selection process, then the algorithm goes to step 108 which deals with the matching process with less than three matched pairs In step 108, the orientations of the normal vectors at the corresponding umbilical points are aligned. First, translate the scaled surface ri by the difference between the positions of the matched umbilical points. Then, align the normal vectors at the umbilical points. The alignment of the normal vectors can be achieved by using the unit quaternion method 3. Let us assume that we have two normal vectors ni and n2 at the corresponding umbilical points for r1 and r2, respectively. The problem of matching the normal vectors can be stated as rotate n around the vector Nn= m1 xna by an angle e formed by ny and n2. The angle e can be calculated by 0= arccos(n1, n2), see 3 for details of rotation in the quaternion frame In step 110, matching of lines of curvature emanating from an umbilical point is performed Depending on the type of the umbilical point, one(lemon) or three(star and monstar) lines of curvature pass through the umbilical point, and each direction can be determined by the structure of the cubic terms C(a, y) as summarized in Lecture 20. The directions can be obtained by calculating angles of the lines of curvature with respect to the local coordinate system at the umbilical point 9, 10. Using the angles, vectors which indicate the directions of lines of curvature at the umbilical point can be obtained. These vectors are calculated at the matched umbilical points on ri and r2. Suppose that the number of the direction vectors on ri is nul and the number of the direction vectors on r2 is nu2. Choose one vector from r2 and align all of the vectors on r1. This process produces nul matched cases among whic one match is chosen that minimizes equation(21. 4). This alignment is achieved by rotation around the normal vector in the tangent plane at the matched umbilical point. Therefore, the rotation method using the unit quaternion can be used in this process 3 Method 2 The rigid body transformation can also be obtained by using the KH method described in Section 4.2 after the scaling transformation is resolved. The algorithm is the same as in Figure 21.4 from step 100 through step 104. After the scaling transformation is resolved, the Kh method can be used to find the rigid body transformation between two object 21.9.2 Optimization method 7 a problem with scaling effects can be solved with an optimization technique. Since there is no quantitative measure that can be used to estimate a scaling value, the solution scheme has to resort to an optimum search method which can narrow down the best estimate from the possible set of candidate solutions The KH method can be treated as a function of the scaling factor which yields a value of in equation(21. 4)when a scaling factor is given. Namely, steps 10, 12 and 14 in the diagram of Figure 21.1 are grouped as a function f such that ∫=重(σ,R,t) where is the expression given in the equation(21.4), o the scaling factor, R the rotation matrix and t the translation vector. Since the rotation matrix and the translation vector are 12selection process, then the algorithm goes to step 108 which deals with the matching process with less than three matched pairs. In step 108, the orientations of the normal vectors at the corresponding umbilical points are aligned. First, translate the scaled surface r1 by the difference between the positions of the matched umbilical points. Then, align the normal vectors at the umbilical points. The alignment of the normal vectors can be achieved by using the unit quaternion method [3]. Let us assume that we have two normal vectors n1 and n2 at the corresponding umbilical points for r1 and r2, respectively. The problem of matching the normal vectors can be stated as: rotate n1 around the vector Nn = n1×n2 kn1×n2k by an angle θ formed by n1 and n2. The angle θ can be calculated by θ = arccos(n1, n2), see [3] for details of rotation in the quaternion frame. In step 110, matching of lines of curvature emanating from an umbilical point is performed. Depending on the type of the umbilical point, one (lemon) or three (star and monstar) lines of curvature pass through the umbilical point, and each direction can be determined by the structure of the cubic terms C(x, y) as summarized in Lecture 20. The directions can be obtained by calculating angles of the lines of curvature with respect to the local coordinate system at the umbilical point [9, 10]. Using the angles, vectors which indicate the directions of lines of curvature at the umbilical point can be obtained. These vectors are calculated at the matched umbilical points on r1 and r2. Suppose that the number of the direction vectors on r1 is nv1 and the number of the direction vectors on r2 is nv2. Choose one vector from r2 and align all of the vectors on r1. This process produces nv1 matched cases among which one match is chosen that minimizes equation (21.4). This alignment is achieved by rotation around the normal vector in the tangent plane at the matched umbilical point. Therefore, the rotation method using the unit quaternion can be used in this process [3]. Method 2 The rigid body transformation can also be obtained by using the KH method described in Section 4.2 after the scaling transformation is resolved. The algorithm is the same as in Figure 21.4 from step 100 through step 104. After the scaling transformation is resolved, the KH method can be used to find the rigid body transformation between two objects. 21.9.2 Optimization method [7] A problem with scaling effects can be solved with an optimization technique. Since there is no quantitative measure that can be used to estimate a scaling value, the solution scheme has to resort to an optimum search method which can narrow down the best estimate from the possible set of candidate solutions. The KH method can be treated as a function of the scaling factor which yields a value of Φ in equation (21.4) when a scaling factor is given. Namely, steps 10, 12 and 14 in the diagram of Figure 21.1 are grouped as a function f such that f = Φ(σ, R, t), (21.13) where Φ is the expression given in the equation (21.4), σ the scaling factor, R the rotation matrix and t the translation vector. Since the rotation matrix and the translation vector are 12