A transformation of the plane is a function that associates to each point a new point . In this chapter we study the “classical” transformations, those that leave something geometrically significant invariant: the isometries (translations, axial and central symmetries, rotations), which preserve distances, and the dilatations (with the special case of the homothety), which multiply them by a fixed factor. We shall see how each transformation acts on the points and how it modifies the equation of a curve, and we shall learn to compose several transformations so as to recognise, for example, an ellipse as a circle first dilated and then translated.
Transformations have a dual nature: they can be thought of as “moving the points” (the point changes, the axes stay fixed) or as “moving the axes” (the reference frame changes, the points stay fixed). The two viewpoints are equivalent up to a change of sign in the parameters. Here we adopt the first, the “active” one.