Soddy circles and plug gages

Measuring the diameter of holes is difficult enough but the problem compounds when the diameter is relatively small.

The sure fire way to get accurate sizes is to use plug gages. However a set of plug gages can be pricey. Despite this, most people fail to realize that plug gages can be used to measure holes larger than the largest plug in the set - thus extending the effective range of the set and adding a bit of justification to the cost. For example, a plug gage set with a maximum plug size of 0.25 can be used to measure holes up to 0.536.

The way to do this depends on a bit of mathematics known as the Soddy circle. If I place three circles of differing diameters such that each is tangent to the other two, it's always possible to draw a circle around these three that is tangent to each of the three. 2000 years ago Apollonius addressed this problem. Much later (1936), a mathematician named Soddy derived a formula for this circle, known now as the 'outer Soddy circle' (yes, there is an inner Soddy circle, but we don't need to go into that here).

The outer circle represents the hole whose diameter we wish to measure. The three inner circles represent the three plugs we'll use to determine the outer circle diameter.

Calculating the Soddy circle diameter given the diameter of the three plugs is a bit of messy mathematics.

d1*d2*d3/(d2*d3+d1*(d2+d3)-2.*sqrt(d1*d2*d3*(d1+d2+d3)))

Unless you're comfortable with this sort of thing you're advised to use the Soddy program included with PLUG.ZIP on my page. You enter the diameters of the three plugs you've fit into the hole you wish to measure and it returns the diameter of the Soddy circle.

The more interesting question revolves around which three plugs to select to use as a gage when boring a hole to a desired size. Sadly, there is no closed form solution to this problem and a computer is needed to do an exhaustive search. The PLUG program that is part of the above mentioned archive does just that.

And that will be tremendous aid to me. Marv, your program is riding to work with me tomorrow!
My plugs are Asian, probably bought around 1990 for a song. Closeout when Carmen Tool, Paramount, CA became Phase II, entire minus set [-.0002] .011 to 1.000. Real size isn't too important, with micrometer grads in tenths. Boring accurately on the mill is when tool and quill are extended the shortest distance possible. That usually precludes bore-mike from entering without moving X, Y, or Z. Gauge pins are only 2'' long. As stated, pre-calculating which to use is flat impractical. My in-process technique became; 1st pin around .6x of diameter by caliper, 2nd about .4x. 3rd is tedious for some, just poking in successively larger pins until slip or very lightest press fit is attained and calculating from there. Ideally, 3 equal [or nearly] pins [120* apart] would best represent roundness/ concentricity and cylindrical form at least in the 2" area pins contact. But monkeying around switching pins won't entertain most employers...When you produce near perfect bore wins him back, at least 'til next time. Aren't doing this while roughing, telescope gauge is nice for angling in under the bit. It's those last few thou/ mils before finish pass AT SAME FEED RATE, finish pass will be. OK on a lathe too, if you have gravity like most places on the planet, very helpful. Not restricted to gauge pins; die leaders, dowels, wrist pins and most any ground OD's work carefully handled.
On a surface plate, a bored through section upright, 1 pin, 1 bearing ball, and whatever test pin fits works too.
Past few days been 60F-ish, home projects are done, well caught up, have't a tool for that,don't feel like it. Cue John Fogerty "Honey Do". LOL's a second there!
Besides, I want my wireway brackets! Going out to get stand for the Roper-Whitney 17 punch finished, from November last.
Sending out plasma cutting, fixture almost done. Arrangements made for spotwelder, halfways toward vibratory tumbling and barrel zinc plate or powdercoat finish.
Need separate pedestals for both benders designed. One was an earlier post coming into HMT.net, the HF mod.
So if anyone isn't busy...