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Of *• e i 
 
 Report No. kk9 
 
 /k^C4 
 
 A PROCEDURE FOR DETECTING INTERSECTIONS 
 AND ITS APPLICATION 
 
 by 
 
 Kiyoshi Maruyama 
 
 May, 1971 
 
 THE LltiKAKY Ul- I HE 
 
 UNlVtRSH Y Oi- ILLINOIS 
 ^j yp 'ama-CHAMPAIGN 
 
Report No. 4^9 
 
 A PROCEDURE FOR DETECTING INTERSECTIONS 
 AND ITS APPLICATION 
 
 by 
 
 Kiyoshi Maruyama 
 
 May, 1971 
 
 Department of Computer Science 
 University of Illinois at Urbana-Champaign 
 Urbana, Illinois 6l801 
 
 This work was supported in part by the Department of Computer Science 
 at the University of Illinois at Urbana-Champaign, Urbana, Illinois; 
 the National Science Foundation under NSF Grant GJ-328. 
 
Ill 
 
 ACKNOWLEDGEMENT 
 The author wishes to thank Dr. J. Nievergelt, Professor of Computer 
 Science at the University of Illinois, for suggesting the problem and his 
 helpful discussion and comments. Thanks are extended to Miss Sue Cook who 
 helped in getting this report finished. 
 
IV 
 
 TABLE OF CONTENTS 
 
 Page 
 
 1. INTRODUCTION ± 
 
 2. INTERSECTION DETECTION PROCEDURES 
 
 FOR GEOMETRIC OBJECTS 2 
 
 2. 1 P-C FUNCTION APPROACH 2 
 
 2. 2 LP AND IP APPROACH 5 
 
 2.3 GEOMETRIC APPROACH 6 
 
 2. 1; SPACE PARTITION APPROACH . „ 6 
 
 3. GEOMETRIC PROCEDURE FOR DETECTING INTERSECTIONS . . . . . . . H 
 
 3.1 PRELIMINARY DISCUSSION . . . „ H 
 
 3-2 FACE-TO-FACE INTERSECTION DETECTION 15 
 
 3. 3 MJRTHER DISCUSSION „ 20 
 
 3. 4 ALGORITHM AND EXPERIMENTAL RESULT . . „ 25 
 
 k. A PROBLEM IN PATH FINDING 29 
 
 4.1 STATE SEARCH AND STATE CONTIGUITY . . . „ . . 30 
 
 4.2 PATH SEARCHING y 4 
 
 if. 3 SEARCH TREE PRUNING 37 
 
 '4. k EVALUATION FUNCTION AS A DISTANCE EVALUATOR 37 
 
 !+. 5 EXAMPLES OF DISTANCE EVALUATION FUNCTION >,0 
 
 4.6 SIMULATION RESULTS OF GOAL SEEKING AUTOMATON Itf 
 
 5. CONCLUSION „ 71 
 
 LIST OF REFERENCES 72 
 
1. INTRODUCTION 
 A problem of both practical and theoretical interest is to determine if 
 an object can be moved from one position a to another position p, inside a 
 complicated geometric structure. In some problems there will be no path from 
 a to p and some others there will be several paths of which perhaps the shortest 
 must be determined. A restricted problem of this type, the two dimensional 
 sofa problem has been studied by Howden [17 ]. 
 
 To solve such a problem it is basically important to have a very 
 efficient algorithm for determining whether two bodies intersect, since a moving 
 object cannot share the same space with some other moving objects or with 
 obstacles at the same time. This intersection detection problem between physical 
 objects has been studied by Comba[l], for convex objects with differentiable 
 surfaces. 
 
 The plan of this report is as follows. In chapter 2 we introduce 
 several intersection detection procedures for geometric objects, and an 
 intersection detection procedure by means of face-to-face intersection analysis ' 
 will be discussed in Chapter 3- As an application of the intersection detection 
 procedure developed in Chapter 3, we will disucss a path finding problem in 
 geometrically constrained space in Chapter If. 
 
I 
 
 NTERSECTION DETECTION PROCEDURES FOR GEOMETRIC OBJECTS 
 
 In this chapter, we describe several procedures to detect intersections 
 hetween objects. The representation of objects and its data structure [8 , 19] 
 are heavily dependent upon the class of objects as well as a detection procedure 
 
 used. 
 
 2.1 P-C FUNCTION APPROACH 
 
 Let us first describe the procedure which has been developed by Comba [1 
 for detecting intersections of convex regions in J-space by means of a pseudo- 
 characteristic (P-C) function. 
 
 Given n objects, each of which is bounded by one or more surfaces, the 
 problem of deciding whether all the objects have a common intersection may be 
 formulated as the problem of finding whether a system of inequalities (in 
 general nonlinear) has a feasible solution. In the approach followed here, 
 the expressions that represent the bounding surfaces of all the objects are 
 
 combined into a single pseudocharacteristic function, G(x,y,z), which has the 
 property that the region where G < 0, if such a region exists, is a close 
 
 approximation to the region where all the objects intersect. The problem is thus 
 
 reduced to that of finding whether there are any points in space where G takes on 
 
 nonpositive values. 
 
 A). An object is the intersection of k > 1 regions % where each Ej. is defined 
 by an inequality Si (x,y,z) < 0, and the function g, has the following properties: 
 
 (i) It is convex 
 
 (ii) It is differentiable 
 
 (iii) It is normalized so that igrad gi | > | on the surface 
 gj_ = 0, and 
 
(iv) Given a bounded region of space, grad gi is bounded there. 
 
 The p-c function G of the intersection of n objects is defined in 
 terms of the g ± functions for the objects by 
 
 V ± = (g* + t 2 ) 1 ^ + g . (i) 
 
 V = )^i (2) 
 
 r 1 (v t2 ^ 
 
 G= 2 (v -~) + c (3) 
 
 where t anc c are small positive numbers. The boundedness of the objects is not 
 a serious restriction from a practical viewpoint. The convexity assumption is 
 the most restrictive, since it implies that only convex objects can be defined. 
 Comba proved that G is a convex function of the g± and hence of the coordinate 
 variables. Convexity and boundedness imply further that, G has exactly one 
 ninimum. it is possible to compute lower bounds for the values of G without 
 actually finding its minimum. The assumption of differentiability makes it 
 possible to use a gradient method in the search for negative values of G and the 
 computation of lower bounds to G. 
 
 B). A sequence of points approaching the minimum of G is constructed by means 
 of a descent procedure. The search is terminated as soon as 
 
 (i) a negative value of G is obtained, Indicating that there is 
 an intersection; or 
 
 (ii) the minimum of G is obtained and if it is positive, there is no 
 intersection; if bounds on the size of the objects are known, it is possible to 
 compute lower bounds for G, and the search can be stopped if a positive lower 
 bound is established. 
 
The central idea of the method is that of applying the transformation 
 (1) to each of the faction gi that define the various regions % . *e Action 
 V ± is always positive and it is an increasing function of gl . If t < 1, V ± 
 
 also has the property that 
 
 f 2 Si + 0(t 2 ) §i > ° 
 
 
 
 V. = 
 
 i 
 
 Consider the expression 
 
 t & i 
 
 0(t 2 ) 6i < ° 
 
 1 t 2 
 G* - | (V - V 
 
 we can easily verify that the transformation from V to G* is the inverse of 
 the transformation from 6i to V V It can he also seen that 
 
 G 
 
 ( < V < t 
 
 = o v = t 
 > o v > t. 
 
 Hence, except for the vicinity of the spaces gi - 0., G* will be negative inside 
 the intersection of the regions g, < and positive outside that intersection. 
 The reader may see [ Oj for further discussion. 
 
 If all of the functions g ± are linear, the intersection detection 
 problem reduces to a linear programming problem, i.e. deciding whether a set of 
 linear inequalities has a feasible solution. In this case one would expect a 
 priori that a linear programing algorithm will solve the problem more efficiently 
 than an algorithm based on the p-c function. 
 
