Soddy circles and plug gages

[mklotz]

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.