Pantograph
What Is It?
A Mathematical Droodle

In the applet below, all the points, except A, are draggable. Of special interest is the behavior of C when B is being dragged and vice versa.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


The applet simulates a linkage device known as Pantograph which is used for copying and rescaling documents. Thomas Jefferson was known to be one of the chanpions of the pantograph patented in the US in 1803 as polygraph. A German polygraph has been constructed in 1603 by Christoph Scheiner.

The theory of the pantograph derives from similarity of triangles.

With the vertex names shown in the applet, assume that

BE||AF and BD||CF, and also that AD / AF = EF / CF = α.

Then A, B, C are collinear and AB / AC = α.

Indeed, let B' be the point of intersection of B'D and AC, B'D||CF. Then - from the similarity of triangles ACF and AB'D - AB'/AC = α.

Let B'' be the point of intersection of B''E and AC, B''E||AF. Then - from the similarity of triangles ACF and B''CE -

AC / B''C = CF / CE,
(AB'' + B''C) / B''C = (CE + EF) / CE,
AB'' / B''C + 1 = EF / CE + 1,
B''C / AB'' = CE / EF,
B''C / AB'' + 1 = CE / EF + 1,
AC / AB'' = CF / EF,
AB'' / AC = EF / CF = α.

We thus have, for two points B' and B'' between A and C, AB'/AC = α = AB'' / AC, implying that B' and B'' are one and the same point, which is denoted B.

Note the above is true regardless of the specific positions of the points. Fix point A and drag point B. Point C will trace a shape similar, in fact homothetic with center A and coefficient AF/AD, to that traced by B.

If C traces a shape then B traces a homothetic shape with the coefficient AD/AF.

Linkages

  1. What Is Linkage?
  2. Peaucellier Linkage
  3. Hart's Inversor
  4. The Fundamental Theorem of 3-Bar Motion
  5. Pantograph
  6. Watt's and Chebyshev's Linkages

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Alexander Bogomolny

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