9.3 Spherical and circular blending in terms of general- ized cylinders For a detailed reference on this topic, see Pegna [12 C ∏I Figure 9.4: Model of milling via a spherical ball cutter(3-axis milling)or a disk cutter(5-axis) The center of the sphere(or spherical cutter) moves on the intersection curve of the offsets of the plane II and cylinder C of radius R by an offset amount equal to Rs, i.e. ellipse E(see Figure 9.4) Let a be a unit vector along the cylinder axis, e a unit vector perpendicular to plane Il, and O the intersection of the cylinder axis and the plane. Also (9.27) Rct Rs Rs+R(Rs+Rc)/cos p R。 Figure 9.5: Definition of the ellipse Find the directrix E, an ellipse. The center of E is [0, -Rs tan Rs= Ro. The major (Rs +Re)/cos o along e2. The minor axis is Rs+Rc in the direction parallel to er 0,-sin o, cos pl (9.29)9.3 Spherical and circular blending in terms of general￾ized cylinders For a detailed reference on this topic, see Pegna [12]. e e e Π ψ φ 1 2 3 O’ O E C a u Rs Figure 9.4: Model of milling via a spherical ball cutter (3-axis milling) or a disk cutter (5-axis). The center of the sphere (or spherical cutter) moves on the intersection curve of the offsets of the plane Π and cylinder C of radius Rc by an offset amount equal to Rs, i.e. ellipse E (see Figure 9.4). Let a be a unit vector along the cylinder axis, e3 a unit vector perpendicular to plane Π, and O the intersection of the cylinder axis and the plane. Also, e1 = e3 × a |e3 × a| (9.27) e2 = e3 × e1 (9.28) a e2 Rs O O’ e3 φ Rc R + c Rs O e2 R ψ e1 R + s Rc (R + s R )/cos c φ ’ Figure 9.5: Definition of the ellipse. Find the directrix E, an ellipse. The center of E is [0, −Rs tan φ, Rs] = RO0. The major axis is (Rs + Rc)/ cos φ along e2. The minor axis is Rs + Rc in the direction parallel to e1. a = [0, − sin φ, cos φ] (9.29) 7