 Comba assumed convex objects with differentiable surfaces but we 
 hereafter assume a more restricted class of objects namely, polygons for 2-space 
 and polyhedra for 3 -space. This restriction is reasonable because many objects 
 
can be represented as the union of (convex or nonconvex) polyhedra in 3- space and 
 detecting intersection of such unions in an immediate extention of the case for a 
 single pair of polyhedra. 
 2.2 LP AND IP APPROACHES 
 
 When one assumes polyhedral objects to be convex or to be unions of 
 convex polyhedra, a set of linear inequalities represents a convex object. To 
 detect an intersection between two objects, this algorithm (i.e. the phase 1 
 
 of Simplex Method) searches for the existence of feasible solutions to the joint 
 
 set of linear inequalities of two convex polyhedra. Therefore, if one object 
 
 2 consists of the union of N convex polyhedra and the other object fi* consists of 
 
 N' convex polyhedra, then it is necessary to solve (N x N') joint sets of 
 
 inequalities to conclude whether or not SI and fi' intersect. Furthermore, if 
 
 there are M such objects in space to be solved for intersection, it is necessary 
 
 to determine M(M-l)/2 object-object intersection detections. 
 
 However, allowing new integer variables (i.e. O-l) we get one composite 
 
 system by Integer Programming formulation. 
 
 Let Q ± , i = 1,2, . . .M, be M objects and let for each £l ± , g ± < 
 
 denotes the corresponding set of inequalities. Since the placement of M objects 
 
 is possible only if 
 
 a ± n fij = <f) for i and j , i f j . 
 
 we have the following Integer Programming problem formulation with the sacrifice 
 of introducing new 0-1 integer variables, 5. . for i / j. 
 
G. . < Bii U, , f or i / j 
 
 1J — x 3 x 3 
 
 M(M-l) , 
 i J 
 
 5. . =0 or 1 
 13 
 
 where G- • denotes a joint set of inequalities of a± and fi., U is an upper bound 
 
 1 j o *■ J 
 
 of G Thus , if the above composite system have a feasible solution then such 
 
 id 
 placement of M objects is impossible, otherwise the placement is possible. 
 
 Unfortunately, even though we get one composite system to solve, if M 
 get large then the formulation to Integer Programming problem become somewhat 
 cumbersome as well as it increase the number of new variables which should be 
 introduced greatly. 
 2.3 GEOMETRIC APPROACH 
 
 This algorithm employs a face-to-face intersection analysis, which 
 first eliminates those faces which cannot possibly intersect the other 
 polyhedra, and then makes a pairwise analysis of the remaining faces. This method 
 will be discussed in the following chapter. 
 2.k SPACE PARTITION APPROACH 
 
 This procedure partitions the bounded space into smaller and smaller cubes, 
 and decides at each stage whether or not there can possibly be an intersection of 
 polyhedra in the cube currently being determined. Many large cubes may usually be 
 disposed of quickly. Those which cannot are partitioned further, if necessary 
 down to the desired limit of resolution. A similar approach has been used for 
 the hidden line problem [lj.1]. 
 
There are three cut planes we consider, 
 
 a). X-cut: a partition of a cube by a plane which is parallel to the 
 Y-Z plane, 
 
 b). Y-cut: a partition of a cube by a plane which is parallel to the 
 Z-X plane, and 
 
 c). Z-cut: a partition of a cube by a plane which is parallel to the 
 X-Y plane. 
 
 The decision procedure of which cut should be chosen for the next partitioning 
 simply depends upon the last two cuts which have been used for the partitioning 
 to get the cube, e.g., if last two cuts are an X-cut and a Y-cut then a Z-cut 
 may be chosen for the next partition. Of course one may choose the next cut 
 plane by considering the characteristics of the cube being cut. 
 
 At each stage of the partition, information about points, edges, and 
 faces in the two partitioned cubes is kept for use in further partitioning as "well 
 as in detecting intersection. The algorithm stops partitioning a cube as soon as 
 the size of the partitioned cubes become smaller than or equal to a predefined 
 cube size, called resolution cube. It then checks whether or not such a cube 
 contains portions of more than one object. If Lt does then the algorithm concludes 
 that those objects have a "pseudo-intersection. " The cube will be disposed from 
 the stack of cubes, SOC, which is shown in Fig. 1. 
 
 The algorithm stops for further partition of a particular cube being 
 determined currently as soon as it detects that the cube contains no object or 
 only a portion of one object. Such a cube will also be disposed from the SOC, 
 where the SOC contains only those cubes which require further partitioning to 
 detect intersections. 
 

 6bject 
 
 name 
 
 F,E,P 
 
 
 
 3) 
 
 c 
 
 I" 
 
 V 
 
 
 Cube k-1 
 
 
 Cube 1 
 
 .-' 
 
 r 
 
 V 
 
 n 
 
 5) 
 
 =n 
 
 Fig. 1 The stack of cubes, SOC 
 
The algorithm terminates completely either when the SOC becomes empty or 
 when M(M-l)/2 results are found, where M is the number of objects in the space. 
 
 It is quite often happens that some cubes in the SOC may contain portions 
 of some objects such that those objects have been determined to intersect in 
 an earlier stage of partitioning of some other cubes which were in the SOC. Thus 
 the algorithm disposes of those cubes in the SOC which only contain objects 
 whose intersections have been determined. 
 
 Fig. 1 illustrates a situation of the SOC at some stage. 
 
 To get an idea of the space partition approach, an example of the 2- space 
 case (3-space case is a relatively simple extension of 2-space), where objects 
 are denoted by polygons, is shown in Fig. 2. Cube 1 contains objects ftp ^ ft 
 and fi 6 while Cube 2 contains objects Sl ± , ft 2 , and ^ after a X-cut partition 
 has been performed. Cube 2 is further partitioned into Cube 2-1 and Cube 2-2 
 by Y* C ut, and the former contains Q ± and P^ while the latter contains only ft 
 thus Cube 2-2 is disposed from the SOC. At this stage there are two cubes, 
 Cube 2-1 and Cube 1 in the SOC. 
 
 We have the following two choices to select for the next cube partitioning 
 from the stack: the cube at the top of the SOC or the cube which has the largest 
 volume in the SOC. For example, for the first case, Cube 2-1 may be partitioned 
 further to Cube 2-1-1 and Cube 2-1-2 at the third stage, while Cube 1 may be 
 partitioned further to Cube 1-1 and Cube 1-2 for the second case as illustrated 
 in Fig. 2. 
 
 It is interesting to notice that the first selection method corresponds to 
 a tree search of depth first (see Chapter if) in endorder or preorder, and 
 the second selection method corresponds to a tree search of topdown in parallel. 
 
10 
 
 Cube 1 
 
 Cube 1-1 
 
 Cube 2 
 
 Cube 2-2 
 
 Cube 2-1 
 
 X 
 
 Fig. 2 An example of the space partition approach for 
 2-space case with polygonal objects. 
 
11 
 
 3- GEOMETRIC PROCEDURE FOR DETECTING INTERSECTIONS 
 
 in this chapter, we discuss a direct approach, called face-to-face inter- 
 section detection procedure, which first eliminates those faces of two objects which 
 cannot possibly intersect each other, and then makes a pairwise analysis on the 
 remaining faces. 
 
 We divide objects which are polyhedra or unions of polyhedra into two 
 classes: objects and obstacles. Objects can move around in the J-space, while 
 obstacles cannot and create a restricted, i.e., constrained, space for objects. 
 3-1 PKELIMHIABY DISCUSSION 
 
 We use a four-tuple n(P, E, F; p) to denote an object (or an obstacle) 
 which is a polyhedron in 3-space, where P is a set of points, E is a set of edges, 
 F is a set of faces and ? is a point called c-point defined as the average of the 
 
 points of the object fi: 
 
 _ n 
 
 P = (2! P-; )/n where P = fp '" p , v ] 
 
 A three-tuple fi(P, E; p) may be used to denote a polygonal object in 
 2-s P ace. Hereafter we consider polyhedral objects because similar properties 
 of the following properties for polyhedral objects are applied to polygonal objects. 
 
