Dilation

A dilation in geometry is, formally speaking, an oriented contact transformation on a plane. It is a type of transformation in which a line moves an arbitrary distance in a direction perpendicular to it (resulting in a parallel line) and in which a circle changes its radius. A circle of zero radius is then seen as a point and conversely a point can be magnified into a circle. An important attribute of this representation is the fact that it preserves touch.

The dilation method is often used to find the tangents of two circles which is useful, for example, while solving Apollonius problems. Here is an example of its use:

Find the tangents of the blue given circles l (S_1, r_1) and k (S_2, r_2).

Solution (in red): Reduce the two circles by the radius of the smaller one (r_2): the circle l thus becomes the circle l_1, the circle k is transformed only into a point (S_2). Between the tangent line and the line connecting the center of the circle with the tangent point is always an angle of 90° therefore we can use the Thales’s theorem to find the point of touch (P_1). Connecting this point (P_1) and the center S_2 creates a line m. Parallel to this line shifted by the radius of the smaller circle r_2 in the direction perpendicular to it away from S_1 is the solution of the problem. The solution has the form of a pair of lines (only one of the two tangents is drawn in the sketch for the sake of simplicity). To find the second pair of tangents we apply the same procedure, but at the start we increase the radius of the circle l by r_2 (again only one of the two solutions is shown in the illustration).