be a degree n planar rational polynomial curve in the x, y plane. The resulting implicit equation of the curve f(X,Y)=0 is a polynomial of degree n. Let a={a1,a2,a be a direction vector. Then the three equations X+ual describe a cylindrical ruled surface. Hence, the implicit surface equation becomes: 了3 This equation can be transformed to the standard form using a symbolic manipu lation program such as Macsyma (c) Implicitization of a conical ruled surface Figure 10.9: Conical ruled surface be the apex of the conical ruled surface and R(t)=[X(t),Y(t) 15Let R(t) = [X(t), Y (t)] be a degree n planar rational polynomial curve in the x, y plane. The resulting implicit equation of the curve f(X, Y ) = 0 is a polynomial of degree n. Let a = [a1, a2, a3] be a direction vector. Then the three equations x = X + ua1 y = Y + ua2 z = ua3 describe a cylindrical ruled surface. Hence, the implicit surface equation becomes: f(x − z a3 a1, y − z a3 a2) = 0 This equation can be transformed to the standard form using a symbolic manipu￾lation program such as Macsyma. (c) Implicitization of a conical ruled surface x y z R(t) u R0 tˆ Figure 10.9: Conical ruled surface. Let R0 = [x0, y0, z0] be the apex of the conical ruled surface and R(t) = [X(t), Y (t)] 15