 A minimal sphere , denoted by S(r) of an object fl is defined as the smallest 
 sphere of center ? which contains a. Its radius is denoted by R. s(r) denotes the 
 a gfanal sphere of center p which is contained in a. Its radius is denoted by r. 
 In general, S(r) may be undefined for a non-convex object fi since p" may not be 
 contained in a. 
 
 S : 
 
 R = maxfdCp^p] 
 p^eP 
 
 N/ 
 
 
 S : 
 
 r = min{d(fj,p)} 
 fjeF 
 
12 
 
 where d denotes the Euclidian distance between two points or between a point and 
 
 a plane. 
 
 Let ft(P, E, F; p) and ft'(P', E', F*; p 1 ) be two objects whose minimal and 
 maximal spheres are S(R), S(r) and S(R'), S(r'), respectively. Then ft and ft« are 
 said to be strongly separated .if D > R + R', and are said to intersect strongly 
 if D < r + r', where D = d(p, p')« 
 
 The minimal box Il(l , I , I ) of an object ft is the smallest rectangular 
 1 x y z 
 
 parallelepiped whose three pairs of faces are parallel to X-Y plane, Y-Z plane, 
 and Z-X plane, respectively, such that it contains the object ft, where 
 
 I = [inf (ft), sup (ft)] 
 x x ' *x 
 
 I = [inf (ft), sup (ft)] and 
 
 j *y <y 
 
 I = [inf (ft), sup (ft)]. 
 
 2 Z Z 
 
 In practical cases we use II (I*, I*, I*) instead of H(l ,1 ,1 ) for the minimal box 
 r x y z x y ^ 
 
 
 of ft, where 
 
 
 
 
 I ci 
 
 X X 
 
 
 I* - 
 
 X 
 
 I = 2t 
 
 X 
 
 I C I* 
 
 y y 
 
 
 I* - 
 
 y 
 
 I = 2t 
 
 y 
 
 I c I* 
 
 X z 
 
 
 i* - 
 
 z 
 
 I = 2t 
 z 
 
 and t > is called a 
 
 toler 
 
 ance. 
 
 
 Property 1. 
 
 
 
 
 Let ft and ft' be two polyhedral objects whose minimal boxes are II (i -,1 ,1 ) 
 
 x y z 
 
 and IT' (i*, I', I 1 ), respectively. Then a necessary condition for ft n ft' {=§ is 
 x y z 
 
 n n .r /£$. 
 
13 
 
 Let us further assume ft jifl', ft • ^ ft , then the property 1 leads us to 
 the following. 
 Property 2. 
 
 ft Oft' ^<J> if and only if there exist at least a pair of faces (f , f ') 
 f e F, f e F' such that f flf ' /$, where f. n (IE n II ' ) /§ and f! D (n nil') /$>. 
 Property 2 implies the outline of our geometric approach to detect intersection 
 between two polyhedral objects. 
 
 It is enough to examine these pairs of faces (f . , f!), where f.eF, 
 f'.eF and f . , f! are at least partially contained in the intersection of two 
 minimal boxes II and H ' , called the solution box and denoted by II, to determine 
 whether or not ft and ft ' intersect. We may expect that the introduction of the 
 solution box reduces greatly the number of pairs of faces which should be considered 
 further for face-to-face intersection analysis, and it makes our analytic approach 
 competetive to the other approaches discussed in the preceding chapter. The solution 
 box II of two minimal boxes II and II' is defined only if the following conditions 
 hold. 
 
 max(inf (ft), inf (ft')) < min(sup (ft), sup (ft')) 
 
 max(inf (ft), inf (ft 1 )) < min(sup (ft), sup (ft 1 )) 
 «y y a y 
 
 and max(inf (ft), inf (ft')) < min(sup (ft), sup (ft'))- 
 
 Now we encounter to the following two questions. 
 
 (a) How to extract those faces of two objects which intersect the 
 solution box? 
 
 (b) How to determine whether or not a pair of faces intersect? 
 
lit 
 
 For the first question, one may use an approach to find "strictly only" 
 those faces of two ejects which intersect the solution box. But this leads us to 
 the detection of intersection between the solution box and each face of ejects. 
 Even though this kind of intersection detection procedure may he implemented 
 s^pler than the one between two given objects, it still has to contain some 
 cumbersome process. Therefore, instead of finding strictly only those faces which 
 have intersection with the solution box, we extract those which may "possibly" 
 intersect with the solution box. We consider the following simple approach to 
 solve the question, where the solution box of the given two objects is denoted by 
 
15 
 
 Il(l x^ I y' I z ) ^' We alS ° aSSUme that each face of Ejects is bounded by its minimal 
 box, called a face-minimal box (c.f object-minimal box). Thus our extraction 
 approach becomes to extract those faces such that the intersection of face-minimal 
 box and II is non-empty. 
 
 Some other properties to reduce the number of faces which should be 
 determined for face-to-face intersection detection are discussed in section 3.3. 
 The second question of determining the intersection of a pair of faces is 
 described in the following section. 
 3.2 FACE-TO-FACE INTERSECTION DETECTION 
 
 Let us think about a polygonal face f in 3- space, where the plane of f, 
 denoted by g f , is the hyperplane which contains the face f. Then it is clear to 
 see that any straight line L on g intersects an even number of edges of f (see 
 Fig. 3). 
 
 Let f i and f.. be two non-parallel polygonal faces whose hyperplanes are 
 
 s f . and g f > respectively. Let us also assume that L n denotes the intersection 
 
 1 J . u 
 
 line of these two hyperplanes (see Fig. 6). By arranging intersections between L 
 
 and edges of f . and f. in geometrical order from the left to the right, we form a 
 
 sequence of two different types of intersections; one for those on f . and the other 
 
 1 
 
 on f.. Then f ± and f are said to have even parity if these two types of points 
 are juxtaposed multiples of two by two each other. In other words, if and 1 
 correspond to intersection points between L Q and edges of f . and L and edges of 
 ty respectively, then such a 0-1 sequence has an even parity if the sequence is 
 contained in (00 + 11)*, otherwise it has non-even parity . 
 
 In Fig. k, examples of an even and a non-even parities on L are shown. 
 The state diagram illustrated in Fig. 5 shows how to recognize those sequences of 
 even parity. Those sequences which end up at state S are even parity and those 
 encountered in state S^ are non-even parity. 
 
16 
 
 -e - 
 
 Fig 
 
 . 3 The plane g~ of a face f . 
 
17 
 
 ■• • — * * — • • X * J* 
 
 (a) . Even parity. 
 
 "• *—• *- 
 
 ■*• — • — K »- 
 
 (b) . Non-even parity, 
 
 Fig. k Even and non-even parities on L 
 
 Fig. 5 The state diagram to recognize even parities, 
 i.e., sequences of the form (00 + 11) . 
 
18 
 
 -• •-* *-* •" 
 
 Fig 
 
 6 A face-to-face intersection: nOn-even parity, 
 
19 
 
 Theorem 1. 
 
 Let f. and f . be two non-parallel faces. Then f. and f . have no intersectic 
 i 3 i 3 
 
 if and only if their parity is even, i.e., they intersect if and only if the parity 
 
 is non-even 
 
 Proof: 
 
 (separation —^ even parity). 
 
 Let us assume that there exist a sequence S whose parity is non-even 
 
 and f. and f . separate, 
 i 3 
 
 S = s s . . . S . . . . s n : even number 
 12 l n 
 
 s . = or 1 for all i, s . = o for f . and 
 i ' l ■ i 
 
 s . = 1 for f ., 
 i 3 
 
 .. Z 
 
 = even, / = even. 
 
 s . =0 s . = i 
 
 i l 
 
 Since S is non-even parity, then there exist at least one subsequence S ' in S 
 
 such that s. = for s.eS' and Is'l is odd. Let us denote such a subsequence of 
 
 3 3 . ' ' 
 
 maximum cardinarity k, where k is odd, by 
 
 S ' = s . s . s s. = 000 
 
 x l x 2 x 5 \ 
 
 k 0's 
 
 Then there exist s. = in S-S' such that either (s.,s. ) or (s . ,s.) forms an 
 3 3 1-l i k 3 
 
 interval in L^ which is contained in f . . Without loss of generality, let (s.,si ) 
 i 3 1 
 
 form such an interval, then it is clear to see that there exist a subsequence S" 
 
 of all l's between s. and s. . This says f. and f . have intersection in the 
 
 3 i-j_ . x 3 
 
 interval corresponds to (s.,s. ), which leads us to a contradiction. 
 
 3 ij_ 
 
20 
 
 (even parity —a separation) 
 
 This is clear from the definition of even parity since intervals formed 
 by two by two selection of neighboring cross points on L Q for t ± and those intervals 
 for f. have no intersection. (Q.E.D. ) 
 
 Theorem 1 shows how to determine intersection between two faces by means 
 of the recognition of even parity. 
 
 To generate the sequence for a pair of faces f^ and f , we first 
 
 evaluate the intersection line L Q of g and g f . Then choose a sufficiently large 
 
 i 3 
 interval which may contain all intersections on L Q with f ± and f . From one end of 
 
 the interval toward the other end, slide a small piece of segment, called a bullet, 
 
 on L and detect intersection between the bullet and edges of f and f . When such 
 *■ J 
 
 an intersection is determined then generate or 1 depending on the bullet intersects 
 an edge of f. or an edge of f .. At each generation of either or 1, the parity state 
 
 1 J 
 
 changes between states which are illustrated in Fig. 5- 'Che bullet may be slided 
 until the other end of the interval if necessary. 
 3.3 FURTHER DISCUSSION 
 
 We introduce some other concepts which may be used to make a set of 
 these pairs extracted by means of solution box, as small as possible. In other 
 words we try to make the number of face pair s, which should be determined for 
 face-to-face intersection to conclude intersection of two given objects, small, and 
 consequently we save the computation time for intersection detection. 
 
 Let fi(P,E,F) and fi^P^E^F') be two polyhedral objects. If there 
 exists a face f eF^F' such that the plane g of f separates fi and fi' into two 
 different half-spaces, then it is clear to see the two objects have no intersection. 
 This existence of such a face is determined by the following way. Without loss of 
 generality, let us assume feF', let II denote the solution box of 0, and fi' aad let 
 P and P' be P <= It and P' c II, respectively, where P cPandP'cP'. Then if one 
 
 s s 
 
 I - g 7 r- x- s _ s _ 
 
21 
 
 of the following conditions hold, 
 
 (a) g f (P i ) > for p ._£?,,, and 
 g^fe '.) < for PleP' 
 
 (b) g f (P i ) < forP.€P s , and 
 g f (pp > for pj € P^ 
 
 then Q, and ft' have no intersection. 
 
 Two polygonal faces f. and f . are said to be mutually divisible if and 
 
 J- J 
 
 only if the plane g of f. divides points of f . into two half-spaces and the plane 
 g of f . divides points of f. into two half-spaces. A collection of such pairs is 
 denoted by: 
 
 S = f(f ,f ) f.eF, f.eF' and f . , f. are mutually divisible). 
 D LV i' y ' i ' 3 10 
 
 Then it is easy to see that if f. and f . are not mutually divisible , then f and f 
 have no intersection. In other words, the mutual divisibility of two faces is a 
 necessary condition that two faces intersect. Therefore, if S D for given two 
 object is empty then they have no intersection. 
 
 Let f be a face of a convex polyhedron ft(P,E,F;p) and p be any point which 
 is not contained in ft. Thenf is said to be visible at p if and only if 
 sgn(g|p)) = -sgn(g f (p)), where g f is the plane of f. 
 
 Let ft(P,E,F;p) and ft' (P' ,E ' ,F'jp' ) be two convex polyhedra. Then a pair 
 of faces (f ,f), f.eF. f.eF', is said to be mutually visible if there exist points 
 p . p', p is in f . and p! is in f!, such that f. is visible at p' and f' is visible 
 
 at p.. Such a collection of pairs is denoted by: 
 
 l 
 
 S = {(f ,f\) | f.eF, f.eF', f.,f. are mutually visible), 
 r i'j'ij i 3 
 
 From the above definition of mutual visibility, it is easy to see that the mutual 
 
 visibility of two faces f. and f. is a necessary condition that t ± and ft intersect. 
 
22 
 
 Theorem 2. 
 
 Let S denote a collection of pairs of faces which are mutually visible for 
 
 two given convex polyhedra Si and 0». Then SI c ft' or Q c ft' if and only if 
 
 Proof: Let us assume Si 3 ft' then there exist no point in any face of ft* that some 
 faces of ft are visible because by the definition of visibility. Thus ^ =$. 
 
 From the definition of mutual visibility, ^ =$ implies either p ± c ft' 
 for all P eP or p' c ft for all p'.eP'. Since ft and a' are convex, we get either 
 
 ficfl' orficfi'. Q.E.D. 
 
 Theorem 3« 
 
 Let ft and ft' be two polyhedra such that ft ft' /£$, ft £ ft' and ft £ ft'. 
 Then there exist at least three pairs of faces in S Q whose parities are non-even, 
 
 where 
 
 S B nS4 if n ' fi ' are 
 
 convex 
 
 S B n S D otherwise, 
 
 S : pairs extracted by solution box 
 B 
 
 S : mutually divisible pairs 
 
 Sfj : mutually visible pairs. 
 Proof: It is easy to see that S c /= § for the given condition. Let us assume neithe: 
 point of fl nor fl' contained in the other, then Si has at least 2 faces to intersect 
 ft' and ft' has at least 2 faces to intersect Si. Thus there are at least k pairs of 
 faces which are non-even parities. 
 
23 
 
 Let us assume fi contains a point p! of ft» then since p! is at least 
 3-points (which means at p' at least three inequalities corresponding to faces 
 become zero), there are at least three pairs of faces whose parities are non-even. 
 (Q.E.D. ) 
 
 The Vein diagram of mutually divisible pairs, mutually visible pairs, and 
 pairs by solution box is illustrated in Fig. 7, where we assumed Cl n Cl 1 £§, 
 Q £ CI * and Cl £ CI ' . 
 
 In some cases one may want to know to what degree two given objects 
 intersect. Therefore, let us introduce some measures of the degree of penetrat ion. 
 To measure the degree of penetration of two polyhedral objects Cl and Cl 1 , we define 
 P-deg as 
 
 P-<teg(ft,fl«) = max(7 v , 7 V .) 
 where V, V and v correspond to the volume of Cl, Cl 1 and Cl n Cl 1 , respectively. 
 Therefore, if objects have no intersection then P-deg(fi,fi') = 0. If one object 
 contains the other completely then it becomes 1, otherwise < P-deg (fi,fi*) < 1. 
 
 Let us assume that space is bounded and digitized into an n x n x n 
 point matrix, then the volume of an object may correspond to the cardinality of 
 the point set. Using this cardinality, we define another measure of the degree 
 of penetration, denoted by P-deg , 
 
 P-deg n (fi,fi') = max ( ^f^- , i^il ) 
 and lim P-deg (fi,fi«) = P-deg(ft,ft' ), 
 
 n 
 n-x» 
 
 where indicates cardinality of an object which is digitized. 
 
2»4 
 
 Pairs of faces with non even parity. 
 
 Fig. 7 Vein diagram of filtering processes on F x F' 
 
25 
 
 J>.k ALGORITHM AND EXPERIMENTAL RESULT 
 
 From the properties discussed in the preceding sections, we get the 
 
 following algorithm PD which detects intersection between two polyhedral objects. 
 
 A minor modification of the algorithm leads to the intersection detection of many 
 
 objects as well as the measurement of the degrees of penetration. 
 
 The main scheme of this algorithm is to first find those pairs 
 
 of faces, i.e., the candidate pairs, which possibly intersect by using the solution 
 
 box (if the number of such pairs is quite large then the further partition of the 
 
 solution box is carried out to minimize the number). Second to eliminate these 
 
 pairs which obviously do not intersect from the candidate pairs by means of mutual 
 
 divisibility and mutual visibility, and then apply face-to-face intersection 
 
 detection on the remaining pairs. 
 
 Algorithm PD. 
 
 Step 1. Evaluate S(R), S(r ) andS(R'), S(r') for fi{P,E,F;£) and fl 1 (P» ,E' ,F';p" ) , 
 
 respectively. Set D = d(p, p'). If D > R + R' then 0, and ft* are strongly 
 
 separated, terminate. If D < r + r' then Q, and ft' are strongly intersected, 
 
 terminate. 
 
 Step 2. Evaluate the minimal boxes II (I ,1 ,1 ) andH'fl ,1 ,1 ) for Q and ft* 
 
 v x ' y' z x' y z 
 
 respectively. Set H =n^H'. If II =<E> then fi and 0' have no 
 
 intersection, terminate. 
 
 Step 5. Evaluate the following sets. F is a subset of F such that each face of 
 
 s 
 
 F possibly intersects H, F' is a subset of F such that each face of F' 
 s ' s s 
 
 possibly intersect II. P and P' are subsets of basic point sets P and P', 
 
 s s 
 
 respectively, such that each point in the set is in n. 
 
26 
 
 If there exists a face f in F gy F^ such that for all P^F,, all p..€P g 
 either one of the following pair of conditions holds, then terminate. 
 
 (a) g f ( P± ) > and g f (p.) < 0, 
 
 (b) g f ( Pi ) < and g f ( Pi ) > 
 
 where g_ is the plane of f. 
 
 Step k. If the addition of two cardinalities, |fJ + |F^|, is larger than M 
 
 (where M is an integer about 10 ), then the following partition of the 
 
 solution box n is carried. Partition II into Jl ± and ig by the similar 
 
 way as used in "Space Partition Approach" described in Chapter 2. 
 
 Evaluate F and F q , F s D F o may not necessarily disjoint, such 
 
 s l 2 1 S 2 
 that each face of F s possibly intersect ]L ± and each face of F s 
 
 1 
 possibly intersect II . Similarly, evaluate F g and F g such that 
 
 each face of F' possibly intersects II and each face of F^ possibly 
 s-j_ - 1 2 
 
 intersects IL , respectively. If the above partition of II carried then 
 
 set S = F x F 1 u F x F' . Otherwise S = F x F\ 
 c s l s l 2 2 
 Step 5 If S =<!>, then fi and fi' have no intersection, terminate. 
 
 Step 6. Choose a pair (f.,fl) from S q , and set S q = S Q - (f^^)- « the pair 
 is not mutually divisible then the pair has no intersection, thus go 
 back to Step 5. (the mutual visibility of the pair is determined if 
 a and ft' are convex polyhedra). 
 Step 7. Determine the parity of the pair, if it is non-even then the pair intersex 
 thus terminate. Go back to Step 5. 
 
 It is easy to understand that if objects and obstacles are restricted to 
 a class of convex polyhedra or disjoint unions of convex polyhedra then some other 
 properties for convex polyhedra which have not been discussed in this paper may be 
 used to minimize the cardinality of the set of candidate pairs which should be 
 
27 
 
 determined for their parities. If objects and obstacles are restricted to a 
 class of polyhedra with only convex faces, then the face-to-face intersection 
 detection procedure, especially parity determination procedure, is greately 
 simplified. 
 
 From the computation time aspect, we found that it is important to 
 minimize the candidate pairs as much as possible since the time required for 
 parity checks is greater than those for the other filtering processes. 
 
 In Fig. 8, we show an example of one object and four obstacles in 3-space 
 which has been used as a data for Algorithm PD. The object called "UFO" is a 
 non-convex polyhedron consists of Ik- faces, 2k edges and 2k points. The obstacles 
 called "HOUSE" and "TREE" are represented by disjoint unions of convex polyhedra. 
 The HOUSE consists of 36 faces, 72 edges and 48 points. The TREE consists of 19 
 faces, 36 edges and 23 points. The obstacle called "ROCK" is a convex polyhedron 
 consists of 9 faces, 16 edges and 9 points. 
 
 In this example, the object UFO was moved and rotated around the 
 obstacles and detected intersection. The average run time for detecting intersection 
 for the illustrated data example was about a quarter to a half second, where the 
 algorithm was written in PL/l and implemented on 360/75. 
 
28 
 
 Fig. 8 An example of the simulated objects for 3- space 
 
29 
 
 k. A PROBLEM IN PATH FINDING 
 
 One of the most basically important studies in artificial intelligence 
 is to find an efficient algorithm for the shortest path in a given state space. 
 This area has been studied for many years and various shortest path algorithms and 
 their applications maybe seen in [3, 4,5,9,10,15,17, 20,21,23,27, 28, 30,31,31*]. 
 
 A problem of both practical and theoretical interest is to find efficient 
 
 algorithms to determine if an object ft can be moved from one position p to 
 
 u s 
 
 another position p t , inside a complicated geometric structure C. In some problems 
 there will be no path from p g to p t and some others there will be several paths 
 of which perhaps the shortest must be determined. A restricted problem of this 
 type, the two-dimensional sofa problem [13, 2 k , 36], has been studied by Howden [17]. 
 
 We are interested in a more generalized formulation of this problem 
 as stated below. There is a given set S of states, which may not necessarily be 
 
 specified explicitely, and a set T = {t^ t ? , , t fc ) of transformation provided 
 
 for n Q on S. A transformation will be applicable to some, but not necessarily all, of 
 the states. Thus the problem is to determine a sequence r of transformations, 
 
 r -vvs X> 
 
 such that an application of r on Q Q moves fi Q from a state a to another state p, the 
 target or the goal, and at each state fi Q is under certain given constraints. 
 
 This problem mainly consists of the following two parts. One is the "state- 
 search" problem where we wish to find a state with some particular property and the 
 other is the "path-search" problem where we seek a sequence of transformations between 
 two states. 
 
30 
 
 We assume an automaton A of ft with no visual information but with whole 
 information of territory as well as the shape of the A Q . It was pointed out by 
 Ernst [9 1 that an automonous object must have a picture of the world, and a notion 
 of its own location and state, in order to predict the consequence of its actions 
 and hence to generate alternative courses of action, and decide among them. An 
 internal representation of the environment has two fundamental characteristics that 
 determine the amount of relevant dimensions of the environment and the resolution 
 along any single dimension. The former although dictated largely by the nature of 
 the environment itself, may also be a function of the given task. 
 
 We hereafter consider an automonous automaton A. of ft with transformations- 
 translations and rotation, and obstacles ft., i = l,2,...,m which form a constrained 
 environment C for ft , where each ft. for i = 1,2,. . . ,m is restricted to an 
 undeformable polygon. The assigned task on the automaton A Q is to find a path from 
 a given starting state a to another given goal state which may be optimal under the 
 satisfaction of the given environmental constraints. 
 k.l STATE SEARCH AND STATE CONTIGUITY 
 
 When an automaton A of ft moves from the present state a ± to the next 
 state a , it must know whether ft^ can occupy a. under environmental constraints, 
 and should also determine whether or not an application of some transformations 
 provided for A achieve the motion from a. to a., in some sense continuously. The 
 first question corresponds to the problem of solving intersection between ft and 
 obstacles ft., i = l,2,...,m, in C. The second corresponds to the problem of 
 solving the question that A of ft can actually move from a. to a. under the 
 satisfaction of environmental constraints, e.g., if some rotational transformations 
 on ft are required to achieve a. such rotations should be contiguous . The contiguity 
 
 ^ J 
 
 relation between two states will be defined later in this section. 
 
