where R(v)=(Rc+Rs)cos a, R+R sin -Rs tan Rs (9.41) COS te = Recos w, R sin g +Rs ( sin y tan o - sin o coS (9.42) cOS o Rs(1-sin o sin l) which simplifies to Re sin y COS U cos o R, tan p(sin y sin -1), Rs (1-sin o sin v)(9.43) Notice that v is in the normal plane to the directrix: v·Rb=0 where Ru=-(Re+Rs)sin v, Rc+Rs cos v,, 0 cOS Hence, the generatrix is the arc of a great circle of the sphere on the plane tc-R, tp-R between t and t. The angle of the arc is 0=cos-(e3 v)=cos"(sin o sin (9.46) in the local n, b system of the directrix q(u, v )=Rssin Oun-cos Bub for v E0, 1 (947 where R t= Rol t (9.50) Setting v= 2Tu for u E 0, 1, the blending surface B(u, u)=R((u))+q(u, v (a)) (9.51) Note that the surface is not a rational polynomial surface This result generalizes to spherical blends of general surfaces. A schematic diagram of pherical blends of surfaces is shown in Figure 9.7. The implementation is procedural and involves intersections of offset surfaces 11, 10 to define the directrix and the projection of a point on a surface to define the generatrix [14]where R(ψ) = " (Rc + Rs) cos ψ, Rc + Rs cos φ sin ψ − Rs tan φ, Rs # (9.41) tc = " Rc cos ψ, Rc sin ψ cos φ + RS sin ψ cos φ − tan φ − sin φ cos φ ! , (9.42) Rs(1 − sin φ sin ψ) # which simplifies to tc = " Rc cos ψ, Rc sin ψ cos φ + Rs tan φ(sin ψ sin φ − 1), Rs(1 − sin φ sin ψ) # (9.43) Notice that v is in the normal plane to the directrix: v · Rψ = 0 (9.44) where Rψ = " −(Rc + Rs)sin ψ, Rc + Rs cos φ cos ψ, 0 # (9.45) Hence, the generatrix is the arc of a great circle of the sphere on the plane tc − R, tp − R between tc and tp. The angle of the arc is θ = cos−1 (−e3 · v) = cos−1 (sin φ sin ψ) (9.46) in the local n, b system of the directrix. q(v, ψ) = Rs[sin θvn − cos θvb] for v ∈ [0, 1] (9.47) where b = e3 (9.48) t = Rψ |Rψ| (9.49) n = e3 × t (9.50) Setting ψ = 2πu for u ∈ [0, 1], the blending surface is: B(u, v) = R(ψ(u)) + q(v, ψ(u)) (9.51) Note that the surface is not a rational polynomial surface. This result generalizes to spherical blends of general surfaces. A schematic diagram of spherical blends of surfaces is shown in Figure 9.7. The implementation is procedural and involves intersections of offset surfaces [11, 10] to define the directrix and the projection of a point on a surface to define the generatrix [14]. 9