Introduction

Affine transformations are transformations that preserve collinearity of points.  In other words, three points that lie on a line will continue to lie on a line after being applied to an affine transformation.  Affine mappings are of the form Ax + b where A is an nxn square matrix and x and b are vectors in .  This site explores five classifications of affine transformations:  Translations and Parabolic Shearing, Scaling by Real Exponentials, Rotations by Complex Exponentials, and (2 more i need to come up with better names for that aren't yet included in the java applet), all of which are complete affine structures on the non-Euclidean space.  is a representation of a 2-dimensional equivalence class /, also known as a torus.  Affine geometry is closely related to projective geometry because the set of affine transformations is a subgroup of the general linear transformations on the projective plane, so affine space is embedded in the projective space of - {0}.

Equivalence Classes

X = X' if X - X' , where is the set of integers.

In other words, when we are talking about equivalence relations, we are talking about elements of mod .  A fundamental domain is needed to establish an equivalence class.

Fundamental Domain

Let have a group G that acts on a subspace X of , then D is a fundamental domain if and only if

1)  Every x X is equivalent to some x' D.  This implies D is a manifold with a boundary.

2)  If two points in D are equivalent, then they are on the edge.

For example, [0, 1] is a fundamental domain for acting on .

If we take a 2-dimensional equivalence class, we can define a coordinate system corresponding to points on a torus.  The horizontal edges of our fundamental domain associate, as do the vertical edges, and all four corners meet at one vertex when mapped to an actual torus.

To illustrate:

Where the first transformation associates the vertical edges of the parallelogram on the left, and the second transformation associates the horizontal edges of the newly formed cylinder.  As the boundary of the square has equivalence relations, neighborhoods of points on the boundary are only partial neighborhoods in the plane, but are associated with other partial neighborhoods in the plane that come together as complete neighborhoods on the torus as illustrated below:

Affine Space

A k-dimensional affine subspace of is a k-dimensional hyperplane.  See the projective geometry page to see why affine spaces are embedded in projective hyperplanes.

Affine Transformations

Affine transformations are in the group of homogeneous linear transformations that take x to ax + b.  Isometries are area preserving maps with |a| = 1.  

Next: Translations and Parabolic Shearing

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