GEOMETRIC CONSTRAINT SOLVING IN < 2 AND < 3. Department of Computer Sciences, Purdue University. and PAMELA J. VERMEER

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1 GEOMETRIC CONSTRAINT SOLVING IN < AND < 3 CHRISTOPH M. HOFFMANN Deartment of Comuter Sciences, Purdue University West Lafayette, Indiana , USA and PAMELA J. VERMEER Deartment of Comuter Sciences, Washington and Lee University Lexington, Virginia 4450, USA ABSTRACT Geometric constraint solving has alications in a wide variety of elds, such as mechanical engineering, chemical molecular conformation, geometric theorem roving, and surveying. The roblem consists of a given set of geometric elements and a descrition of geometric constraints between the elements. The goal is to nd all lacements of the geometric entities which satisfy the given constraints. In two-dimensions, several dierent aroaches have been examined and imlemented, while the three-dimensional roblem has been much less exlored in revious literature. In light of the diverse alicability of the roblem, we have three objectives in this aer. First, we rovide a brief overview of the basic aroaches to geometric constraint solving. Second, we review a secic solution to constraint solving for two-dimensional geometric roblems. Finally, we resent develoing work in extending the solution technique for the two-dimensional roblem to geometric constraint solving for elements in three-sace. 1. Introduction Geometric constraint solving is a roblem with alications in many arenas, such as mechanical engineering, chemical molecular modeling, and surveying. In each of these communities the roblem has been aroached in a variety of ways and with diering levels of success. The roblem consists of a given set of geometric elements and a descrition of geometric constraints between the elements. The goal is to nd all lacements of the geometric entities which satisfy the given constraints. For examle, the set of elements might be a set of three lines, with the constraints that the rst two lines must be erendicular and the third must make a secied angle with the rst line. This articular roblem has innitely many solutions, and an additional constraint such as the length of one segmentbetween the intersections of two airs of lines would tie down a articular solution. A roblem is well-constrained if there are a nite number of solutions to the roblem, while a roblem with an innite number of solutions is underconstrained. A roblem is overconstrained if one constraint can be deleted yet the constraint system

2 still has a nite number of solutions. In the examle of the three lines, if the angle that the third line must make with the second line were given as another constraint, the roblem would be overconstrained, since that angle is already determined by the rst two angle constraints. An overconstrained roblem may have a solution when the additional constraints are consistent with revious constraints, but often overconstrained roblems have no solution. One alication for geometric constraint solving is in the area of Comuter Aided Engineering, articularly in the branch of mechanical engineering design. A method for designing an object with the comuter should be strongly visual in order to rovide an intuitive interface, but must also rovide a way to roduce a careful, detailed descrition. One aroach for reaching these two disarate goals is to aly geometric constraint solving to the roblem of geometric data inut. Through a grahical user interface, the user can sketch a rough outline of the object to be constructed. By adding constraints such as the length of an edge of the object or the angle between two edges, a recise descrition of the object is obtained. Such a system has been designed and imlemented with oints, lines, and circular arcs in the lane as allowable geometric entities, and constraints such as an angle between entities, distance from one entity to another, incidence, and tangency (in the case of circular arcs) 4. The object obtained as the solution to the constraint roblem can then be swet or rotated to obtain a three-dimensional object. Additional features can be added by sketching on a two-dimensional lane of the resulting object, and extruding the sketched region, or cutting a slot or hole in that shae through the object. As an examle of this alication, consider Figures 1,, and 3, which demonstrates the design of a control arm. These gures were generated using the featurebased modeling system of Chen 11, which interfaces with the two-dimensional constraint solver of Fudos 5 for its geometric inut. Figure 1 on the left shows a role which has been sketched, dimensioned, and solved by the constraint solver. Once the user is satised with the role, it is extruded, generating the solid on the right of the gure. To add another feature to the model, a lane of the current solid is selected in which to sketch and dimension the new feature. Here, the to face has been selected as the reference face, and the role of a new feature sketched, dimensioned, and solved, as shown on the left of Figure. This new role is then extruded, yielding the solid shown on the right of the gure. Several other features, are added in similar fashion, and the nal control arm is dislayed in wire frame and shaded in Figure 3. A second alication area for geometric constraint solving is in geometric theorem roving. Often the assumtions and conclusions of such theorems can be exressed in terms of constraints between the geometric elements under consideration. Solving the constraint system entails that the relationshis are ossible, hence constitutes a roof of the theorem. An examle of this alication can be found in Chater 7 of Homann 7. Yet another alication of geometric constraint solving is in molecular conformation in chemistry and biology. This entails ositioning atoms, reresented by

3 Figure 1: A two-dimensional constraint roblem is solved and extruded to obtain a threedimensional object. Not shown are the tangency constraints between the line segments and the arcs. Figure : A feature is added using the role sketcher and a further extrusion. An additional constraint not shown in the diagram is that the two arcs on the right are concentric. Figure 3: Final control arm

