bisector. The intersection point can be calculated by the following formula:

A1 x B1 y C1  F1 (t1  h) 0



A2 x B2 y C2  F2 (t1  h) 0

(21)

Thus, the coordinate of the intersection point M is

xm  (B1C2  B2 C1 ) / T (t1  h)(B1 F2  B2 F1 ) / T



ym  ( A2 C1  A1C2 ) / T (t1  h)( A2 F1  A1 F2 ) / T

(22)

By calculating a series of intersection points, the bisector of U and V can be acquired.

4.2 Trochoidal machining for corners of a cavity

Material is more easily gathered at a cavity corner, especially corners with smaller angles, while milling corners with trochoid realize smaller tool loads. The path elements involved are mainly clockwise and anticlockwise arcs and linear segments, which may be combined to produce a total of nine types of corners as mentioned by the paper of Choy and Chan (2003). Trochoidal machining of cavity corners can be realized in three ways, as shown in Fig. 12,

 Corner trochoidal machining with variable circles. Material at the formed corner of each contour ring can be gradually removed by the method of variable trochoidal circles.

 Corner trochoidal machining with isometric circles. Material at the formed corner of each contour ring can be gradually removed by the method of isometric circles.

The first and second method generate a trochoidal path at the corner of each ring in the contour-parallel path.

 Perforative corner trochoidal machining along the medial axis. This uses the method of Voronoi diagrams to calculate the medial axis of the cavity and arrange the perforative trochoid according to the medial axis.

The computation algorithm for trochoidal machining includes the following steps:

(1) Calculate the engagement angle or the milling force at the straight line area of the contour-parallel cutting path, which is regarded as a basic reference for the trochoidal calculation.

(2) Compute the guiding curve. If the first or second method of trochoidal machining is adopted, analyse the corner type and calculate the corner bisector or the offset curve

of the corner edge. If the third method of trochoidal machining is adopted, calculate the medial axis of the cavity.

(3) Determine the initial circle and calculate the size and location of the trochoidal circle step by step, guided by the curves in Step 2 so that the engagement angle or milling force can satisfy the control requirements presented in Section 3.

(4) Add the new trochoidal circle and the middle curve between two adjacent circles into the stored linked list.

4.2.1 Corner trochoidal machining with variable circles

As shown in Fig. 12a, a few variable circles may be applied to mill the material of the remaining corner. The circle (c1) on the end position of the corner is very specific and is selected as the initial trochoid circle. Other trochoid circles will be computed one by one until the corner materials are cut out. Finally, all of the trochoid circles are stored in the trochoid list in reverse order.

The calculation of the next trochoid circle constitutes the key sub-algorithms. Fig. 13 shows that the current trajectory is U and V, and the trajectory with the offset distance r (tool radius) is UW and VW. The circle centred by N and tangential to UW and VW is the offset circle of the current trochoid circle. The computation on the next trochoid circle involves two steps: