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