THE CONCAVITY PARAMETERS

The surface (S) divided by b is certainly a good measure of convexity. It does not discriminate, however, between a triangular shape and a saucer shape. To eliminate scale differences, S is divided by b.

The height (h) divided by b is another measure of convexity; it has the same inability to discriminate between triangular and saucer shapes as does S.

The best-fitting circle, as defined previously, seems to be one of the best parameters to describe convexity. Of course the curve is not really part of a circle; maybe an ellipse would be a better approximation, but since the surface S is only a small part of the circle, another geometrical curve would only give the illusion of additional precision that is far beyond the accuracy of drawing the curve.

The convexity index C may be defined as the curvature of the circle, 1/r, as a function of b, w, or h, e.g.: C(b2r) = b/2r, C(w2r) = w/2r, C(hr) = h/r.

C(b2r) is infinitely small for completely flat molars, and its maximum is 1, when the diameter of the circle is identical to the length of b. An alternative is C(w2r), also infinitely small for completely flat molars, but C(w2r) may be larger than 1, especially in specimens where the top of the protocone lies far away from the lingual border of the tooth (in occlusal view), or, in other words, when the lingual slope of the protocone has an important horizontal component.

Another possibility is C(hr). In this case h is divided by r, instead of 2r, to force the maximum to be 1. As will be shown, C(hr) is probably the best concavity index.

In some cases the circle method gives an unsatisfactory result, e.g., when the profile is very asymmetric, when it is completely flat or even convex, or when the profile is formed by two almost straight lines, and approximation to a circle is poor.