Affine geometry

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring the ideas of distance and angle. This means affine geometry focuses on properties that do not depend on how far things are or how they are angled.

One of the main ideas in affine geometry is parallel lines. These are lines that never meet, no matter how far they are extended. A key rule in affine geometry, called Playfair's axiom, says that for any line and a point not on that line, there is exactly one line parallel to the first line that passes through the point.

Comparisons of shapes in affine geometry are made using affine transformations. These are special ways of moving or changing shapes that keep lines parallel and points in line with each other. Affine geometry can be studied in different but equally valid ways, either through synthetic geometry or using linear algebra.

History

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

Later, after Felix Klein’s Erlangen program, affine geometry became known as a broader version of Euclidean geometry. In 1918, Hermann Weyl used affine geometry in his book Space, Time, Matter to explain ideas about vectors.

Systems of axioms

Several ways to explain affine geometry using rules have been suggested.

Pappus' law

Because affine geometry looks at parallel lines, one rule about parallels noted by 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. A plane has the "minor affine Desargues property" when two triangles in parallel perspective, with two parallel sides, must also have their third sides parallel. If this 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 is linear and 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 maintain the ratios of distances along those lines. They can stretch, shrink, or shift shapes but will always keep parallel lines parallel.

Some important facts about shapes stay the same even after these transformations. For example, in any triangle, the point where lines from each corner to the middle of the opposite side all meet is an example of something 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 about the study of spaces where we look at points and lines without measuring distances or angles. These spaces, called affine spaces, can be described using numbers, but they can also be studied just by looking at how points and lines relate to each other. One key feature of these spaces is that they include special lines called parallel lines, which never meet. This helps us understand many important properties in geometry.

Projective view

Affine geometry sits between Euclidean geometry and projective geometry. Think of it as Euclidean geometry without the ideas of distance and angle. It can also be seen as projective geometry with a special line or plane set aside to show points at infinity. In affine geometry, parallel lines behave in a special way, following the parallel postulate. Affine geometry helps build the rules for Euclidean geometry when we define perpendicular lines and also supports Minkowski geometry with the idea of hyperbolic orthogonality. An affine transformation is a kind of projective transformation that keeps regular points separate from points at infinity.