(d) Type-IV (Intersectional  shape)。

Fig。 4。 To facilitate the distinguishing of shapes, the obtained trajectory was rotated by θrot to have the maximum width。

2。2。3。 Geometrical characteristics of each type of trajectory

The geometrical characteristics that distinguish each shape type are the first-order and second-order derivatives of the coupler curve of a four-bar linkage and the radius of curvature of some sections of the trajectory。 These characteristics are described as follows:

First-order  derivative (slope):

Fig。 5。 Infinite  points  and  zero-slope  point  of  trajectories。

Second-order  derivative  (change  in  angle  of slope):

If the derivative values are plotted according to the input angle θ1, there are only two points at which the first-order derivative is infinite for all types。 The infinite points pide the trajectory into an upper side and lower side, as demonstrated in Fig。 5。 In the case of Type-I, there is one zero-slope point for each part, where the first-order derivative is zero。 In addition, the sign of the second-order derivative at the zero-slope points is different for each side of the trajectory。 The sign of the upper zero-slope point is  negative,  and  that of  the lower side is  positive。

In Type-II, there are one or three points on the flat side and one on the other side。 Since the flat side does not include a straight line, the number of zero-slope points varies from one to three。 Thus, the radius of curvature is used to distinguish this type  from  the others。 If the  radius of curvature  of  one side  is almost  infinite,  the  shape must be classified  as Type-II。

In the case of Type-III, there are three zero-slope points on one side and one on the other side。 In addition, the signs of the second-order derivative of each side are the same。 Finally, in Type-IV, there are two zero-slope points per side。 Table 1 gives a summary of these  geometrical  features。

2。2。4。 Classification results and discussion

Using the method described in the previous section, 6889 coupler curves were examined, and the results are shown in Table 2。 Among the coupler curves of the atlas, Type-I elliptical shapes account for 59。22% of the total coupler curves。 Types-II, III, and IV account for 10。77%, 21。55%, and 8。36%, respectively。 0。1% of all coupler curves cannot be categorized into one of the four types。 In particular, two of them had an elliptical shape, but there were five zero-slope points on one side。 Another one had an intersec- tional shape, but the number of zero-slop points was the same as Type III。 Thus, these exceptional samples have two intersectional points。

Similar results can be observed with the second methodology。 A total of 10,165 coupler curves were  examined  using the  sec- ond method。 Among all trajectories of randomly generated four-bar linkages, Type-I accounts for 66。47%, while Types- II, III, and IV account for 5。90%, 19。69%, and 7。91%, respectively。 In this case, there are only three exceptions。 One is the same as the latter exception of the first method。 Another case has four infinite points, and the third one has an intersectional shape, but it has four zero-slope  points on one side of the    trajectory。

Because of the difference in the boundaries of each link length, it seems that the results of each method are slightly different。 But each result shows the same trends。 Despite the exceptional trajectory shapes, the results show that four shape types can cover 99。95% of all of the trajectories。 The 10 exceptional cases are not enough to be considered as a single category, so they were as- sumed  to  be negligible。