4 d 1 α β d γ Figure 4: A well-constrained sketch for a generic solver oints in three-sace, so that they satisfy certain distance relationshis. Clearly, this involves solving a three-dimensional geometric constraint roblem. As these examles of alications demonstrate, there is a wide variety of uses for geometric constraint solving in two and three dimensions. In light of the diverse alicability of the roblem, we have three objectives in this aer. First, we rovide a brief overview of the basic aroaches to geometric constraint solving. Second, we review a secic solution to constraint solving for two-dimensional geometric roblems. Finally, we resent develoing work in extending the solution technique for the two-dimensional roblem to geometric constraint solving for elements in three-sace.. Basic Aroaches to Constraint Solving Beginning with a set of geometric elements and certain constraints between the elements, there are two basic strategies for solving the roblem. The rst, an instance solver, immediately uses the exlicit values of the given constraints to determine the ossible geometric congurations which satisfy the constraints. The second, a generic solver, determines whether the given geometric elements can be laced using the given constraints, indeendent of the values which are assigned to the constraints. That is, the constraints have a symbolic rather than numerical value. The determination of secic lacement of the geometric elements in a generic solver takes lace only after a decision has been made about whether or not the roblem is generically well-constrained. As an examle of the two dierent aroaches, consider the sketch of Figure 4. Here the constraints are given symbolically, rather than with actual values. A generic solver would be able to reort that the conguration is well-constrained, and would be able to determine a method for constructing the ossible congurations without needing the actual values of the constraints. An instance solver requires that the values of the constraints be given before it makes any solution determination. Generic solvers are often more elegant and more ecient than instance solvers. They also allow more exibility in the choice of the underlying method alied to determine the actual ositions of the geometric elements. However, generic solvers usually are not able to handle the case of overconstrained but consistent

5 Figure 5: A well-constrained sketch may yield fundamentally dierent solutions. systems, while most instance solvers can nd the solution. In the examle above, a generic solver may be able to determine that there is a solution, yet may not be able to construct it, in the secial case that the values of the constraints force the conguration to be an (overconstrained) triangle. An underlying rincile fundamental to most constraint solvers is the fact that the osition of the geometric elements can be exressed as (nonlinear) algebraic equations, with the constraints as arameters in the equations. This means that awell-constrained role may have an exonential number of distinct solutions, deendent on the number of geometric elements. For examle, Figure 5 shows three ossible solutions of a well-constrained role that dier in the interretation of the function of the arc, the tangency tye, and the right angle constraint between two segments. From the user's ersective, only one solution will be correct. Constraint solvers select what they consider the intended solution by deducing certain toological and metric roerties from the user's sketch of the role. The deductions are based on a few heuristic rules that succeed under normal circumstances with high robability. These rules are aroriate when the user sketch is, in a technical sense, \close" to the intended solution. This may ormay not be the case, and is often not the case when the dimensional constraints of a role are changed in value, a common occurrence in redesign. Surrisingly, most solvers have almost no rovisions that would allow the user to select a dierent solution if the solver's heuristics fail. Develoing eective aradigms for redirecting a solver interactively is an imortant roblem, and is addressed in aers by Homann and Fudos 4;5. In the following sections, we consider four dierent methods for constraint solving, with secial emhasis on the grah-based method, the aroach we take in our solver, and which we describe in greater detail in Section Numerical Algebraic Comutation Numerical constraint solvers function by rst translating the constraints into a system of algebraic equations. This system is then solved using an iterative

6 technique such as the Newton-Rahson method. Clearly, numerical methods are an examle of instance solvers. A ositive feature of this aroach is that it is able to handle overconstrained but consistent roblems which other techniques may not be able to solve, assuming convergence. Moreover, the solvers are very general. For this reason, many constraint solvers fall back on iterative techniques when the native method is not sucient to solve a given conguration. However, there are some serious drawbacks with the numerical aroach. First is the roblem that of the otentially exonential number of solutions, iterative methods can roduce only a single solution. Also, the solution to which it converges deends strongly on the initial conguration. Furthermore, because of the multile solutions and the large number of arameters, the constraint solving roblem is often ill-conditioned, making convergence dicult or imossible... Symbolic Algebraic Comutation Once again, the constraints are formulated as a system of algebraic equations. However, instead of alying numerical techniques to determine a solution, general symbolic comutations are undertaken to nd the solution to the system of equations. Methods such as Grobner basis 3 or Wu-Ritt 10 techniques can be alied to nd symbolic exressions for the solutions. This aroach is an instance solver if numerical coecients are used in the system of equations. However, if the system can be solved with symbolic coecients, a generic solution to the constraint system is found. The generic solution can be evaluated with secic constraint values to nd the actual hysical congurations ossible for the given constraint roblem. One otential roblem with this method is that certain equations in the basis may be algebraically deendent on one another when evaluated with secic constraints values. Thus at the generic solver level, the solver may determine that a solution exists, yet it will not be able to nd any of the secic congurations satisfying the constraints. A further handica of this method is that solving symbolic systems of equations can be extremely comute-intensive. For this reason, restrictions are often laced on the tyes of geometric entities allowed, as well as the tyes of constraints between them which may be secied..3. Logical Inference and Term Rewriting This aroach alies general logical reasoning techniques to the geometric roblem of constraint solving. This aroach has been taken by Aldefeld 1 and Bruderlin, among others. As an examle of this method, consider the system described by Bruderlin. Geometric entities are restricted to oints, lines, vectors, and triangles, and the constraints allowed are distances between oints, angles between lines, or two angles of a triangle. These geometries and constraints are incororated into a set of redicates for the system. A set of allowable congruence relationshis for the geometries used are established, and then rules of Euclidean geometry for ruler and comass constructions are alied. These rules are set u