31 
 
 A procedure for the detection of intersection between ft and ft , 
 
 i' 
 
 i = 1,2,... ,m, which are polygonal obstacles, is a simplification of the 
 intersection detection procedure developed in Chapter 3. Let II and II., 
 i =l,2,...,m, are the minimal boxes for ft and ft . , i =l,2,...,m, respectively. 
 If n o nn i =$ for all i then ft Q has no intersection with any of obstacles. If 
 n niI i £® for some 1 > then extra °t those edges of ft and ft. which may intersect 
 with the solution box JL Q ^ = I^fl n . If the number of edges which are extracted by 
 
 the above process is quite large, then n is further partitioned and the procedure 
 
 i 
 minimizes the number of pairs of edges which should be determined for intersection. 
 
 Let (e , e ) is a pair of edges such that e.ni 1$ and e fin l§ , 
 
 where e eE Q of ft Q and e k eE i of fl Then the mutual divisibility between e. and e 
 
 is determined to detect intersection between & n and ft . 
 
 1 
 
 To discuss state contiguity, let us first consider an example which is 
 
 illustrated in Fig. 9 , where the automaton A can occupy the next state a from a 
 
 (J 3 2 
 
 with a translation and rotation of degree it/2. But it is easy to see that the 
 
 rotation cannot be achieved either at a, nor at a, , i.e., even though a satisfies 
 
 d 3 3 
 
 the given constraints, transformation on ft A from au to a_ is not achieved. Therefore, 
 
 u d 3 
 
 one way to achieve a , ft Q should be rotated by an angle jr/2 at the previous state 
 
 Q^ before A Q applies translation on ft to a . 
 
 . Two states Q^Cx^y^,^) and a 2 (x 2 ,y 2 ,0 2 ) are said to be contiguous if A 
 
 has a translation t ± : (x^y^ -> (Xg,y g ) with t" 1 : (x 2 ,y 2 ) -» (x.^) and 
 
 9 i n ®2 ^®> where and 9 denote admissible range of rotations at (x ,y ) and 
 
 (Xp>y,>)> respectively. Let a_(x,,y_,0_) be another state and assume a. and a 
 ^ d 3333 23 
 
 are contiguous. It is easy to see that, in general, the transitivity relation of 
 contiguity does not hold, i.e., cl and a contiguous and a and a contiguous does 
 not imply a^ and a are contiguous through a . Fig. 10 shows the admissible range 
 
32 
 
 / / / 
 
 *L 
 
 / 
 
 / 
 
 L-^-^ 
 
 n 
 
 • 
 
 I 
 
 / 
 
 // 
 / 
 
 <\ 
 
 /////// 
 
 oft 
 
 Fig. 9 State contiguity. 
 
33 
 
 at(X. 
 
 atoC 
 
 at 
 
 °s 
 
 Fig. 10 Contiguity on rotational transformations. 
 
5U 
 
 of rotations at Q^ Ofe, and ^ which correspond to the illustration of Fig. 9. 
 If G 2 consists of a single interval of admissible rotation, then ^ a 2 contiguous 
 and a 2 , ^ contiguous imply the contiguity between o^ and ^ through ofe, i.e., 
 the distance of transformation from C^ to ^ through a^ under the satisfaction 
 of environmental constraints. This contiguity relation should he determined for 
 each state if a transformation from one state to another require rotations. 
 
 4.2 PATH SEARCHING 
 
 We will consider heuristic tree search approaches while a straight forward 
 
 exhaustive search procedure has been taken by Howden [17 ]. 
 
 A heuristic tree search [3, 30, 31] consists of the following two 
 parts: one is constructing a path tree for admissible states given by state search, 
 and second, an evaluation function which estimates a future path distance (or the 
 number of transformations which may be required to move automaton A Q of fi Q from 
 the state being considered) to the goal state. 
 
 Let us first consider the problem of searching a tree to find a given 
 goal state. If it is necessary to search the whole tree, i.e. , to find the node j 
 which is optimal in some sense (by optimal we mean the sequence V of transformations 
 with minimal cardinality), the sequence in which the search is carried out is 
 unimportant. If, however, we are content to discover Just one state with the given 
 property the sequence in which states are searched may be of great interest. 
 We will consider the following tree searching. 
 Search Controlled by an Evaluation Function 
 
 in this method of search we may think of a "frontier" of those nodes whose 
 successors have not been examined. This frontier is extended by choosing any node 
 in it and examining its successors. The node chosen may be that which has the 
 
35 
 
 minimum value of some functions called "evaluation functions" normally chosen on 
 heuristic grounds. (Such a function may be stated as the maximum desirability 
 and sometimes called "reluctance function". ) 
 
 This method of search has been extensively discussed by Doran and 
 Michie [3] who used it in their program called the "Graph Traverser". They have 
 shown that given a suitable evaluation function it is an effective method of 
 trackling a number of problems. 
 
 The main point to notice is that the next node whose successors are 
 examined is not necessarily one of those produced at the previous move, i.e. 
 the frontier will be pushed forward for a while in one region, but if the evaluation 
 function indicates that further advances here are not as profitable as had been 
 anticipated, some other part of the frontier may be extended. 
 Measures of Performance [35] 
 
 The three most interesting measures of tree search programs performance 
 over a particular search of a graph are 
 
 (i) the length of the path produced ( |r | ) 
 (ii) the total number of nodes developed (D) 
 (iii) the total number of nodes added to the search tree (n). 
 Since every node on the constructed path except the last must have been developed, 
 it follows that |r I < N. Denote the minimal path length for a given start and 
 goal by |r*|. This will also be the minimum number of developments which must be 
 made to find a path from the start to the goal. Then we define the path efficiency 
 as |r*|/|r|. Similarly, we define the development efficiency as jr*|/D. The 
 calculation of both these efficiency requires a knowledge of |r*|. Typically, 
 however, |r*| will be unknown. The ratio |r|/D, which is the fraction of the 
 total number of nodes developed which are incorporated into the actual path 
 found, can always be calculated, and is of considerable importance. It will be 
 
56 
 
 (a) . low penetration 
 
 (b) . high penetration 
 
 Fig. 11 Two search trees which illustrate the "penetration" 
 of a search tree. An "elongated" tree has a high 
 penetration , and a "husky" tree has low penetration. 
 
37 
 
 called the "penetration" of a search, or partial search, and may "be thought of 
 as representing the degree to which the search tree is elongated rather than 
 "bushy" (see Fig. 11 ). 
 4.3 SEARCH TREE PRUNING 
 
 There is a limit to the size which the search tree of the automaton 
 A Q can attain. Either this will be fixed by the storage capacity of the automaton 
 upon which the search tree program is running, or else it will be determined by 
 the time taken to work with a large tree. The reaction of the program when all 
 the available storage space (about 10 to 30 nodes) has been filled by the tree 
 is described as the partial path is fixed upon, all the tree except a single 
 node is erased, and a new partial search begins (this scheme is illustrated in 
 Fig. 12 (a)). This is unnecessarily catastrophic. There is no need to throw 
 away so much hard-won information and it seems unreasonable to do so. 
 
 Another version of pruning commits itself only to a single additional 
 step along the path (or some specified number of steps : 1 to k) , and only 
 that part of the tree thereby rendered valueless is discarded. This scheme is 
 illustrated in Fig. 12 (b). 
 
 Of course, it is easy to imagine more complicated pruning schemes, e.g., 
 It would be possible to detect high-valued branches 'without committing the program 
 to any additional path segment, however, we use the second approach of search 
 tree pruning of developed nodes about 10 to 30, while [5"] used J+0. 
 k-.k EVALUATION FUNCTION AS A DISTANCE EVALUATOR 
 
 An evaluation function estimates the distance from a state to the goal 
 state. Let us now define a distance estimator as a function which when applied 
 to any two states, specifically estimates the distance over the provided space 
 from the first to the second. Notice that we may discuss good or bad distance 
 estimators for a particular state space (or graph) quite independently of any 
 
38 
 
 % f 
 
 h i 
 
 (a) . static tree. 
 
