Exploring methods to sharpen serrations I came to a conclusion which I haven't seen discussed: many factory serrations may not be cylindrical even when they appear to be.
Consider the bread knife serrations in fig.1 below. Each serration appears to roughly be a circular arc. This shape however is not fit using a cylindrical rod as the intersection of a cylinder and an oblique plane (either face of the knife) is an ellipse, not a circle. (Fig.2) The lower the angle (in knife edge terms) the greater the eccentricity. If a cylinder of large enough radius is used and the cut is shallow, limiting the intersection to a fraction of its diameter, a circular arc may be approximated (fig. 3) but deeper circular serrations cannot be produced. If instead a rod, wheel edge, or stone is itself given an oblate elliptical profile, this shape will be "stretched" across the plane of intersection, producing a circular arc when the angle exactly compensates. (Fig. 4)
The formula for the "stretching" ratio is (1 + cot(a)^2)^(1/2) where a is the edge angle in radians. For example a 30° edge will create a 2:1 transform, while a 40° edge will give 1.56:1.
Figure 1
Figure 2
Figure 3
Figure 4
Consider the bread knife serrations in fig.1 below. Each serration appears to roughly be a circular arc. This shape however is not fit using a cylindrical rod as the intersection of a cylinder and an oblique plane (either face of the knife) is an ellipse, not a circle. (Fig.2) The lower the angle (in knife edge terms) the greater the eccentricity. If a cylinder of large enough radius is used and the cut is shallow, limiting the intersection to a fraction of its diameter, a circular arc may be approximated (fig. 3) but deeper circular serrations cannot be produced. If instead a rod, wheel edge, or stone is itself given an oblate elliptical profile, this shape will be "stretched" across the plane of intersection, producing a circular arc when the angle exactly compensates. (Fig. 4)
The formula for the "stretching" ratio is (1 + cot(a)^2)^(1/2) where a is the edge angle in radians. For example a 30° edge will create a 2:1 transform, while a 40° edge will give 1.56:1.
Figure 1

Figure 2

Figure 3

Figure 4

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