The equation for the resulting blending surface is ove Bezier B-sp B(,t)=∑∑ Bke Bk.4()N4(t) (9.21) k=0E=0 Bn(,t)=3∑∑(Bk-Bk-1)B3()N4(t) (9.22) E=0k=1 From position and derivative continuity we get B B (9.24) B1t=R(+Q/3 (9.25 B R (2) R2-Q2/3 (9.26) The disadvantage of approximation is the increase in data and the resulting storage re- quirements. The advantage is that the same class of functions is used which makes it easy to include in a geometric modeler and easy to transfer between CAD/CAM systems Special attention needs to be given to the correspondence of parametrization of linkage curves and this may necessitate reparametrization of linkage curves. See Bardis and Pa- trikalakis [1, and Hansmann [7] for more details on these issues Figure 9.3: Blending surface cross-link curves
The equation for the resulting blending surface is: B(w,t) = X 3 k=0 Xn `=0 Bk,` B´ezier z }| { Bk,4(w) Above B-spline z }| { N`,4(t) (9.21) Bw(w,t) = 3 Xn `=0 X 3 k=1 (Bk,` − Bk−1,`)Bk,3(w)N`,4(t) (9.22) From position and derivative continuity we get: B0,` = R (1) ` (9.23) B3,` = R (2) ` (9.24) B1,` = R (1) ` + Q (1) ` /3 (9.25) B2,` = R (2) ` − Q (2) ` /3 (9.26) The disadvantage of approximation is the increase in data and the resulting storage requirements. The advantage is that the same class of functions is used which makes it easy to include in a geometric modeler and easy to transfer between CAD/CAM systems. Special attention needs to be given to the correspondence of parametrization of linkage curves and this may necessitate reparametrization of linkage curves. See Bardis and Patrikalakis [1], and Hansmann [7] for more details on these issues. t t Figure 9.3: Blending surface cross-link curves. 6
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 generalized 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
The equation of the ellipse is(see Figure 9.5): R(v)=Ro +e(Rc+ R)cos v +e 2 Re+ Rs sin t (9.30) os .The projection of the center of the sphere(or projection of the ellipse)on the plane is: tp=(R+ Rs)cos ve1+ Re+Rs 9.31) cOs o tc: Projection of R (equation of the ellipse that is the center of the sphere) to the P: Projection of R onto the axis a P-R (unit normal) 9.32 P-RI R Ro R。:R Figure 9.6: Side view Observe that Ro =((R-Ro) a a (9.33) a(-1)tan o sin (Rc+Rs P-R=(P-Ro)-(R-RoN (9.35) --(Re+Rs)tan o sin 0, -sin o, cos ol (9 (Re+ Rs)cos p, sin g (9.37) P-R=(Re+Rs )[cos -sin cos o, -sin o sin (9.38 P-R P-R tc=r+Rsv (9
• The equation of the ellipse is (see Figure 9.5): R(ψ) = RO0 + e1(RC + Rs) cos ψ + e2 Rc + Rs cos φ sin ψ (9.30) • The projection of the center of the sphere (or projection of the ellipse) on the plane is: tp = (Rc + Rs) cos ψe1 + " Rc + Rs cos φ sin ψ − Rs tan φ # e2 (9.31) • tc: Projection of R (equation of the ellipse that is the center of the sphere) to the cylinder. P: Projection of R onto the axis a. v = P − R |P − R| (unit normal) (9.32) Π a R O’ P tc Rs Rs R Rc v p t Figure 9.6: Side view. Observe that: P − RO0 = ((R − RO0) · a)a (9.33) = a(−1)tan φ sin ψ(Rc + Rs) (9.34) P − R = (P − RO0) − (R − RO0) (9.35) = −(Rc + Rs)tan φ sin ψ[0, − sin φ, cos φ] − (9.36) (Rc + Rs)[cos ψ, sin ψ cos φ , 0] (9.37) P − R = (Rc + Rs)[− cos ψ, − sin ψ cos φ, − sin φ sin ψ] (9.38) v = P − R |P − R| = −[cos ψ,sin ψ cos φ,sin φ sin ψ] (9.39) Hence, tc = R + Rsv (9.40) 8