7 B a a A d A d B Figure 6: A set of geometric elements with constraints, and the corresonding constraint grah; d denotes a distance constraint. in Prolog, and Prolog rewrite-rules are used to solve the system. The result is a construction technique for solving the inut constraint system. All ossible hysical solutions can be found using the Prolog backtracking mechanism. While this aroach has the otential to be a generic solver, as imlemented the system of Bruderlin is an instance solver, since the redicates and rules use the actual constraint values throughout the deductive rocess. The major advantage of this method is that it avoids translation of the system into comlex algebraic equations. The limitations are that only constructive geometries can be handled, and that the method is not very ecient for large systems of constraints. These disadvantages are common among this tye of solver, and hence they are not often alied in commercial constraint solvers..4. Grah-based Construction Sequences Grah-based algorithms for solving geometric constraint roblems have two hases, the rst an analysis hase and the second a construction hase. The grahbased aroach begins by rst constructing a grah reresentation of the roblem. Each node in the grah reresents a single geometric element, so that a line segment a of length d delimited by two oints A and B would have three nodes. An edge between two geometric entities indicates a constraint between the elements. The tye of constraint is indicated by a label on the edge. This simle examle is shown in Figure 6, where edges which assert incidence are unlabeled in the grah. Once a constraint grah has been obtained from the given geometric entities and the constraints, the grah is analyzed to determine whether the roblem is wellconstrained. If the grah is well-constrained, this hase also determines a sequence of stes for solving the roblem. The second hase of the grah method takes the construction sequence determined from the rst hase and erforms the necessary construction stes to actually lace the geometric elements. Since the rst hase does not deend on the values of the constraint but only on the number and tye of constraints between the geometric elements, this is a generic method of constraint solving. The actual values of the constraints only come into lay in the second hase when the construction stes are carried out. There are a variety of ways to handle the analysis hase of the grah-based method. 1;13 One aroach is to look for a sequence of construction stes such that the next construction ste deends only on reviously laced elements. Not all

8 congurations can be handled in this way, however. A dierent aroach looks for collections of geometric elements whose members can be laced with resect to one another based on constraints between them. These collections are then laced relative to one another, thus forming new, larger collections of elements, until all constraints have been rocessed and the locations of all the elements are known. The aroach to constraint solving which we have imlemented in two dimensions, and which we are develoing in three dimensions, is a grah-based technique which uses the recursive analysis hase just sketched. In the next section we rovide more details of our two-dimensional constraint solver, as a basis for our discussion of the extension of the method to three-dimensional constraint solving. 3. Two-Dimensional Constraint Solving Our aroach to geometric constraint solving is a recursive analysis, grahbased method. This aroachisfavored for two reasons. First, it allows determination of whether the roblem is well-constrained or not in quadratic timein the worst case. Second, it decoules the constraint solving roblem into grous of smaller systems of equations which can be solved indeendently and then merged, rather than framing the roblem as a single, large algebraic system to be solved. In this section, we rovide more in-deth exlanation of the method as imlemented for the two-dimensional roblem, uon which our extension to the threedimensional roblem is based, by summarizing recent work of Bouma, Cai, Fudos, Paige and Homann. Comlete details about the two-dimensional constraint solver and references to further works can be found in. 4; Geometric Entities Considered The geometric elements considered in the two-dimensional constraint solver are oints, lines and circles of xed radius. A oint is reresented by two coordinates ( x ; y ). A line is reresented by a signed, unit normal and the distance of the line from the origin, (n x ;n y ;d). Equationally, the line can be reresented as the set of oints (x; y) satisfying n x x + n y y, d =0and n + n =1. A circle is reresented by x y its center (c x ;c y ) and its radius c r.for simlication uroses, we require that the radius of a circle be xed, that is, the radius cannot be varied to satisfy constraints. The constraints between these geometric entities are incidence of two entities, distance between two oints, distance between a oint and a line, distance between two arallel lines, angle between two lines, tangency between a circle and a line, and concentricity of circles. However, because the circles are restricted to having xed radius, they can be treated as oints by transforming the constraints in which they are involved into distance and incidence constraints of their center oints only. In fact, all constraints can be transformed into distance and angle constraints only, which greatly simlies the lacement roblem. Since the roblem is reduced to lacing oints and lines, any geometric element is xed (u to nitely many ositions) by knowing its relationshi to two