 (b) . dynamic tree 
 
 Fig. 12 Illustration of the operation of static and 
 dynamic search trees. 
 
39 
 
 problem associated with that space. Nevertheless, given a good distance estimator, 
 a fortiori we have a good evaluation function for use in any particular search 
 over space. 
 
 Now we define the distance estimator as a linear convex sum of m 
 
 heuristic functions, 
 
 m 
 
 H(a.,a t ) = Z w k h k (a.,a t ) 
 
 k=l 
 
 m 
 
 / w. = 1 
 k=l K 
 
 where h^a ,0!^) is the kth function based upon heuristic grounds between a. and 
 a t in space C (examples of evaluation function are shown in the next section), 
 
 \ = ^( a ±> fi o' C ^ is the wei S ht for the kth heuristic function and is a function 
 of the present state a ± of the automaton A in space C. In general, the weight 
 \ for \( a ±> a is deter mined by A Q considering the local and grobal 
 
 environment at the state a. . 
 
 l 
 
 The idea of the linear convex sum of some heuristic functions is 
 intuitive since it is almost always impossible to find a quite simple single 
 function which can be applied to any given geometrical space and works effectively 
 at any state of the given space. In other words, some functions are efficient for 
 certain geometrical configuration while some others may be very inefficient for 
 such configuration. Our distance estimator H(a. , a.) can be interpreted as 
 a simple structure of the reticular formation in animals [ 39]. 
 
 In the previous sections we discussed several trees whose devlopment 
 is so called "uni-directional. " According to our assumption that the given 
 
 automaton A Q knows the picture of the world C, we consider a version of so-called 
 "bi-directional" search tree [31], which may be called "rendezuous. " This is an 
 
 approach that from both present state a. and A^ and the 
 
 1 
 
>.'0 
 
 present state p. of the image of A Q , denoted by A« of n • , the automaton 
 develops two search trees. One towards p ± from a. and the other from ^ to a., 
 and try to intersect £l Q with the image Jj£. 
 
 Let a. and p. be the ith states of A Q and A£, respectively. Then our 
 
 distance evaluator may be modified to: 
 
 m 
 H(a., p,) = ^ * k \ (c V Pl ) 
 
 k=l 
 
 where * k = y?(a., p., n Q , fl£, C) , for all 
 
 k. 
 
 The decision process of which search tree should be developed at ith stage 
 depends on the local environments of the automaton and the image of itself, 
 and it is not specifically discussed in this paper. 
 k.5 EXAMPLES OF DISTANCE EVALUATION FUNCTION 
 
 It is desirable to consider a function to evaluate the distance between 
 
 a and p with straight line segments as the one illustrated in Fig. 15. However, 
 
 i i 
 this may not be efficient on computational aspect unless very fast algorithm for 
 
 such a scheme is easily implemented. In fact, if C consists of many obstacles 
 
 and if large numbers of obstacles are located around a ± and p ± , it may quite 
 
 often lead to a very inefficient procedure to estimate distance between a. and p.. 
 
 Since we may not strictly require the automaton A Q to take one of the optimal 
 
 paths if any exists, but rather hope that it may take a path which is close to an 
 
 optimal path as well as the path seeking time is small, we may have some functions 
 
 which are not quite like the one in Fig. 13, but yet they are easy to implement 
 
 as well as in some sense "good" evaluation functions. 
 
 Some of simple heuristic distance estimation functions for given two 
 
 states a and p are illustrated in Figures Ik, 15, l6, IT, and 18. For example, 
 
 h (a. , p. ) function considers those obstacles which intersect the line segment 
 
m 
 
 
 . . . , 
 
 
 //// / 
 
 
 ///// 
 
 
 ///// 
 
 
 ///// 
 
 
 
 // 
 
 
 
 '//// 
 
 // 
 
 M 
 
 / / 
 
 ///// 
 
 ■'-/ 
 
 '////, 
 
 Fig. 13 Distance estimator with straight line segments 
 
>!2 
 
 a 
 
 Fig. Ik Distance evaluation function h^(Gt^, ^), around 
 those obstacles on the line segment. 
 
') 3 
 
 a 
 
 Fig. 15 Distance evaluation function h^ (a. , 6.), around 
 
 2 1 1 ' 
 
 only the nearest obstacle to Qt.. 
 
kk 
 
 <%i 
 
 Fig. 16 Distance evaluation function h (a ± , 3 i ), around 
 
 only the minimal box of the nearest obstacle to a 
 
1*5 
 
 p i 
 
 Fig. 17 Distance evaluation function h, (a. , 6.), around 
 
 the union of the minimal boxes of those obstacles 
 on the line segment. 
 
h6 
 
 a. 
 
 h 
 
 Fig. 18 Distance evaluation function h (a ± , $ ± ) , around 
 the minimal boxes of these obstacles on the line 
 
 segment. 
 
 
 
 
>i7 
 
 CH.,B. and take the direction corresponds to the minimum path of all 2 paths, 
 
 where M is the number of obstacles on the line segment. The function h„(<x , B. ) 
 
 2 1 1 
 
 considers only one obstacle which is nearest to the state a. and take the direction 
 
 i 
 
 of the minimum of two possible paths, and so on for h , h , and h . Of course, 
 there may be some other functions other than h through h which can be easily 
 implemented. Some of these illustrated distance estimation functions may be ill- 
 defined depending upon where a. and 8. are located with respect to these obstacles 
 
 on the line segment (X..&.. 
 
 A procedure called BSA, bullet -shooting algorithm, is implemented in the 
 
 automaton A_ to detect the obstacles on the line segment ex. ,6. as well as the 
 i 7h i 
 
 first obstacle which is hit by the bullet, if there exist any obstacles on the 
 segment. A bullet of length 5 is shot from a. toward p. on the line segment, 
 
 x = (x! - x. )T + x. 
 i l i 
 
 y = (yJ - y i )T + y ± 
 
 o < t < 1 
 
 a ± = (v y ± ) 
 
 Pi = (^, y:) 
 
 with a step motion V, < V < 8, and detect intersection between the bullet and 
 obstacles, if any are on the line segment. If there are no obstacles on the line 
 segment, the automaton moves toward p. . 
 
 When the automaton is located in a region with many obstacles, called 
 "bush", then it may be desirable for the automaton A to have a decision process 
 whether A should first get away from such a bush and then seek for his image 
 state p. rather than seeking the image through the bush. Which means it is 
 desirable for him to have an ability to decide whether the bush is local or global. 
 
'48 
 
 Fig. 19 Direction parities: L-parity and R-parity which assume 
 . either even or odd. 
 
1*9 
 
 Let us assume an automaton is located in the interior of an obstacle, 
 this situation is illustrated in Fig. 19- Then, sometimes or quite often, it takes 
 direction to seek the goal in which it never achieves the goal until after such a 
 space is almost exhaustively searched and then it realizes the direction which 
 has "been taken before was wrong. For example, in Fig. 26, the automaton never 
 achieves the goal if he started to look for the goal in the right space with 
 
 respect to the line a. ,8.. This situation can be easily avoided by the following 
 
 approach. 
 
 For the left and the right direction from <X toward p. , we define 
 
 direction parities as the number of intersection of edges on the line segment 
 
 a ,8.. L-parity is found by the left edge following and R-parity is found by 
 i ; l 
 
 right edge following. The L-parity and R-parity assume either even or odd 
 values, thus the automaton searches for the goal in the space whose parity is 
 even. This condition is illustrated in Fig. 19. 
 k.6 SIMULATION RESULTS OF GOAL SEEKING AUTOMATON 
 
 In this section, we present some computational results of our goal 
 seeking automaton A of shape fi n under the assumption that A has complete 
 information of the world C. Q, of A and obstacles ft., i=l, ..., M are assumed 
 to be polygons. 
 
