mathematical actions. Every affine transformation is the composition of a linear transformation and a translation. A linear transformation is a function that can rotate, scale, reflect, or shear a shape. However, a purely linear transformation is restricted because it must always keep the origin, or the zero point, in its original place.

A translation solves this problem by acting as a simple slide that moves every point by a specific vector. By combining these two steps, an affine transformation can move a shape anywhere in space and change its orientation or size.

There are many different types of affine transformations that can be applied to a space. Common examples include translation, which slides an object, and scaling, which changes its size. Rotation spins an object around a point, while reflection flips it over a line. Another type is called a shear mapping, which slides parts of an object in a way that tilts its shape.

You can also have a homothety, which is a specific type of scaling. Furthermore, you can combine these transformations in any sequence to create complex new movements. Because these transformations are invertible, you can always perform the exact opposite movement to return to the original state.

Mathematically, these transformations are described using the language of affine spaces and vector spaces. An affine space is a set of points where you can subtract one point from another to find a vector. An affine map between two different spaces is a function that acts linearly on these vectors.

If the dimension of the space is at least two, a semiaffine transformation is a specific type of bijection. This type of map must send every $k$-dimensional subspace to another $k$-dimensional subspace. For example, it must send points to points, lines to lines, and planes to planes. In spaces where the field has at least three elements, this is often referred to as a collineation.

In modern mathematics, we represent these complex movements using augmented matrices. In standard vector algebra, a matrix handles linear maps and a vector handles translations. To combine them, mathematicians use a technique involving homogeneous coordinates. They append a "1" to the end of every vector and add an extra row and column to the matrix.

This allows a single matrix multiplication to perform both the linear change and the translation at once. This method effectively treats the original space as a subset of a higher-dimensional space. This higher-dimensional view is a core concept in projective geometry.

These mathematical tools are highly significant in technical fields like computer graphics and robotics. Using augmented matrices allows computers to combine many different transformations into one single operation. Instead of calculating ten different moves separately, a computer can multiply ten matrices together to get one final instruction.

This efficiency is vital for real-time rendering in video games and for helping robots navigate through physical environments. Computer vision researchers also rely on these principles to help machines interpret and process visual data from the real world.

Beyond engineering, affine transformations reveal beautiful patterns in nature and mathematics. One striking example is found in fractals, such as the Barnsley fern.

In such a fractal, each small part of the shape is related to the whole through a series of affine transformations. This creates a sense of self-similarity, where the pattern repeats at different scales. By studying these transformations, we gain a deeper understanding of how complex structures can be built from simple, repeatable geometric rules.