Classical Geometry: Euclidean, Transformational, Inversive, and Projective

CHAPTER 11

HOMOTHETIES

11.1 The Pantograph

Isometries provide a dynamic way of dealing with congruency. In this chapter, we study a transformation which serves the same purpose for the notion of similarity.

Without employing photography, photocopying, or computer graphics, it is not a simple matter to produce an enlarged or reduced copy of a figure. There is, however, a physical instrument called a pantograph that allows us to accomplish this.

A pantograph is formed from four thin fiat rods of equal length that are joined together by four hinge pins P, Q, A, and B so that APBQ is a parallelogram and OA = AP. The instrument lies flat on the drawing board and is fixed to the board at the pivot point O. Pencils are attached to the instrument at points P and P′.

If an enlargement is desired, the pencil at P is used to trace the original feature. As this is being done, the pencil at P′ draws a copy magnified by a factor equal to OQ/OA. If a reduction is desired, the pencil at P′ is used to trace the figure so that the pencil at P draws the reduced copy.

To see why this works, note that OAP and OQP′ are similar triangles by sAs. It follows that O, P, and P′ are collinear. Moreover,

and thus the figure that is traced by P may be considered a “contraction” towards O of the figure that ...