Affine space

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. Affine space provides a framework for studying geometry without needing to measure distances or angles directly. Instead, it focuses on how points and lines relate to each other, especially through parallelism.

The fundamental objects in an affine space are called points. These points have no size or shape and can be used to form lines and planes. Through any two points, a straight line can be drawn, and through three non-collinear points, a plane can be formed. One key feature of affine space is the idea of parallel lines—lines in the same plane that never meet. For any given line, it's possible to draw a parallel line through any other point in the space.

Unlike in a vector space, an affine space does not have a special point called the origin. This means you cannot add points together or multiply them by numbers directly. However, you can still describe movements from one point to another using something called displacement vectors. These vectors show how far and in what direction you need to move to get from one point to another. By combining points in specific ways called affine combinations, you can find new points that lie within the same space.

Informal description

An affine space is like a vector space where we don't know which point is the starting point, or "origin." Imagine two friends, Alice and Bob. Alice knows a certain point is the origin, but Bob thinks a different point, called p, is the origin. When they add vectors together, they might get different results because they use different starting points.

However, if they use a special kind of combination where the total of their numbers adds up to 1, they will both end up at the same point. This special combination is called an affine combination, and it helps them agree on where points are, even though they disagree on where the origin is. This idea of using affine combinations is what makes up an affine space.

Definition

An affine space is a special kind of geometric space. It is made up of points and vectors. Think of points as locations — places in space without any size or shape. Vectors tell us how to move from one point to another.

In an affine space, we can add a vector to a point to get another point. For example, if you start at point A and add vector V, you reach a new point. This works like moving in a straight line. The space has rules that make sure movements are consistent and can be reversed, which helps us understand how points relate to each other without needing to measure distances or angles.

Affine subspaces and parallelism

An affine subspace is a special part of an affine space. Think of it like a straight line or a flat plane inside the space. These subspaces are important because they help us understand how points and directions relate in the space.

Two affine subspaces are called parallel if they have the same direction. This means they are aligned in the same way, even if they are in different places. For any point in the space, there is exactly one affine subspace with a given direction that passes through that point. This idea helps us describe how things are positioned relative to each other in the space.

Affine map

An affine map is a special kind of function between two affine spaces. It keeps certain geometric properties, like how points line up and distances between points along straight lines, but it doesn't need to keep exact distances or angles.

One key idea is that an affine map can be described by choosing a single point in the starting space and then using a linear map — which deals with vectors and directions — to figure out where every other point goes. This makes affine maps very useful in areas like computer graphics and physics.

Main articles: Affine transformation and Affine group

Vector spaces as affine spaces

Every vector space can also be viewed as an affine space. In this view, each element of the vector space can be seen as either a point or a vector. The zero vector, which represents the starting point, is often called the origin.

When we have another affine space using the same vector space, picking any point in that space lets us match it perfectly with the vector space. This matching happens by aligning the chosen point with the origin, making the two spaces essentially the same in a clear and natural way.

Relation to Euclidean spaces

Euclidean spaces, like the lines, planes, and spaces we learn about in school, are special types of affine spaces. In modern math, a Euclidean space is defined as an affine space with extra rules that let us measure distances and angles using something called an inner product.

An affine property is a feature that works in affine spaces without needing to measure distances or angles. Things like parallelism and what it means for a line to be a tangent to a shape are affine properties. Properties that need angles, like a normal line, are not affine properties. Affine properties stay the same even when we move or stretch the space in certain ways, called affine transformations.

Affine combinations and barycenter

In an affine space, we can combine points using special rules. If we have points and certain numbers that add up to zero, we can create a special vector that doesn’t change no matter where we start from.

When the numbers add up to one, we can find a special point called the barycenter. This point is a mix of the original points, weighted by those numbers. It’s like finding a balanced center point among several locations. We also call this a affine combination of the points.

Examples

When children solve sums like 4 + 3 or 4 − 2 by counting on a number line, they are using a one-dimensional affine space. Time can also be seen as a one-dimensional affine space, where specific moments are points and durations are distances between them.

The space of energies is another example of an affine space. While we can talk about differences in energy, the idea of absolute energy isn't always meaningful. Physical space is often modeled as an affine space in both everyday and relativistic physics.

Affine span and bases

In an affine space, the affine span of a set of points is the smallest space that includes all those points. It is made by combining the points in specific ways.

A set of points is affinely independent if no point can be made from combining the others too closely. An affine basis is a smallest set of points that can generate the whole space. For a space with dimension n, you need n + 1 points to form a basis.

Coordinates

There are two main kinds of coordinate systems used in affine spaces.

Barycentric coordinates

Barycentric coordinates describe a point as a weighted average of other points. If you have several points in the space, you can assign weights to each point. The point you are describing is the "center of mass" of these weighted points. This helps in understanding shapes like triangles and other polygons.

Affine coordinates

Affine coordinates use a reference point (the origin) and direction vectors. To find the coordinates of any point, you see how far along each direction vector you need to go from the origin to reach that point. This is similar to how we use coordinates on a grid to find exact locations.

Both types of coordinates help us describe positions in space without needing to measure distances or angles directly.

Properties of affine homomorphisms

An affine transformation is a way to change the position and size of shapes in space while keeping certain properties, like parallelism, the same. Unlike simple straight-line changes, affine transformations can also shift the whole space by moving a specific point.

One important example of an affine transformation is projection. This is like shining a light on an object and looking at its shadow on a wall. In geometry, projecting means mapping points from one space to another in a way that keeps things parallel. This idea is very useful in studying shapes and spaces.

Axioms

Affine spaces can be studied using different methods, such as coordinates or by listing rules called axioms. One way to describe affine spaces is by using certain rules. For example, in a flat space, any two points are connected by exactly one line. Also, for any point and any line, there is exactly one line that passes through the point and is parallel to the given line. Finally, there must be at least three points that are not all on the same line.

There are many types of affine spaces that follow these basic rules, including those that are not as straightforward as the usual flat spaces. These ideas help mathematicians understand the relationships between points and lines in space.

Main article: Affine geometry

Relation to projective spaces

Affine spaces are connected to projective spaces. For example, you can get an affine plane by taking a projective plane and removing one line and all its points. You can also go the other way: take an affine plane and add a special line at infinity to get a projective plane. These ideas work in higher dimensions too.

Certain changes to projective space that keep the affine space the same also give us changes to the affine space. Also, every affine transformation can be expanded to a projective transformation, meaning the group of affine transformations is a part of the larger group of projective transformations.

Affine algebraic geometry

In algebraic geometry, an affine variety is a special kind of shape inside an affine space. It is made up of points where certain polynomial equations are true. To work with these polynomials, we need to choose a system of coordinates, called an affine frame. This helps us turn the problem into one about solving equations with variables.

When we pick these coordinates, we can match the affine space with a more familiar space of lists of numbers. This makes it easier to study the shapes defined by the polynomials. The whole affine space itself is also an affine variety because it satisfies the equation of the zero polynomial.

Ring of polynomial functions

By choosing coordinates, we can match up polynomial functions on the space with regular polynomials in several variables. This creates a special kind of mathematical structure called a k-algebra, which behaves nicely under changes of coordinates. The way these polynomials are organized by their degrees does not depend on how we choose our coordinates.

Zariski topology

Affine spaces have a special way of looking at "close" points called the Zariski topology. In this view, the basic closed shapes are exactly the affine varieties themselves. This topology works for any field of numbers and lets us use ideas from topology to study these geometric shapes.

Cohomology

One important feature of affine spaces is that, in a certain mathematical sense called cohomology, they are very simple. This simplicity extends to many other affine varieties as well.