Affine space — Safekipedia Adventurer

In mathematics, an affine space is a geometric structure that helps us study shapes and positions. It is like a simpler version of Euclidean spaces. In an affine space, we don’t worry about measuring distances or angles. Instead, we look at how points and lines relate to each other, especially when lines are parallel.

In ⁠ R 3 {\displaystyle \mathbb {R} ^{3}} ⁠, the upper plane (in blue) P 2 {\displaystyle P_{2}} is not a vector subspace, since 0 ∉ P 2 {\displaystyle \mathbf {0} \notin P_{2}} and ⁠ a + b ∉ P 2 {\displaystyle \mathbf {a} +\mathbf {b} \notin P_{2}} ⁠; it is an affine subspace. Its direction (the linear subspace associated with this affine subspace) is the lower (green) plane ⁠ P 1 {\displaystyle P_{1}} ⁠, which is a vector subspace. Although a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } are in ⁠ P 2 {\displaystyle P_{2}} ⁠, their difference is a displacement vector, which does not belong to P 2 , {\displaystyle P_{2},} but belongs to vector space ⁠ P 1 {\displaystyle P_{1}} ⁠.

The main things in an affine space are called points. These points have no size or shape. We can use them to make lines and planes. If we pick any two points, we can draw a straight line between them. If we pick three points that aren’t in a straight line, we can make a plane.

One special idea in affine space is about parallel lines. These are lines in the same plane that never meet each other. For any line, we can draw a parallel line through any other point.

Unlike in a vector space, an affine space does not have a special starting point called the origin. This means we can’t add points together or multiply them by numbers. But we can still describe how to move from one point to another using something called displacement vectors. These vectors tell us how far and in which direction to move. By using special rules called affine combinations, we can find new points that stay in the same space.

Informal description

Origins from Alice's and Bob's perspectives. Vector computation from Alice's perspective is in red, whereas that from Bob's is in blue.

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 is the origin.

When they add vectors together, they might get different results because they use different starting points.

But if they use a special 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. 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 has points and vectors. Points are places in space with no size or shape. Vectors show how to move from one point to another.

In an affine space, you can add a vector to a point to find another point. For example, if you start at point A and add vector V, you reach a new point. This is like moving in a straight line. The space has rules that keep movements consistent and reversible. This helps us understand how points relate to each other without measuring 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 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 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 rule that works between two affine spaces. It keeps some important geometric ideas, like how points line up and distances along straight lines, but it doesn’t have to keep exact distances or angles.

One important idea is that an affine map can be explained by picking one point in the starting space and then using a linear map — which works with vectors and directions — to find where every other point should go. This makes affine maps useful in things like computer graphics and physics.

Main articles: Affine transformation and Affine group

Vector spaces as affine spaces

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

If we have another affine space that uses the same vector space, we can pick any point in that space. By matching this point to the origin, the two spaces become the same in a clear 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 an affine space with extra rules. These rules help us measure distances and angles using something called an inner product.

An affine property is a feature that works in affine spaces. We do not need to measure distances or angles to see these properties. 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 mix points using special rules. If we have points and numbers that add up to zero, we can make a special vector that stays the same, no matter where we begin.

When the numbers add up to one, we can find a special point called the barycenter. This point is a balanced mix of the original points, based on those numbers. It’s like finding the middle point among several places. We also call this an 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. We can make this space by mixing the points in special ways.

A set of points is affinely independent if no point can be made from the other points. An affine basis is the smallest group of points that can create 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, you can give each point a weight. The point you are describing is like the "center of mass" of these weighted points. This helps when studying 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 move and change shapes in space. It keeps some things the same, like which lines are parallel.

One example of an affine transformation is projection. This is like shining a light on an object and seeing its shadow on a wall. In geometry, projecting means moving points from one space to another while keeping lines parallel. This idea helps us study 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 create an affine plane by starting with a projective plane and removing one line and all its points. You can also do the opposite: take an affine plane and add a special line at infinity to get a projective plane. These ideas work in higher dimensions as well.

Certain changes to projective space that do not affect the affine space also give us changes to the affine space. Every affine transformation can be expanded to a projective transformation. This means 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 shape inside an affine space. It is made of points where certain polynomial equations are true. To study these polynomials, we need to choose a system of coordinates, called an affine frame. This helps us solve the equations.

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

Ring of polynomial functions

By choosing coordinates, we can match polynomial functions on the space with regular polynomials. This creates a special mathematical structure called a k-algebra. The way these polynomials are organized 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 the affine varieties themselves. This topology works for any field of numbers.

Cohomology

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