[12 can be employed. The problem of NGWS type can be solved by the moment method using the principal moment of inertia and ratio of areas or volumes. No attempt, however, has been made to solve the problems of NPWS type. For a more detailed overview, see Ko 6 21.4 Problem statement 21. 4.1 Distance metric The Euclidean distance between two points p1 and p2 is defined as d(p1,P2)=|p1-p2 We also define the minimum distance between a surface r and a point p as follows (r, p)=minde(p, pi), Vpi Erl 21.4.2 Distance between a point and a parametric surface Let us assume that we have a point p and a parametric surfacer=r(u, u),0<u, v<l.Then the squared distance function is defined as follows D(u,)=|p-r(u,)|2 (p-r(u,)·(p-r(u,) Finding the minimum distance between p and r is reduced to minimizing(21.3) within the square 0<u, v<l. Therefore, the problem needs to be broken up into several sub-problems which consider the behavior of the distance function at the boundary and in the interior of the bound(10. The sub-problems are summarized as follows: Find the minimum distances(1)in the interior domain,(2) along the boundary curves and (3)from the corner points. Among those minimum distances the smallest one is chosen as the minimum distance between the point p and the surface r. A robust calculation of the minima of the distance function(21.3 can be achieved by the Interval Projected Polyhedron(IPP)algorithm[13, 10, 14 21, 4.3 Distance metric function A function can be defined using the squared distance function(21.3) to formulate a matching problem. Suppose that we have a NURBS surface rB and an object ra represented in discrete points or surfaces. Then, a matching problem can be stated as finding the rigid body trans- formation(a translation vector t and a rotation matrix R)so that a global distance metric function ∑d(rB,(GRP+t) becomes minimal. where g is a scaling factor[12] can be employed. The problem of NGWS type can be solved by the moment method using the principal moment of inertia and ratio of areas or volumes. No attempt, however, has been made to solve the problems of NPWS type. For a more detailed overview, see Ko [6]. 21.4 Problem statement 21.4.1 Distance metric The Euclidean distance between two points p1 and p2 is defined as de(p1, p2) = |p1 − p2|. (21.1) We also define the minimum distance between a surface r and a point p as follows: dsp(r, p) = min{de(p, pi), ∀pi ∈ r}. (21.2) 21.4.2 Distance between a point and a parametric surface Let us assume that we have a point p and a parametric surface r = r(u, v), 0 ≤ u, v ≤ 1. Then the squared distance function is defined as follows: D(u, v) = |p − r(u, v)| 2 , = (p − r(u, v)) · (p − r(u, v)). (21.3) Finding the minimum distance between p and r is reduced to minimizing (21.3) within the square 0 ≤ u, v ≤ 1. Therefore, the problem needs to be broken up into several sub-problems which consider the behavior of the distance function at the boundary and in the interior of the bound [10]. The sub-problems are summarized as follows: Find the minimum distances (1) in the interior domain, (2) along the boundary curves and (3) from the corner points. Among those minimum distances, the smallest one is chosen as the minimum distance between the point p and the surface r. A robust calculation of the minima of the distance function (21.3) can be achieved by the Interval Projected Polyhedron (IPP) algorithm [13, 10, 14]. 21.4.3 Distance metric function A function can be defined using the squared distance function (21.3) to formulate a matching problem. Suppose that we have a NURBS surface rB and an object rA represented in discrete points or surfaces. Then, a matching problem can be stated as finding the rigid body trans￾formation (a translation vector t and a rotation matrix R) so that a global distance metric function Φ = X ∀p∈rA dsp(rB,(σRp + t)) (21.4) becomes minimal, where σ is a scaling factor. 4