Course MT3818 Topics in Geometry

Previous page
(Full finite symmetry groups in 3 dimensions)
Contents Next page
(Affine theorems)

Affine Geometry

Recall from an earlier section that a Geometry consists of a set S (usually Rn for us) together with a group G of transformations acting on S.
We now examine some natural groups which are bigger than the Euclidean group. Although the geometry we get is not Euclidean, they are not called non-Euclidean since this term is reserved for something else.

Definition
An affine transformation or affinity of Rn is one of the form Ta comp L with Ta a translation and L belongs GL(n, R).
The group of all such transformations is called the Affine group and is written A(Rn).

Examples

  1. (x, y) goesto (1 + 2x, 1 + 2y)


  2. (x, y) goesto (1 + x + y, 2 + y)


Note that affine transformations do not preserve distance or angle. Area or volume is not preserved either.

They do preserve some geometric properties.

  1. Collinearity

    If A, B and C are collinear, so are their images under any affine map.

    More generally, we have:

    Definition

    A translation of a linear subspace of Rn is called an affine subspace.

    For example, any line or plane in R3 is an affine subspace.

    Theorem
    Affine transformations map affine subspaces to affine subspaces.

    Proof
    This follows from the fact that linear maps map linear subspaces to linear subspaces.

  2. Parallelism

    Theorem
    Parallel lines are mapped to parallel lines.

    Proof
    Two parallel lines are lines in an affine plane which do not meet. Since affine transformations preserve planes and incidence, their images lie in an affine plane and do not meet. Hence they are parallel.

  3. Ratios

    Theorem
    Ratios of lengths of intervals on any line are preserved.

    Proof
    This follows because such ratios are preserved by linear maps and by translations.

    In fact ratios of lengths on pairs of parallel lines are preserved.
    The property of intermediacy (one point being between two others on a line) is also preserved.



Previous page
(Full finite symmetry groups in 3 dimensions)
Contents Next page
(Affine theorems)

JOC March 2003