I'd like to understand the math behind how a pantograph scales. The
purpose is to construct a computer driven unit with motors and lead
screws in the x y axis and then be able to scale the output within a
range.

A pantograph, as I am sure you are aware, is an instrument that can be
used to scale a copy of an image.  I used to have one of these as a
child and used it to enlarge copies of cartoons!

The mathematics behind how a pantograph works is very simple to
explain with a piece of paper and a pencil but it is much more of a
challenge to do it without pictures.  I think therefore it would be
better if you first took a look at the following web pages who explain
the theory behind the pantograph in simple detail:

http://www.cs.smith.edu/~orourke/DTS/HowaPantographWorks.html
http://www.ies.co.jp/math/products/geo1/applets/panta/panta.html

Put as simply as I can without a picture the pantograph is constructed
of two longer pieces and two shorter pieces.  The two shorter pieces
fit to form a parallelogram with the two longer pieces.  (One short
piece is parallel to one large piece.)  The scale of enlargement is
determined by how near the smaller pieces are to the ends of the
longer pieces.

I hope the above pages satisfy your curiosity, if not please ask for
further clarification and I will try to explain them in more detail.

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Request for Answer Clarification by latheid-ga on 26 May 2004 10:48 PDT

http://www.softhook.com/draw.htm this is one site that if the rotary
motion was replaced with leadscrews would be simular to what I'd like
to understand. The equations are the key. But I just don't have enough
math background to develop what I need from the first URL given. I
believe that it has something to do with the ratio of the long and
short sides and the two attachment points to the lead screws. Am I on
the right tract?

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Clarification of Answer by palitoy-ga on 26 May 2004 11:10 PDT

You are on the right lines (pun intended)...

If you can image the below diagram as you pantogram:

A         B           C
-----------------------
          |           |
          |           |
          |           |
          ------------------------
          D           E          F

The enlargement factor is the ratio of the length A to C divided by A
to B.  So if the distance from A to C was 10 notches and the distance
from A to B was 4 notches, the enlargement factor would be 10/4 (or
2.25x).

Let me know if this is still unclear and I will try and explain it a little more.