The classification in this paper has two advantages compared to exist classification [19]。 First, because four categories are easy to distinguish from each other, a designer can easily choose a category that involves the shape of desired trajectory by visual in- spection。 Second, the geometrical properties used for the classification are simple to calculate。 This means that the classification process in computer simulation is so fast that if this classification is used for the design of four-bar linkage, whole design time can  be reduced。

3。 New approach for four-bar linkage design

The first and second-order derivatives of the coupler points include the geometrical features used to distinguish the coupler curves, which we prove in this section。 Furthermore, the new methodology based on these features is presented。

Table 1

Geometrical properties of each shape type。

Type I II III IV

Num。 of Infinite points 2 2 2 2

Zero-slope  point ratio 1:1 1:1 or 1:3 1:3 2:2

Sign of second derivative at zero slope point Upper - - or + + +/-

Lower + + + -/+

Radius of curvature Finite Infinite Finite Finite

Table 2Classification results with methods 1 & 2。

First method (visual inspection) Second method (using geometrical features)

Type I 4081 (0。5922) 6757 (0。6647)

Type II 742 (0。1077) 600 (0。0590)

Type III 1485 (0。2155) 2001 (0。1969)

Type IV 574 (0。0836) 804 (0。0791)

Exception 7 (0。0010) 3 (0。0003)

Total 6889 (1。0000) 10,165 (1。0000)

3。1。 Geometrical meaning of first-order derivative (slope) of coupler point

The slope is frequently used to distinguish  coupler  curves。 Based  on this, we propose  the following  geometrical  interpretation of  the first-order  derivative  of  the coupler point:

Proposition 1。 If two plane figures  have the same first-order  derivative  profile, they can be called  mathematically    similar。

Proof of Proposition 1。 In the discrete system and based on Fig。 6,

If  φ1,1 =φ2,1, then Δ O p1,1  p1,2  and Δ O p2,1  p2,2  are  similar。  The similarity  ratio  is  n1 r1   ¼  n1。

The tangent values of angles φ1,1  and φ2,1  are described as follows: