Affine space

Points are tiny spots in space.

They have no size or shape.

You can draw a straight line through two points. You can also make a flat plane with three points.

Lines can run side by side. These are called parallel lines. They will never meet.

In this space, there is no special starting point. We call this a zero point the origin. In an affine space, we forget where the origin is.

We can move a point to a new spot. This move is called a translation.

Imagine you are looking at a map. You can see many points. These points are just locations. They have no size or shape.

In a normal math space, there is a special starting point. We call this the origin. But in an affine space, we forget the origin. We do not pick one spot to be the center. This makes the space feel very free.

Even without an origin, we can still move around. We do this using translations. A translation is a move from one point to another. We use a vector to show this move. A vector is like an arrow that shows direction and how far to go.

In this space, we can also find parallel lines. These are lines that stay the same distance apart. They will never meet, even if they go on forever. We can also find flat shapes called planes. A plane is a two-dimensional surface. You can make a plane using three points. You can also make a line using two points. These parts of the space are called affine subspaces.

An affine space is a special kind of geometric structure. It is built to focus on certain ideas while ignoring others. In many math spaces, we care about how long a line is or what the angle is between two lines. An affine space does not need those measurements to work. Instead, it focuses on things like parallelism and the ratio of lengths. It is the setting for a type of math called affine geometry.

The main objects in this space are called points. These points are just locations with no size or shape. They are zero-dimensional, which means they have no width or height.

In this space, we can build many different shapes. If you take two points, you can draw an infinite straight line through them. This line is one-dimensional. If you take three points that are not in a single line, you can draw a two-dimensional plane. You can even create higher-dimensional shapes called affine subspaces. These subspaces are flat parts of the larger space. One important rule in an affine space is about parallel lines. These are lines that lie in the same plane but never meet each other.

To understand how it works, we must look at how it differs from a vector space. In a vector space, there is always one special point called the origin. In an affine space, we forget which point is the origin. There is no predefined way to add two points together. However, we can use something called a translation to move between points. A translation uses a vector, which is like an arrow showing a direction and a distance. When you add a vector to a point, you get a new point.

There is a famous way to think about this using two people named Alice and Bob. Imagine Alice thinks one point is the origin, but Bob thinks a different point is the origin. If they try to add vectors, they might get different answers because they start in different places. However, they will always agree if they use something called an affine combination. This is a special kind of math where the coefficients, or the numbers used, must add up to exactly one. When this happens, Alice and Bob will always arrive at the same point. This shared understanding is what makes the affine structure work.

Many other math concepts are built on top of these ideas. For example, a Euclidean space is a type of affine space. A Euclidean space is a more specific version that includes measurements for distance and angles. In finite dimensions, an affine subspace can be found by solving a specific kind of math problem called an inhomogeneous linear system. The directions of these subspaces are found using a different system called a homogeneous linear system. This shows how the simple idea of moving through space connects to much larger math rules.

An affine space is a geometric structure that generalizes the properties of Euclidean spaces. It focuses on specific geometric relationships while ignoring others. Specifically, an affine space is independent of concepts like distance or the measurement of angles. Instead, it preserves properties related to parallelism and the ratio of lengths for parallel line segments. This structure serves as the fundamental setting for affine geometry.

The primary objects in an affine space are called points. These points are considered zero-dimensional locations that possess no size or shape. From these points, one can construct various higher-dimensional structures. Any two points define an infinite straight line, which is a one-dimensional set. Any three points that are not collinear define a two-dimensional plane. In general, points in a general position can form a k-dimensional flat, also known as an affine subspace.

A defining characteristic of an affine space is its notion of parallelism. Within a single plane, parallel lines are pairs of lines that never meet. Given any line and any point in the space, exactly one line can be drawn through that point that is parallel to the first. All lines that are parallel to one another are said to share a common direction. This direction is represented by a linear subspace. While points themselves cannot be added together, they can be moved using translations. A translation is a vector that moves a starting point to a new resulting point.

To understand the mechanism of an affine space, it helps to compare it to a vector space. In a vector space, there is a distinguished point called the origin, which is the zero vector. An affine space can be thought of as a vector space where the origin has been forgotten. Because there is no fixed origin, you cannot arbitrarily add two points together. Instead, you use the associated vector space to find the difference between points. This difference is a free vector, often called a translation or displacement vector. Adding such a vector to a point results in a new point in the same affine space.

A helpful way to visualize this is through the perspectives of two people, Alice and Bob. Suppose Alice believes point A is the origin, but Bob believes point B is the origin. If they attempt to add vectors, they may calculate different results because their starting points differ. However, they will always agree if they use an affine combination. An affine combination is a weighted sum of points where the numerical coefficients sum to exactly one. Under this specific condition, the choice of origin does not change the final result. This shared result is what defines the affine structure.

Mathematically, an affine space is defined by a set of points and an associated vector space. The relationship between them is a transitive and free action of the vector space's additive group on the set of points. This means that for any two points, there is a unique vector that connects them. This process is often described using Weyl's axioms. These axioms involve the existence of unique translations and the satisfaction of the parallelogram property. The parallelogram property ensures that the geometric relationships remain consistent across the space.

Affine subspaces, also called flats or linear varieties, are subsets of the affine space. A subset is an affine subspace if there is a point such that the set of vectors formed by its points is a linear subspace. The linear subspace associated with an affine subspace is called its direction. Two subspaces are considered parallel if they share the same direction. In finite dimensions, these subspaces can be identified as the solution sets of inhomogeneous linear systems. The directions of these subspaces correspond to the solutions of the related homogeneous linear systems.

Finally, affine spaces connect to many other mathematical fields. A Euclidean space is a specific type of affine space where the associated vector space is a real inner product space. In such a space, you can finally define distances and angles using a positive-definite quadratic form. Additionally, any vector space can be viewed as an affine space by simply treating the zero vector as an origin. This allows mathematicians to switch between viewing elements as points or as displacement vectors depending on the problem they are solving.