How is a circle defined? Something like

“A shape that is the set of all points on a plane equidistant from a given point (the centre).”

This effectively describes a circle as a shape having a constant radius. It is tempting to assume that a shape with constant diameter is therefore also a circle but this is not necessarily true. For example, a 50p coin has a constant diameter (so vending machines can identify them) but is clearly not a circle. If you don’t believe this, put a 50p on a piece of paper and draw two parallel lines that just touch it. Now rotate the coin. It will still fit between the lines. For a lot more on this see Kevin Houston’s blog

From a structural engineering perspective, is this anymore than a curiosity? Yes, for two reasons, one theoretical the other practical. From a theorectical perspective we often require structures to be circular and have specifications about just how accurately circular they should be. These are particularly important for very thin shell structures, such as silos or tanks, which can be very sensitive to small imperfections. Care must be taken to define definitions of “out-of-roundness” in terms of radius rather than diameter if certain types of imperfection are not to be missed (not all constant diameter shapes are as obviously non-circular as a 50p) . I have discussed this problem in full in this paper,

The related practical problem is one of measurement. Measuring diameters is easy – just put a tape measure right across. But measuring a radius is harder. Where is the centre? This is a problem that mechanical engineers have known about for sometime and they have well-developed means of measuring circularity based on radius for small objects. However, it is only recently the problem has been noted in structural engineering and care should be taken to measure correctly – a method is proposed here.

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Guillermo ReinThe entry is most interesting Martin. I also found the time to read your paper, but it would be difficult at this moment. What is the basis of the method you proposed?

martingilliePost authorThanks Guillermo. The approach I took was to take the centre of a nominally circular structure as the centre of gravity of the actual cross-section. This is not perfect but is close enough for practical purposes. Then by measuring radii from this point, the degree of out-roundness-can be determined. This general approach has been used for some time in mechanical engineering but cast in a form that requires “the workpiece” to be mounted on a turntable. Clearly this isn’t possible with many structures. I have copied the relevant paragraph from the paper below.

“To define a circle, a datum is required, and it is because measures

of out-of-roundness based on diameter do not use a datum that

they are unsuitable. Because radial displacements are of the most

interest in structural engineering, defining an out-of-roundness

parameter based on deviations in radius is logical. To do this, the

center of the nominal cross section of an imperfect circular shell

must be located. It is not clear where this is for an out-of-round

section, so any method determining out-of-roundness must include

a means of locating it. A simple approach is to identify the approximate

center of a section by taking an arbitrary diameter and locating

the midpoint. Using this point as the origin of a coordinate system,

the cross section may be mapped by identifying the coordinates of a

suitable number of equally spaced points around its circumference

(Fig. 5). In practice, these coordinates will be most easily measured

in polar coordinates, but conversion to a Cartesian system will be

desirable for the following manipulation. The center of the circle

that best fits the mapped shell section will lie, to a good approximation,

at the center of mass of the set of measured points”