How a Pantograph Works

A pantograph has one fixed point O (the “Origin”), and two special points P and Q. It has the property that Q traces an enlarged, or "scale" copy of whatever P traces.  

See Figure 1a.

(a) (b)

Figure 1

There are five essential ideas to understanding why Q traces a scale copy of P.

    1. The short links form a parallelogram with the subparts of the long links.  A

parallelogram is a quadrilateral with opposite sides parallel.  Thus each

short link is parallel to one of the long links. See Figure 1b.

    2. In any one position of the pantograph, one can lay down a "grid" of lines

all parallel to the two directions determined by the links.  This grid determines a "coordinate system," something like the streets in a planned city.  See Figure 2.

Figure 2

3. One can think of P as reached by adding together two vectors A and B.  A

vector can be represented by an arrow:  it is directed at some particular angle, and it has some particular length.  Following A amounts to walking down the street the appropriate length.  Turning and following B lands one at point P. See Figure 3.

Figure 3

4. Reaching Q is accomplished by adding two vectors A' and B', each parallel to A and B:  walk down A', turn and walk down B'.  

Again see Figure 3.

5. If O, P, and Q all lie on one line (as they do; see Figure 1), then A' and B' are each scale versions of A and B, by the same scale "factor" s.   In our example, this scale factor is s=5/3 =1.67:  A' is 1.67-times as long as A', and B' is 1.67-times as long as B'.
   
   

Finally, because this scale relationship holds for any particular position
of the pantograph, the movements of Q are just s times the movements of P.
So Q always traces a figure s-times larger than the figure traced by P.

The mathematics of vectors is called "Linear Algebra," and is generally
studied in the second year of college, after Calculus.
The property exhibited by the pantograph is called the distributive
property of scalar multiplication in vector spaces.