9 d 1 l 1 l 1 d 3 d 1 d 3 3 d 15 1 l 1 l 5 l 3 5 α 45 α 34 α 3 l 3 3 d 15 l 5 α α 3 45 α 34 l 4 l l 4 d d 45 4 Figure 7: A well-constrained sketch and its constraint grah other reviously xed objects. This fact lays an imortant role in the analysis hase of the algorithm, which we describe in the next section. 3.. Initial Cluster Formation The rst hase of the grah-based method for constraint solving involves analysis of the constraint grah. This analysis determines if the roblem is generically well-constrained or not, and it determines a sequence of stes for lacing the geometric elements if the roblem is well-constrained. The basic idea, as described above, is to build u collections, or clusters of geometric elements which can be laced relative to one another, and then to merge these clusters into larger collections using rigid body transformations. Cluster formation begins by selecting any two nodes of the grah which are connected by a constraint edge. These two entities can be laced in some generic osition, deending on the tye of geometries and the tye of constraint between them. These two entities are then considered known. The cluster is then made as large as ossible by adding to the cluster any node which is connected by a constraint edge to exactly two nodes already in the cluster. There must be at least two known nodes to which the unknown node is related because each of the geometric elements has two degrees of freedom. There cannot be more than two, because otherwise the roblem is overconstrained. When no more nodes in the grah can be added to the cluster, the cluster is considered comlete. All constraint edges used in forming the cluster are deleted from original grah, and a search for a new cluster is carried out in the subgrah. This rocess continues until there are no more constraints from which to make any clusters. For examle, consider the sketch on the left in Figure 7. Its constraint grah is shown on the right of the gure, where incidence is shown by unlabeled edges. If we start the rst cluster with 1 and l 1,we can add to the cluster, and then can go no further, since no other node in the grah is connected to two nodes of

10 U l 1 1 d 1 d 3 3 l 3 W d 15 l 5 α α 3 45 α 34 l 4 l d V Figure 8: The constraint grah forms three clusters. the cluster. We delete the three constraint edges used to form this cluster, and look for a new cluster, erhas beginning this time with and l.to this new cluster we can add 3, and subsequently l 3, since it is connected to l and 3. This cluster is now comlete. We begin our third cluster with 4 and l 3, and add l 4, 5, and 1, in that order. All constraint edges have now been used, so cluster formation is comlete. The constraint grah is shown again in Figure 8, with the three clusters indicated. Note that clusters may have nodes in common; in fact, that roerty is essential for the next ste of the analysis hase. In order for the roblem to be well-constrained, the clusters must be able to be merged together in some way so that a single structure results. Geometrically, this amounts to using rigid body transformations to bring the clusters into correct relationshi with resect to one another. Three clusters, each of which share exactly one node with each of the other two clusters, can be brought into alignment with one another using the shared elements. In our examle, the oints 1 and and the line l 3 are shared in this way between the clusters. We can comute the distance from 1 to l 3 within cluster V and the distance from to l 3 within cluster W. The distance between 1 and is already known, so these three elements can be laced relative to one another, thus merging the three clusters into one larger cluster. If other clusters have two elements in common with the new cluster, they can be merged into it as well. When the merged cluster can be grown no longer, the clusters are searched for another set of three clusters which can be merged. This rocess continues until all the clusters have been merged into a single cluster, or until no more clusters can be merged. If there is a single cluster at the end of the merging stage, the roblem is well-constrained. In that case, the stes for constructing the conguration are detailed by the order of the cluster formation.

11 If multile clusters are obtained, then the algorithm cannot solve the constraint roblem. In that case, the roblem may bewell-constrained but requires coonstruction stes the algorithm cannot erform, or the roblem is not wellconstrained. A comlete theoretical characterization of generically well-constrained oint sets with distance constraints exists. 14 It leads to a nondeterministic algorithm, and no variant isknown that achieves ecient running times. Consequently, ecient constraint solving algorithms require restricting the class of constraint roblems. An imortant oint about the cluster formation rocess is that it is not unique. Any two nodes with a constraint between them can be chosen to begin a cluster, and if more than one node could be added to a cluster at a given ste, any of them can be selected and added. However, for a well-constrained roblem, it has been shown that no matter what clusters are formed, the nal geometric solutions determined by the order of construction from the clusters are congruent 6. The cluster formation hase of the solution does not do any actual lacement of geometric elements. Rather, its function is to analyze the structure of the relationshis between the geometric elements based on the constraints between them. The lacement of the elements itself is done in the second hase of the solution. In the next section, we describe geometrically how to nd the osition of an unknown geometric element, given its relationshi to two known elements Basic Construction Stes The lacement of one geometric element relative totwo others is accomlished by solving small systems of algebraic equations. Because of the restriction on geometries and constraints, these equations have degree at most two. We use the following notation throughout the discussion. q The Euclidean distance norm of avector v =(v x ;v y ) is denoted k v k = v + x v. Let y 1 and be two oints and l 1 (n 1 ;r 1 ) and l (n ;r ) be two lines, where n i is the unit normal of l i, and r i is the signed distance from l i to the origin. We then can write algebraic equations for the geometric constraints between two entities in the following manner: The distance between the two oints 1 and is d 1 : k 1, k= d 1 The signed distance between the oint 1 and the line l is d 1 : 1 n = r + d 1 The signed angle between the two lines l 1 and l is 1 : n =(n x cos 1, n y sin 1 ;n y cos 1 + n x sin 1 ) where n 1 =(n x ;n y )

