11.2.2 Classification of singularities

There are two types of singularities on the offset curves of a regular progenitor curve, irregular points and self-intersections. Irregular points include isolated points and cusps. A point on a curve is called an isolated point of if there is no other point of in some neighborhood of . This point occurs when the progenitor curve with radius is a circle and the offset is . A cusp is an irregular point on the offset curve where the tangent vector vanishes. Cusps at can be further subdivided into ordinary cusps when and extraordinary points when and [102]. An isolated point and a cusp occur when , which using (11.6) reduces to

Note that in (11.6) and (11.7) changes abruptly from -1 to 1 when the parameter passes through at an ordinary cusp, while at extraordinary points does not change its value (see Fig. 11.9 (b)).

Figure 11.9: (a) Offsets to a parabola , (thick solid line) with offsets =-0.3, -0.5, -0.8 (adapted from [94]), (b) at =-0.3, -0.5 the tangent and normal vectors of the offset have the same sense as the progenitor, while at they flip directions

Offset curve/surface may self-intersect locally when the absolute value of the offset distance exceeds the minimum radius of curvature in the concave regions (see Fig. 11.10 (a)). Also the offset curve/surface may self-intersect globally when the distance between two distinct points on the curve/surface reaches a local minimum (i.e. the presence of a constriction of the curve/surface as illustrated in Fig. 11.11). These local and global self-intersections can be visualized as machining a part using a cylindrical/spherical cutter whose radius is too large for -D/3-D milling. It is an essential task for many practical applications to detect all components of the self-intersection points/curves correctly and generate the trimmed offset curve/surface. If the cutter follows the trimmed offset, there will be no overcut or gouging, however we are left with undercut regions which must be milled with a smaller size cutter (see Fig. 11.10 (b)).

Self-intersections of offset curves include nodes and tacnodes. A node is a point of curve where two arcs of pass through and the arcs have different tangents. A tacnode is a special case of a node whose two tangents coincide, as illustrated in Fig. 11.11. Self-intersections of an offset curve can be obtained by seeking pairs of distinct parameter values such that

Figure 11.10: Self-intersection of the offset curve of a parabola (adapted from [254]): (a) offsets to the parabola with and cutter path with gouging, (b) trimmed offsets to the parabola with and cutter path with undercut

Example 11.2.1. (see Figs. 11.9 and 11.10)
Given a parabola , , the unit tangent and normal vectors are given by

The curvature and its first and second derivatives are given by

Thus a stationary point of curvature occurs at . Since , is a maximum with a curvature value . It is evident that the offset distance has to be negative to have a cusp, since is always positive for any . Now let us solve for which yields

We can easily see that if , there is no real root. This means that there is no singularity as long as the magnitude of the offset distance is smaller than . If , there exists a double root , while if there exist two symmetric values of . When , at , we have , , therefore is an extraordinary point, while when , , so at points there are ordinary cusps on the offset curve.

The offset to the parabola is given by

Therefore the equations for self-intersection of offset curve to the parabola in and components become

It is readily observed that the offset is symmetric with respect to -axis, which implies that the pair of distinct parameter values forming the self-intersection must satisfy . The component results in identity, while the component yields

Finally, the non-trivial solutions are .