On regularity, and the irregularity at surfaces

01. The idea that regularity is a possibility derives directly from the foundations of Euclidean geometry. We allow by it such concepts as the straightness of a straight line, the flatness of a plane, the sphericality of a sphere. We can extend the notion to allow in the imagination such other things as, the exact ellipticality of a plane section through an ellipsoid, or the exact equi-angularity at the apices of some regular polyhedron. Extending the meaning of regularity into the realm of joints, we can envisage if we wish, and please refer to figure 6.01, that the convex conical surface at the joint element on link 1 and the flat plane surface at the joint element on link 2 can make, with one another, continuous line contact. We can similarly imagine that the convex surface of the accurately constructed spherical ball on 2 can make, with the concave surface of the matchingly constructed, equi-radial, spherical socket sunk in 3, continuous surface contact. These and other such imagined, continuous contacts between a pair of joint elements in mechanism – they are not single point contacts or even multiple point contacts but contacts involving ∞1or ∞2 of points – can exist in our imaginations only by virtue of the wholly idealised, but well established concepts due to Euclid.

02. But we know that no mould can ever exactly fit its pattern, that no casting can ever exactly fit its mould, that no welded construction ever matches its production drawing, and that no subsequent machining is ever done on any job accurately.