12 The sign of the distance and angle measures are determined from the way in which the user inuts the data. For examle, a line segment has an orientation in the direction from the rst end oint to the second. When a oint is inut and a distance to the line segment assigned as a constraint, the sign is determined from the side of the line on which the oint is initially laced by the user. Again for angle constraints, the orientations of the two line segments are used to determine which region between the segments should be aected by the constraint. Having this algebraic understanding of the geometry of the constraints allows us to evaluate the following cases: Case 1 : ( 1 ; ) ) 3 The oint 3 is to be constructed from two given oints 1 and, where i has distance d i3 from 3. The coordinates of oint 3 must satisfy two quadratic equations arising from the two distance constraints. Geometrically this corresonds to intersecting two circles, one centered at 1 of radius d 13, the other centered at with radius d 3,asshown in Figure 9. The circles either intersect transversally, resulting in two solutions, tangentially, resulting in one solution, or not at all, resulting in no real solutions. In the gure, a situation with two solutions is shown, with both ossible solutions labeled 3. Notice that we can assume oint 1 is at the origin and that lies on the ositive x-axis a distance d 1 from 1, since any other valid conguration could be obtained as a rigid motion of this conguration. Case : ( 1 ;l ) ) 3 The oint 3 is to be constructed from the given oint 1 and the given line l, where 3 has distance d 13 from 1 and signed distance d 3 from l. Without loss of generality, assume that l coincides with the x-axis and that 1 lies distance d 1 from the origin along the ositive y-axis, as shown in Figure 10. Then the oint 3 must lie on a circle centered at 1 of radius d 13 and must lie on a line arallel to l and distance d 3 away from l. Since the distance between 3 and l is signed, there is exactly one line satisfying these conditions. The intersection of the circle and the line rovide the ossible locations for 3. As before, there can be two, one, or no solutions deending on the tyeofintersection. Algebraically, the coordinates of 3 can be found by solving a air of equations, one of which is linear and one of which is quadratic. Case 3 : (l 1 ;l ) ) 3 The oint 3 is to be constructed from the given lines l 1 and l, where 3 has signed distance d i3 from l i. Assume that l 1 coincides with the x-axis and that the angle between l 1 and l is 1, as shown in Figure 11. Since the distances between 3 and the lines are signed, 3 must be the intersection of two lines, one oset arallel to l 1 by d 13 and the other oset arallel to l by d 3.Ifwe ignore the case of arallel lines, there is exactly one solution in this case. Algebraically, this is equivalent to solving a air of linear equations in the coordinates of 3. Case 4 : ( 1 ; ) ) l 3

13 y 3 d 13 d 3 x 1 3 Figure 9: Placing a oint relative to two known oints y d d 3 l x Figure 10: Placing a oint relative to a known oint and line y l d 3 α 1 3 d 13 x l 1 Figure 11: Placing a oint relative to two known lines

14 A c A i c C a a i i C i Figure 1: A grah transformation in this case would limit the solutions to the roblem. Case 5 : ( 1 ;l ) ) l 3 These two cases are identical to cases and 3, resectively, since airwise constraints exist between all three entities. We can therefore easily transform these cases to the revious cases by changing the roles of the geometric entities from xed to unxed, and vice versa, as necessary. Case 6 : (l 1 ;l ) ) l 3 In this case, the angle between each air of lines must be given as constraints. Since the three lines must dene a triangle, the three angles will be either consistent, and determine an innite family of triangles, or redundant, and have no solutions. We consider this case to be overconstrained because the constraints are not indeendent Grah Transformations What we have resented above are the basic elements of a two-dimensional constraint solver. There are many extensions ossible, some of which we have already considered, and others which remain to be exlored. One extension which has roved useful is grah transformations which may increase the constraint information available, thus allowing alternative clusters, ossibly with easier constructions. For examle, if two angle constraints and are given between three lines, a third angle constraintof180,, can be imosed between the air of lines not involved together in the rst two constraints. Grah transformations must be alied judiciously, however, as some transformations may limit the generality of the solution. Consider the examle from 5 shown in Figure 1. If the incidences shown are required, and in addition oint A is constrained to lie on line a, and oint C on line c, the constraint grah is as shown on the right in the gure. This imlies that either a and c are coincident, or A and C are. However, adding either of these incidence relationshis to the constraint grah would eliminate the other ossibility, and therewith some of the solutions to the original roblem. 4. Three-Dimensional Constraint Solving Our basic aroach to constraint solving in three-dimensional sace is analogous to that of constraint solving in two-dimensional sace. We begin by construct-

15 ing a grah which secies the entire constraint system, with the nodes reresenting the geometric elements and an edge between two nodes describing a constraint between the two geometric entities. Based on the information encoded in the grah, the geometric elements are groued into clusters which are lacements of a subset of the elements relative to one another. As we shall see below, forming a cluster in the three-dimensional case requires as a starting conguration three geometric entities which are mutually constrained. Thus it is ossible that not all constraints and geometric elements are used in the cluster formation stage. These remaining nodes and edges form degenerate clusters. The clusters and degenerate clusters are then combined using a recursive technique, resulting in a valid lacement for all the geometric elements. As before, there are in general multile solutions to a given well-constrained roblem Geometric Entities Considered In two-dimensional constraint solving, the geometric entities whichwere considered were oints, lines, and circles of xed radius. All three of these tyes of elements need two constraints to be comletely xed. This roerty is fundamental to the cluster formation and combination method used in the two-dimensional case. The obvious generalization of geometric entities for three-dimensional constraint solving would be oints, lines, lanes, and sheres. Note, however, that oints, lanes, and sheres of xed radius all require exactly three constraints to be laced, while lines require four constraints. In order to kee the construction of clusters as simle as ossible, we do not consider lines at this time, and we further simlify matters by eliminating sheres from our current consideration. Thus the geometric entities which are allowed are oints and lanes. A oint is reresented by its Cartesian coordinates, : ( x ; y ; z ). A lane P is reresented by the direction cosines of the unit normal and the signed distance from the origin, P : (n x ;n y ;n z ;d), where n + x n + y n = 1. The constraints allowed are distance z between two oints, distance between a oint and a lane, and angle between two lanes. Note that xed-radius sheres can be used nevertheless as geometric rimitives because constraints on them can be translated into equivalent constraints on their centers. 4.. Initial Cluster Formation The rst ste of the construction is to form clusters of geometric elements which are laced with resect to one another. Because each geometric element has three degrees of freedom, lacing a new element requires that it be constrained by three known elements. Thus to begin a cluster, a set of three airwise constrained nodes is necessary. These three geometric elements are laced into a standard osition and the resulting conguration is xed u to a rigid motion in sace. Subsequently, a node is added to the cluster if it is incident to three nodes already in the cluster.