 Transformations provided for A are R and either T, or T Q : 
 
 \ = <V V v V 
 
 T Q = {t^ t 2 , t y t^ t 5 , t 6 , t ? , t Q } 
 R = {r:n} 
 
 where T. and T are two sets of translations, and R consists of rotations. The 
 clockwise rotation of ft at its center point p is denoted by positive r where 
 r = it/n and the counter clockwise rotation by negative r. T, and T Q are illustra- 
 ted in Fig. 20. 
 
50 
 
 
 K 
 
 (a) T 
 
 X 
 
 (b) T c 
 
 Fig. 20 Sets of translations: T, and T Q . 
 
51 
 
 Let ft Q (P, E; p) andp^(P', E'; p 1 ) denote the polygons of the automaton 
 A and its image A^ from the goal state, respectively. In general, we have the 
 following linear convex sum function as a distance evaluator. 
 
 m 
 
 H(p,p') =w n d(p*,p-') + wh (p,p') + (1/n) Z Z wh (p.,p.') 
 
 3 k=l i=l k k x x 
 
 p.eP, p.'eP' , |p| = IP 1 I = n 
 1 ' i 'it ii 
 
 m n 
 
 w + w + (l/n) Z Z w = 1 
 
 --L U . -. . _ K 
 
 k=l 1=1 
 
 where d denotes Euclidian distance function. Since it is natural to set 
 dfop 1 ) = h (p,p f ) if there are no obstacles on the line segment pTp 7 , and 
 moreover if some obstacles are on the line segment then we want to set w =0, 
 we implement d function in each h., j = 1, 2, . .., m. 
 
 J 
 
 We use the following simplified form of (l) as our distance evaluator 
 
 n 
 H(p,p T ) =wh k (p,p') + (1 -u))/ n Z VP^Pp (?) 
 
 i=l 
 
 This evaluation function can be interpreted as follows. The first term 
 contributes mainly to translation of Ci Q while the second term contributes to the 
 rotation of Q Q . Thus the contribution to either translation of fi or to rotation 
 of a Q is controlled by the value of w , < u < 1. If w is small then rotational 
 transformations may be applied on fi in the earlier stages of path seeking and 
 later translations may be applied on o . On the other hand, if w is quite 
 large then translations are considered more important than the rotation, thus we 
 may inspect that rotations of o will be remained until the end of path seeking. 
 It is also important to notice that the second term in (2) also contributes to 
 translations of (] ( . If w = 1 for example, fi Q is translated into the goal with no 
 relational transformation, while for w = C both rotation and translations are 
 applied on n to lead to the goal. 
 
52 
 
 / - 
 
 obstacles 
 
 H(5, p') = wh(p, p') + ((l-w)A)\ h( Pi ,pp 
 
 Fig. 21 .Distance e valuator H(p, p'). 
 
53 
 
 Develop Tree. 
 State Search. 
 
 Path Search 
 
 Tree Pruning 
 
 Advance Automaton 
 
 •>- Stop 
 
 Fig. 22 Diagram of the automaton A 
 
 0' 
 
5k 
 
 A simple example of this distance evaluator H(ir,p" ) is illustrated in 
 
 Fig. 21. 
 
 We describe a simplified diagram of our goal seeking automaton A Q in 
 
 Fig. 22, where the function of each block is almost as same as discussed in the 
 
 previous sections. 
 
 Under the above assumption, we let our automaton A Q of n Q seek the 
 specified goal, denoted by P t , from another specified starting position P g in 
 various world C. Some of the results obtained by A Q are illustrated in Fig. 23 
 through Fig. 36. Fig. 23, for example, the world C consists of two obstacles 
 fl and fi 2 which are bounded by ft^ Q Q of A Q is a triangle and h fc of H(p,p') 
 is simply chosen as ^ of Fig. 15, and the weights is set to 1 which means the 
 second term of H(p-,p*) is omitted thus the path attained by A Q consists of only 
 translations from T. . Fig. 2k shows the same constraints as the one in Fig. 23 
 except h, is chosen as h of Fig. Ik, and the provided translations are T Q . As 
 we can see easily that for T Q the seeked path is smoother than the one achieved 
 by T , and the choice of h eliminates the "zig-zag" walk which happened by h 2 
 
 in Fig. 23. 
 
 Fig. 25 through Fig. 33 show the results under the similar assumptions 
 applied in cases of Fig. 23 and Fig. ?k in different world C. In Fig. 3U and 
 Fig. 35, the weight u is set to less than 1 and rotational transformations 
 R = { r; 6}, are applied to achieve the goal. As we can see from these figures, 
 the change of the value of w between and 1 indicates that the necessary 
 rotations are distributed on the path sequence and such a distribution of rotation: 
 can be controlled by changing the value of w . Fig. 36 shows the case where 
 u) =0.5 and rotational transformations, some of them are redundant, are 
 distributed on the seeked path sequence almost evenly. 
 
55 
 
 Fig. 25 h^ = h 2 , w = 1 
 
 and T, 
 
56 
 
 Fig. 2>i \ - h 1 , u =1 and T^ 
 
57 
 
 Fig. 25 h^. = h , w = 1 
 
 and T, 
 
58 
 
 
 Fig. 25 \ = h 2 > w = 1 and T 6 
 
59 
 
 Fig. 2& 
 
 \ = V W = 
 
 1 and T, 
 
60 
 
 Fig. 27 \ = \> w = 1 and T £ 
 
6l 
 
 Fig. 28' 1^.= h 2 , w = 1 
 
 and T, 
 
62 
 
 I I 
 
 Fig. 29 h. = h_, w = 1 and T c 
 k 2 c 
 
63 
 
 Fig. 3 h^ = _h 2> u> = 1 and T u 
 
6>. 
 
 Fig. 51 \ = h 2 , w = 1 and T £ 
 
65 
 
 Fig. 32 h^ = h 2 , w = 1 
 
 and T 
 
66 
 
 Fig. 35 h. - h. , U - 1, T, 
 
67 
 
 Fig- 3 1 ' \ = \>"u = °- 2 > T s and R = ^ r ' 6 ^ 
 
68 
 
 Fig. 35 \ = V W = °' 8 ' T 8 and R =jLr -^ 
 
69 
 
 Fig. 36 h^ = h 2 i w = 0.5, T Q and R = {r;6} 
 
70 
 
 The average |r*|/|r| is found to be greater than 85 percent and 
 |N|/|r*| are located between k and 16, depending on the value of w as well as 
 the world C. 
 
 The program was written on PL/l language and implemented on 360/75, 
 the average run time of each step is about 30 to 80 mili seconds. 
 
71 
 
 5- CONCLUSION 
 We have developed an intersection detection procedure for polyhedral 
 objects by means of face-to-face intersection analysis. It first eliminates 
 those faces of two objects which cannot possibly intersect each other. Such an 
 elimination of faces, called "filtering", has been done by solution box approach 
 and then by mutual divisibility between two faces as well as by mutual visibility 
 if considered objects are convex. It then makes a pairwise analysis, such as parity 
 mode detection, on the remaining faces. 
 
 From our experiment of running the Algorithm PD on various data, we 
 conclude that the procedure developed in Chapter 3 is efficient not only for 
 the determination of the placement of physical objects, but also for the 
 simulation of moving objects in physically constrained space. 
 
 As an application of the intersection detection procedure developed, 
 a modification of the Algorithm PD for polygonal objects was made and applied to 
 solve the path finding problem in two-dimensional space. The method used for 
 solving the problem may be modified relatively easily to solve the path finding 
 problem in three-dimensional space. 
 
 Another approach for the solution of the path finding problem as well 
 as for the solution of the original sofa problem [13, 2k, 36] by means of 
 "sequential pattern" representation of polygonal regions is our current interest 
 and is being considered. 
 
 Each algorithm described in this report has been programmed in PL/l 
 and implemented on a 360/75 at the University of Illinois. 
 
72 
 
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