Affine Transformations, Polynomials, and

In the articles [1] and [3], standard tools and techniques of calculus are used to estab-lish a variety of proportionality results concerning areas defined by the lines tangent to a cubic curve, and by the lengths of certain arcs of a parabola, where the arcs them-selves are determined by an area-proportionality criterion. We demonstrate here that these results can be viewed as consequences of some basic facts about affine transfor-mations in the plane. 1. AFFINE TRANSFORMATIONS. A splendid source of information about affine transformations is Appendix A of [2]; here are the properties we need. Definition. An affine transformation of the plane is a function of the form T (x, y) = (ax + by + e, cx + dy + f), where a, b, c, d, e, and f are constants and ad − bc is not 0. Properties. [2, Appendix A] An affine transformation T has the following properties: (i) T takes lines to lines and parallel lines to parallel lines; that is, if L is a line, then T (L) is also a line, and, if two lines L1 and L2 have the same slope, then the lines T (L1) and T (L2) also have the same slope (not necessarily the sam