16 r t s r 3 u v 1 1 t 4 w u 1 w s 5 v Figure 13: A three-dimensional gure and its constraint grah. When there are no further nodes to be added to the cluster, the edges belonging to the cluster, i.e. the constraints between nodes in the cluster, are deleted from the original grah. Cluster formation is then alied recursively to this subgrah. That is, the subgrah is searched for three nodes which are airwise constrained to start a new cluster, the cluster is grown as far as ossible, and then the edges of the cluster are deleted from the subgrah, resulting in a smaller subgrah. This rocess of forming a cluster and subsequently deleting the cluster's edges from the remaining constraint grah is carried out as long as ossible. Because the origination of a cluster requires three airwise constrained elements, there may eventually be unused constraints in the remaining subgrah, yet no new cluster can be started. When this oint is reached, any remaining constraint and its two incident nodes forms a degenerate cluster. For examle, consider the roblem of lacing the six vertices of the threedimensional object shown on the left in Figure 13, if the lengths of the edges between the vertices are the constraints. We begin the rst cluster using nodes r, s, and t. No other node of the grah is incident to three elements of this cluster, so cluster 1 is comleted. Its edges are labeled 1 in the constraint grah shown on the right in Figure 13, and dislayed in dotted line. We then form a second cluster with nodes u, v, and w. Again, this is a comlete cluster, labeled in the gure. Now none of the remaining constraints are art of an initial cluster, so they must each roduce a degenerate cluster, labeled 3, 4, and 5 in the gure, and dislayed in dashed line. Thus cluster formation is comleted with two full clusters and three degenerate clusters. In order to build these clusters, we need to be able to lace a geometric element from three known elements. In the next section, we resent a case-by-case analysis of how these lacements are executed. Then, with clusters formed, the nal conguration will be determined by recursively merging the clusters. The issues and techniques involved in that rocess are detailed in Section 4.4. Throughout this section, oints are denoted i and lanes P i (n i ;r i ),orp i for short, where n i is the unit normal of P i, and r i is the signed distance from P i to the origin. We assume throughout that distances between oints are non-zero. The

17 G 4 G G 3 G 1 Figure 14: Cluster growth entails a tetrahedral structure in the constraint grah. distance between a oint and a lane, however, may be zero, meaning the oint lies on the lane Basic Construction Stes As discussed in the revious section, the three degrees of freedom of the oints and lanes under consideration means that the lacement of a new oint or lane can be made relative to three already laced oints and lanes. The structure of a region of the constraint grah containing an element which is being added to a cluster is tetrahedral, for the original three elements must be airwise constrained, forming the base of the tetrahedron, and the new element must have a constraint edge between itself and each of the rst three nodes, forming the aex of the tetrahedron. This subgrah structure is shown in Figure 14, where each geometric element G i, i = 1; :::; 4 is either a oint or a lane. Because of the symmetry of the tetrahedron, any three elements can be selected as the base, or known elements, and the remaining aex then is considered to be the unknown element. This allows exibility inchoosing the easiest method for lacing the next element and reduces the number of lacement roblems which must be considered. This interchangability of known and unknown elements may be ossible directly only in the early stages of forming a cluster. For examle, consider the following construction: The beginning of the cluster is a triangle with elements (G 1 ;G ;G 3 ), to which the an additional vertex is added to form a tetrahedron (G 1 ;G ;G 3 ;G 4 ). Three more elements G 5, G 6, and G 7 are added as three tetrahedrons whose bases are faces of the original tetrahedron. Then a new element G 8 could be added to the cluster by having constraints between G 8 and each ofg 5, G 6, and G 7.However, there need be no direct relationshi given between elements G 5, G 6, and G 7, so that a choice of which trile of elements to begin with for adding G 8 is not ossible. This roblem can be overcome by using the information imlicit in the cluster formed so far to determine the relationshis between G 5, G 6, and G 7. For examle, if these three elements were oints, the airwise distances between them could be comuted from within the cluster. Subsequently, adding G 8 could be handled by choosing any three elements of the G 5, G 6, G 7, and G 8

