From: wdr@world.std.com (William D Ricker)
Subject: Re: How do you draw a straight line?
Keywords: Linkages, Mathematica instruments
Organization: The World Public Access UNIX, Brookline, MA
Date: Thu, 15 Oct 1992 21:36:30 GMT

pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <jim.719007306@st-michael> jim@cs.UAlberta.CA (Jim Easton) writes:
>>geg@beep.mis.semi.harris.com (Gregory E. Garland) writes:
>>> Do you forbid gears?
>>I don't know - 
>Ideal gears (or friction drives) and ideal belt-drives give easy
>solutions.  But apart from taking all the challenge out the problem,
>the usual objection to these solutions is that there is no such thing
>as an ideal gear (slightly irregular motion as the teeth slide over
>each other) or belt-drive (same problem as the chalk-line solution...

Subject: Mathematical Instruments and their linkages
   Was: Drawing a straight line
Contents:
    Levers to transmit rotations (long)
    Chain linkages for transmiting rotations (short)
    Instruments which Measure area or integrate (short)

                              -- * --

Levers to transmit rotations

On a tangent to the questions about transmiting rotation and angles
for definition-based mathematical drafting machines, I find
(inadequate) documentation of the following mechanism for transmitting
rotary motion between parallel, contra-rotating shafts via levers
alone:

"Knight's noiseless gearing [sic] for to shafts running in opposite
directions; each shaft has two equal cranks at right angles, which are
coupled by links to rocker arms, which are also coupled in pairs."

[Sorry, I don't have a scanner or a decent postscript drawing package.]

The Illusatration shows that the cranks are short, that the long
rocker arms are paired in acute angles, whose pivots separated by more
distance than the shafts are; and that the diagonaly opposite
rocker-arms are cross-connected by short links.  I suspect that the
connection is via a pin at the center of each cross-link which serves as the
pivot for two long links, one to a crank of each of the shafts.  The
leading cranks' links join on one cross-link and the trailing cranks'
on the other, if I am reading the diagram correctly.  I am uncertain
wether what I am calling the cross-links are what the quote calls the
coupling of the rocker arms, or whether the quote is meant to imply
that the angle between co-pivoted rocker-arms is fixed.

Source: Engineers' Illustrated Thesaurus, Chemical Pub., 1952:
Basic Mechanical Movements: Circular & Reciprocating Motion (cont'd)
Plate 30c, figure S.

I can not begin to hypothesize whether this transmits angles accurately
and linearly, or whether it only transmits gross displacement and/or
full rotations.  Practical use might depend on the non-zero momentum
of the practical implementations (which is why toothed gears work at
speed in many applications inspite of the theoretical tooth-engagement
slop mentioned before, although it may make fine positioning
difficult).

If you wish to hypothesize that it might, these contstraints should be
sufficent to begin ...

                             -- * --

I also wonder if the chain-and-sprockets transmission should defeat
the tooth-to-tooth echangement slip (due to differing radii of
contact, I guess?) that was asserted by prior postings -- in chain
gearings, the motive force is at constant radius, and should be always
spread over many teeth, if well machined. But you have the difficulty
that this gets one parallel, co-rotating shafts, NOT contra-rotating
(possibly divergent) shafts as with gears (bevel gears & crown gears).
(Because twisting the chain in figure-8 probably re-introduces the
engagement stresses, which destroy linearity of angle transmission.)

                             -- * --

BTW, the device which measures the area of a curve by tracing its
volume is called a planimeter; they came in varieties to evaluate
closed curves or strip-charts.  Similar but subltly different devices
would trace a function curve and either draw its integral curve or
evaluate the integral.  Plates 44 & 45, Kustner and Kastner, The VNR
Concise Encyclopedia of Mathematics, Van Nostrand 1975.  A good but
old encyclopedia (eg Brit.14th) and some calculus books will have a
physical explanation of their principles.  If I recall correctly, by
no means certain, they all use differential gearings which some
posters would outlaw in definition-based-instruments; and they certain
all use precision tracing of a figure, which is not error-free unless
guided by another definition-based-instrument which does not mind
back-pressures. Some seem to use friction between wheels and the
drawing surface to derive one or both inputs to the differential gear.
-- 
Bill Ricker wdr@world.std.com      "The freedom of the press belongs 
(formerly wdr@wang.com)             to those who own one."