machine#construction: foundations: whitworth

Whitworth's Method of Generating Accurate Surfaces

  • Using plates
  • using straight edges
    • (using cylinders?)


How a saddle shape can survive the whitworth method

stolen shamelessly from Jim Ash
(note to self- please verify that the whitworth method doesn't already include ninety degree turns. )

The easiest way to explain it is by example. Although this appears
long-winded, bear with me; you can do it in 5 minutes (maybe 10 if you read
as slow as I do).

Take 3 pieces of paper, all the same size and shape (preferably square).
We'll use the side you're about to scribble on to simulate the precision
surface. In the middle of each piece, mark them A, B, and C so each one is
unique. Place a dot on the upper left corner of each piece so you can
identify its orientation (rotation). Now, put an H (for high) right next to
the dots on the upper left corner of each piece. Put another H on the lower
right corner of each piece. Put an L (for low) on the remaining corners
(upper right and lower left). We'll assume the middle of each piece to be
halfway between high and low.

If you look at the cross-sections from the upper-left to lower right
corners (high corners) of each piece you'll end up with concave bowls. If
you look at the cross-sections through the other corners, you'll end up
with convex humps. This shape is mathematically known as a saddle of one
fold. If you care that much, I can scrounge up an equation for you from one
of my math books, but I'm too tired right now.

Now for the cool part.

Take any two pieces and lay them side-by-side next to each other on a
table, oriented the same way. 'Fold' one over on top of the other, like
you're closing a book, so that the 'precision' surfaces contact each other.
If you look at your marks, you'll find that every corner marked with an H
is mated to a corner marked with an L. The middles, being halfway between H
and L, also mate up. You now have two non-plane surfaces that mate
perfectly. Given all three pieces of paper are 'shaped' the same, any two
will mate up as long as they're oriented the same way. The conclusion is
that three saddles can pass the spotting test this way.

Now, the fix:

When you match the plates up, choose one as a master and the other as the
one to be 'adjusted'. Assume the first (if A-B, then it would be A), to be
your master. Now, after running through the 3 match-ups, A-B, B-C, C-A, go
through the sequence again, but rotate the masters 90 degrees from their
original positions. All of a sudden, the highs are going to be opposite
highs, and the lows will be opposite lows. Your spotting will be
incomplete, indicating a problem. Technically, you should only see spotting
where the highs meet the highs. Saddles fail the test. The only surfaces
that can survive this part of the testing will be planes.

There are saddles of more than one fold, and I've tried to find one that
would survive the 90 degree test also, but unsuccessfully. I came up with a
crude proof a couple months ago that none exist, but I didn't really write
anything down.

Jim Ash

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Last-modified: Thu, 14 Dec 2006 19:07:51 GMT (1489d)
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