18 P 3 P 3 d 13 d1 d 3 1 Figure 15: There are two generic congurations for two oints and one lane. as the known elements and lacing the remaining element with resect to the chosen three. The tetrahedron constructed then must be brought into line with the reviously constructed comonent of the cluster via a rigid body transformation. Just as in the rocedure of lacement of geometric entities in the twodimensional case, the generic ositioning of the initial elements deends on the orientation of the geometric elements determined from the user inut. The measurement of a signed distance from a lane is made in the direction of the lane normal if the sign is ositive, and in the oosite direction if it is negative. The region of measure of an angle is determined by the mutual orientation of the two lanes in the region. By incororating this orientation information, a generic initial osition is determined from each three element conguration. For three oints 1,, and 3 with airwise distance constraint d ij between i and j, i<j,nochoice of generic osition is necessary, since only a single generic osition exists. We lace 1 at the origin, distance d 1 along the ositive x-axis, and 3 at either intersection oint ofthexy-lane and the sheres centered at 1 and with radii d 13 and d 3, resectively. This is essentially the oint lacement routine for two-dimensions when two oints are known, but because we are in < 3, the two solutions of < are equivalent u to a rigid body motion. For two oints 1 and, and a lane P 3 with airwise distance constraint d ij between entities i and j, i < j, two generic ositions exist. The lane can be laced as the xy-lane q and 1 at (0; 0;d 13 ). The second oint is then laced at (l; 0;d 3 ), where l = d 1, (d 13, d 3 ). The sign of d 3 will determine whether 1 and lie on the same side of P 3 or on oosite sides. Geometrically, the distance constraint between the lane and the two oints imlies that the lane must be tangent to the sheres centered at 1 and with radii d 13 and d 3, resectively. The enveloe of all such lanes consists of two cones. The normals to the cones are the ossible normals to the desired lane. If the oints are inut oriented oositely with resect to the lane, the generic osition is given by a lane tangent to the cone which crosses between the two sheres. If the oints are inut oriented the same with resect to the lane, the generic osition is given by a lane tangent to the cone exterior to the two sheres. In Figure 15 a rojection

19 P 3 d α 3 d 1 P Figure 16: Two lanes and a oint have two generic ositions. of the situation is shown, with the two ossible locations of the lanes relative to the oints highlighted. As before, any other otential solution can be found by a rigid body motion of the relevant one of these two generic congurations. Again, for one oint 1 and two lanes P and P 3, with distance constraints d 1 and d 13 between the oint and the two lanes, resectively, and angle constraint 0 < 3 < 180 between the two lanes, two generic osition are ossible, deending on the inut orientations. Plane P can be laced as the xy-lane, and lane P 3 ositioned so that the intersection of P and P 3 coincides with the y-axis. Then if the inut orientation of 1 with resect to P and P 3 coincides with the inut orientation of the angle between P and P 3, 1 will be laced in the region containing 3. Otherwise, 1 will be laced in the comlementary region. In either case, 1 can be laced at the intersection of the two signed oset lanes, with y =0. These congurations are shown rojected into the xz-lane in Figure 16. The four lanes shown in dashed line are the four ossible oset lanes, deending on orientation of the distance between 1 and the given lanes. The two intersections below P relate to other combinations of orientations of the signed distances and angles and can be obtained by a rigid body motion of one of the shown cases, thus are not considered generic ositions. Finally, when three lanes are given, and the airwise angles ij between them, two generic ositions exist. In this case, P 1 is laced as the xy-lane and P is laced so that the intersection with P 1 is the y-axis. Then P 3 can rst be laced with resect to P 1 so that the angle between P 1 and P 3 is 13, and so that the intersection between P 1 and P 3 is the x-axis. This makes the intersection of the three lanes coincide with the origin. This rst lacement ofp 3 results in a lane with normal (0;n 3 ;n 33 ), and the four ossible combinations of signs of n 3 and n 33 yield two distinct ositions for P 3. The aroriate lane is chosen deending on the inut orientation of the lanes. Once selected, P 3 can be rotated about the z-axis until the angle between P and P 3 is 3. This derivation of the two generic ositions can also be obtained algebraically: Let the normal of P 1 be n 1 =(0; 0; 1), and the normal of P be n =(n 1 ; 0;n 3 ). These are xed by the inut orientation of P 1 and P. Let the normal of P 3 be

20 Figure 17: Degenerate cases for lacing a oint relative to three known oints. n 3 =(n 31 ;n 3 ;n 33 ). Then n 3 must satisfy n 33 = cos 13 n 31 n 1 + n 33 n 3 = cos 3 n 31 + n 3 + n 33 = 1 Since n 33 and n 31 are comletely determined by the rst two equations, there are two dierentvalues fo n 3 from the third equation corresonding to the two dierent orientations of P 3. We now assume that three known elements have been ut into the single ossible generic osition, based on their signed constraints. Based on these generic ositions, we can now roceed to lace a fourth element with resect to three known elements. Case 1 : ( 1 ; ; 3 ) ) 4 The oint 4 is to be constructed from three given oints 1, and 3, where k has distance d k4 from 4. The coordinates of oint 4 must satisfy three quadratic equations arising from the distance constraints. Geometrically this corresonds to intersecting three sheres. The intersection of two sheres is a circle, and is equal to the intersection of a shere with a certain lane. If the two sheres have the equations (x, u 1 ) +(y, v 1 ) +(z, w 1 ) = d 14 (x, u ) +(y, v ) +(z, w ) = d 4 then this lane is (u, u 1 )x +(v, v 1 )y +(w, w 1 )z + U = d 4, d 14 where U = u 1 + v 1 + w 1, u, v, w. Thus, we can intersect two lanes and a shere instead. There will be two solutions in general. The degenerate situations in this case are shown in Figure 17. On the left of the gure is a rojection of the case where the three oints are collinear. In this case two of the sheres determine a circle on which 4 must lie, and the third shere is either redundant or inconsistent. Thus this case either is overconstrained or has no solutions. In the gure is shown a case where there is no solution.

