Affine Geometry - Affine Transformations

Affine Transformations

Geometrically, affine transformations (affinities) preserve collinearity: so they transform parallel lines into parallel lines and preserve ratios of distances along parallel lines.

We identify as affine theorems any geometric result that is invariant under the affine group (in Felix Klein's Erlangen programme this is its underlying group of symmetry transformations for affine geometry). Consider in a vector space V, the general linear group GL(V). It is not the whole affine group because we must allow also translations by vectors v in V. (Such a translation maps any w in V to w + v.) The affine group is generated by the general linear group and the translations and is in fact their semidirect product . (Here we think of V as a group under its operation of addition, and use the defining representation of GL(V) on V to define the semidirect product.)

For example, the theorem from the plane geometry of triangles about the concurrence of the lines joining each vertex to the midpoint of the opposite side (at the centroid or barycenter) depends on the notions of mid-point and centroid as affine invariants. Other examples include the theorems of Ceva and Menelaus.

Affine invariants can also assist calculations. For example, the lines that divide the area of a triangle into two equal halves form an envelope inside the triangle. The ratio of the area of the envelope to the area of the triangle is affine invariant, and so only needs to be calculated from a simple case such as a unit isosceles right angled triangle to give i.e. 0.019860... or less than 2%, for all triangles.

Familiar formulas such as half the base times the height for the area of a triangle, or a third the base times the height for the volume of a pyramid, are likewise affine invariants. While the latter is less obvious than the former for the general case, it is easily seen for the one-sixth of the unit cube formed by a face (area 1) and the midpoint of the cube (height 1/2). Hence it holds for all pyramids, even slanting ones whose apex is not directly above the center of the base, and those with base a parallelogram instead of a square. The formula further generalizes to pyramids whose base can be dissected into parallelograms, including cones by allowing infinitely many parallelograms (with due attention to convergence). The same approach shows that a four-dimensional pyramid has 4D volume one quarter the 3D volume of its parallelopiped base times the height, and so on for higher dimensions.