Affine geometry

In mathematics, affine geometry is what we have left from Euclidean geometry when we ignore ideas about distance and angle. Affine geometry looks at things that stay the same, no matter how far apart they are or how they are turned.

One big idea in affine geometry is parallel lines. These are lines that never meet, even if you stretch them out forever. A rule called Playfair's axiom tells us that for any line and a point not on that line, there is exactly one line parallel to the first line that goes through that point.

We compare shapes in affine geometry using affine transformations. These are special ways to move or change shapes while keeping lines parallel and points lined up. Affine geometry can be studied in two ways: through synthetic geometry or using linear algebra.

History

In 1748, Leonhard Euler first used the word affine in his book Introductio in analysin infinitorum. In 1827, August Möbius wrote about affine geometry in his book Der barycentrische Calcul.

After Felix Klein’s Erlangen program, affine geometry was seen as a bigger idea than Euclidean geometry. In 1918, Hermann Weyl used affine geometry in his book Space, Time, Matter to talk about vectors.

Systems of axioms

There are different ways to explain affine geometry using rules.

Pappus' law

Affine geometry studies parallel lines. One rule about parallels from Pappus of Alexandria is important:

• Imagine points A, B, C on one line and points A', B', C' on another. If the lines AB' and A'B are parallel and the lines BC' and B'C are parallel, then the lines CA' and C'A are also parallel. (This is the affine version of Pappus's hexagon theorem).

The full set of rules uses point, line, and line containing point as basic ideas:

• Two points are always on one line.
• For any line L and any point P not on L, there is exactly one line that passes through P and does not touch L. This line is called parallel to L.
• Every line has at least two points.
• There are at least three points that are not all on the same line.

According to H. S. M. Coxeter, these rules are interesting because they can lead to many ideas that work not only in Euclidean geometry but also in Minkowski's geometry of time and space.

Ordered structure

An explanation of plane affine geometry can be created from the axioms of ordered geometry by adding two more rules:

1. (Affine axiom of parallelism) For a point A and a line r not passing through A, there is at most one line through A that does not meet r.
2. (Desargues) For seven different points A, A', B, B', C, C', O, where AA', BB', CC' are different lines through O, and AB is parallel to A'B', and BC is parallel to B'C', then AC is parallel to A'C'.

The idea of parallelism in affine geometry creates an equivalence relation on lines. Because the rules of ordered geometry include ideas that match the real numbers, these ideas also apply here, making this a description of affine geometry using real numbers.

Ternary rings

Main article: Planar ternary ring

The first non-Desarguesian plane was noticed by David Hilbert in his Foundations of Geometry. The Moulton plane is a common example. To explain such geometries and those where Desargues theorem is true, the idea of a ternary ring was created by Marshall Hall.

In this method, affine planes are built from ordered pairs from a ternary ring. If a certain property holds in the affine plane defined by a ternary ring, there is an equivalence relation between "vectors" from pairs of points in the plane. Also, the vectors form an abelian group under addition; the ternary ring follows right distributivity:

( a + b ) c = a c + b c . {\displaystyle (a+b)c=ac+bc.} !{\displaystyle (a+b)c=ac+bc.}

Affine transformations

Main article: Affine transformation

Affine transformations are special changes in shapes that keep lines parallel and keep the same ratios of distances on those lines. They can stretch, shrink, or move shapes, but parallel lines will always stay parallel.

Some important facts about shapes stay the same even after these changes. For example, in any triangle, the point where lines from each corner to the middle of the opposite side all meet is one thing that doesn’t change with affine transformations. These unchanging facts help make calculations easier. For instance, the way areas of certain lines inside a triangle relate to the whole triangle’s area is the same for all triangles, no matter their shape.

Affine space

Main article: Affine space

Affine geometry is the study of spaces where we look at points and lines. We do not measure distances or angles in these spaces. These spaces are called affine spaces. We can describe them using numbers, or we can study how points and lines relate to each other. One important feature of these spaces is that they have special lines called parallel lines. These lines never meet. This helps us understand many important ideas in geometry.

Projective view

Affine geometry is a type of math that fits between Euclidean geometry and projective geometry. It is like Euclidean geometry but without ideas about distance and angles. You can also think of it as projective geometry with a special line or plane set aside for points at infinity. In affine geometry, parallel lines act in a special way, following the parallel postulate. This geometry helps create the rules for Euclidean geometry when we talk about perpendicular lines. It also supports Minkowski geometry with the idea of hyperbolic orthogonality. An affine transformation is a type of projective transformation that keeps normal points separate from points at infinity.