21 The other degenerate situation, shown in rojection on the right in Figure 17, occurs when two of the sheres are tangent. In this case again the third shere is either redundant or inconsistent. Shown is the case where it is redundant. Note that both degenerate cases can occur simultaneously as well, that is, the three oints are collinear and two sheres, or ossibly all three, are tangent to each other. Case : ( 1 ; ;P 3 ) ) 4 The oint 4 is to be constructed from two given oints 1 and, and from the given lane P 3, at resective distances d k4 from entity k. The coordinates of oint 4 must satisfy one linear and two quadratic equations which geometrically corresonds to intersecting two sheres and a lane. Note that the lane is the locus of oints at signed distance d 34 from P 3. As before we intersect instead two lanes and a shere, obtaining two solutions in general. There is only one degenerate case: If the two sheres meet tangentially, then as in Case 1 above, the roblem is either overconstrained with a single solution, or there is no solution because the constraints are inconsistent. Case 3 : ( 1 ;P ;P 3 ) ) 4 The oint 4 is to be constructed from a given oint 1, and from two given lanes P and P 3, at resective distances d k4 from entity k. The coordinates of oint 4 must satisfy one quadratic and two linear equations, corresonding to intersecting a shere and two lanes. Because the distances between oints and lanes are signed, there are two solutions in general. A secial case arises if the shere meets both lanes tangentially, as then there must be only a single solution. This case is degenerate when the two lanes are arallel or coincident. If P and P 3 are arallel, then 1 is only constrained to lie on a lane also arallel to the original two lanes. Assuming that the original three entities are consistently constrained, so that such a lane exists, the distance constraints between 4 and the original three entities are either redundant or inconsistent. For if the distances to P and P 3 are consistent, then they determine a lane on which 4 must lie, and the intersection of this lane with the shere centered at 1 of radius d 14 determines an entire circle on which 4 may lie. This occurs because of the redundancy in the distance constraints between P and 4 and between P 3 and 4. If P and P 4 are coincident, again the constraints between 4 and the three given entities determine either a circle of oints for 4 or no oints at all. Case 4 : (P 1 ;P ;P 3 ) ) 4 The oint 4 is to be constructed from three given lanes P 1, P and P 3, where P k has distance d k from 4. The coordinates of oint 4 must satisfy three linear equations corresonding to the intersection of three lanes. There will be one solution in general, because of the orientation of the lanes. As in the revious case, degeneracies occur when two or more of the lanes are arallel or coincident. Unlike before, however, in this case there are no consistent overconstrained solutions ossible.

22 Case 5 : ( 1 ; ; 3 ) ) P 4 Case 6 : ( 1 ; ;P 3 ) ) P 4 Case 7 : ( 1 ;P ;P 3 ) ) P 4 These three cases can be converted to cases, 3, and 4, resectively, by swaing roles of known vs. unknown between aroriate elements, as discussed earlier. As ointed out in that discussion, this role swaing may require a rigid body motion to bring the new elements in line with reviously laced elements of the cluster. Case 8 : (P 1 ;P ;P 3 ) ) P 4 This case is always an underdetermined situation. Here, the angles from two lanes determine the direction of the sought lane (two ossible solutions). The third angle constraint is redundant or inconsistent. For consistent constraints, the lane has an additional degree of freedom that remains undetermined Cluster Merging Once initial clusters have been formed as described above, clusters which share geometric elements can be laced relative to one another. The goal in cluster merging is to combine clusters in such away as to form a rigid body, unique u to rotation and translation in sace. In the two-dimensional setting, the general merging rule is to combine any three clusters each of which shares a geometric element with the other two. In the three-dimensional case, the necessary relationshis between clusters is considerably more comlicated. A cluster has in general six degrees of freedom, three rotational and three translational. Excetions include degenerate clusters such as a lane with an incident oint, which has only ve degrees of freedom, since one degree of freedom is lost by symmetry. To x a cluster in sace, we must determine how to lace certain elements in the cluster with resect to other known clusters based on elements shared between the cluster being laced and the known clusters. However, it is not sucient to lace the cluster based on only one or two elements in the cluster. Fixing a lane alone in the cluster leaves two translational and one rotational degree of freedom, while xing a oint alone leaves three rotational degrees of freedom. Furthermore, xing any combination of two oints or lanes in a cluster is also insucient to x the cluster itself. Fixing two distinct oints or a oint and a lane leave one rotational degree of freedom, while xing two lanes in general osition leaves one translational degree of freedom. Therefore, three searate geometric entities in the cluster must be laced with resect to other known clusters in order to x the cluster itself. In the case of degenerate clusters, the two geometric elements of the cluster must be shared with another cluster in order to x the cluster. Note that degenerate clusters can only be xed u to symmetry, because they only contain two elements.