The radius of the previous offset circle is set as R1, the radius of the current offset circle is set as R2, and the tool radius is set as r. Obviously, |CD|=r，|CE1|=r，

|O1D|=|O1S|= R1, |O2E|= R2, R1=rtro1+r,  R2=rtro2+r.

To avoid a tedious calculation at the circular intersection, a geometric modelling method for the engagement angle in trochoidal machining is proposed as follows. It mainly includes two sections (Figs.3 b-c).

(1) Geometric modelling on the first section SE

Calculating angle  ∠E1CO1

Fig. 3b shows that the cutter rolls along the curve SE and that the tool centre trajectory is S’E’. The coordinates of the cutter centre C(xc,yc) can be computed using

the tangency point E1  and vector

E1C .

The length of CO1  is

CO1   

In addition, the length of E1O1   is

E1O1    

Obviously, from the geometric relationship, ∠E1CO1 can be calculated by the following formula:

cos E1CO1

( CO1

 CE1

 E1O1

) /(2 * CO1   * CE1 )

(2)

( CO1

 r 2

 E1O1

) /(2 * CO1   * r)

Calculating the angle  ∠DCO1

∠DCO1 is an interior angle of the triangle △DCO1, and its range is 0<∠DCO1<π. It can be obtained using the cosine theorem:

cos∠DCO1=(|CO1|2+r2–R12)/(2*|CO1|*r) (3)

Calculating the engagement angle   ∠DCE1

Obviously, the engagement angle ∠DCE1 satisfies the following geometric formula:

atro=∠DCE1=2π–∠E1CO1–∠DCO1 (4)

When SE1 increases in the variation range, the engagement angle with corresponding change may be calculated using this geometric calculation.

(2) Geometric modelling of the second section EGHE

Establishing the trajectory equation for tool centre

The moving trajectory of the cutter centre C is the arc centred by O2 (xO2, yO2) with radius R2-r. Therefore, the following equations for the variation in C(xc, yc) may be established.

xc=xO2 +(R2–r)*cosφ

yc=yO2 + (R2–r)*sinφ (5)

Calculating the angle  ∠DCO1

Obviously, ∠DCO1 is an interior angle of the triangle △DCO1, and its range is  0<

∠DCO1<π.

From the geometric relationship, |CD|=r，|O1D|=R1 are known, and the length of   CO1 is h2=( xc–0)2+( yc–0)2,

According to the cosine theorem,

cos∠DCO1=(h2+r2–R12)/(2*h*r) (6)

The angle ∠DCO1 can be obtained by the inverse cosine.

Calculating the angle  ∠E1CO1

The angle ∠E1CO1 may be calculated as the vector dot product of

O2 C

and

CO1  .

However, because ∠E1CO1 is likely to be larger than π (see Fig. 3c), the relationship of the vector cross product is required to identify whether ∠E1CO1 is greater than or less than π.

The vector quantity

CO1   is set as a=[ax，ay，0], and

ax  = –xO2 – (R2–r)*cosφ

ay  = –yO2 – (R2–r)*sinφ

The vector quantity