Affine Transformations

Up to this point we have studied in modern format mostly the geometry of Euclid. We now turn to transformations that were first introduced by the great mathematician Leonhard Euler (1707–1783). (Euler was introduced in the preceding chapter following Theorem 14.16.) From the meaning of the word affine, we must define an affine transformation as a collineation on the plane that preserves parallelness among lines. So, if l and m are parallel lines and α is an affine transformation, then lines α (l) and α(m) are parallel. However, if β is any collineation and l and m are distinct parallel lines, then β(l) and β(m) cannot contain a common point β(P) as point P would then have to be on both l and m. Therefore, every collineation is an affine transformation. Hence, affine transformations and collineations are exactly the same thing for the Euclidean plane.