a degree n planar rational polynomial curve on the a, y plane. Its implicit equation is a degree n polynomial. The equation of the resulting conical ruled surface is x=x0(1-)+Xu +Yu (1-u) Eliminating u=1-2 and solving for x, r yields 0 This equation can be transformed to the standard form using a symbolic manipu- lation program as macsyma 2. Newton's method Solve ro To(u, u), yo= yo(u, U) and verify the third equation. Use a linear approxi- mation to start the process. Preprocessing using convex bounding box should always be used, coupled with some level of subdivision within a rectangular subdomain of the following function(see Figure 10 02U<1 or 3. Convex box and possibly subdivision followed by minimization in0 <u F(u,v)=R(u,t)-Ro2≥0 a point uo u0, vo)=0 yields Figure 10.10: Distance function squared In order to solve this minimization problem, we need to compute of F(u, u)(Fu= F,=0)in doa degree n planar rational polynomial curve on the x, y plane. Its implicit equation f(x, y) = 0 is a degree n polynomial. The equation of the resulting conical ruled surface is x = x0(1 − u) + Xu y = y0(1 − u) + Y u z = z0(1 − u) Eliminating u = 1 − z z0 and solving for X, Y yields: f  z0 z0 − z x − x0 z0 − z z, z0 z0 − z y − y0 z0 − z z  = 0 This equation can be transformed to the standard form using a symbolic manipu￾lation program such as Macsyma. 2. Newton’s method: Solve x0 = x0(u, v), y0 = y0(u, v) and verify the third equation. Use a linear approxi￾mation to start the process. Preprocessing using convex bounding box should always be used, coupled with some level of subdivision. 3. Convex box and possibly subdivision followed by minimization in 0 ≤ u, v ≤ 1 or within a rectangular subdomain of the following function (see Figure 10.10). F(u, v) = |R(u, v) − R0| 2 ≥ 0 A point u0, v0 where F(u0, v0) = 0 yields the solution. u v F(u,v) O Figure 10.10: Distance function squared In order to solve this minimization problem, we need to compute • Minimum of all local minima of F(u, v) (Fu = Fv = 0) in domain; 16