Point-hyperplane frameworks, slider joints, and rigidity preserving transformations
Yaser Eftekhari, Bill Jackson, Anthony Nixon, Bernd Schulze, Shin-ichi Tanigawa, Walter Whiteley
PPoint-hyperplane frameworks, slider joints, andrigidity preserving transformations
Yaser Eftekhari ∗ , Bill Jackson † , Anthony Nixon ‡ , Bernd Schulze § ,Shin-ichi Tanigawa ¶ and Walter Whiteley (cid:107) Abstract
A one-to-one correspondence between the infinitesimal motions of bar-joint frame-works in R d and those in S d is a classical observation by Pogorelov, and furtherconnections among different rigidity models in various different spaces have beenextensively studied. In this paper, we shall extend this line of research to includethe infinitesimal rigidity of frameworks consisting of points and hyperplanes. Thisenables us to understand correspondences between point-hyperplane rigidity, clas-sical bar-joint rigidity, and scene analysis.Among other results, we derive a combinatorial characterization of graphs thatcan be realized as infinitesimally rigid frameworks in the plane with a given set ofpoints collinear. This extends a result by Jackson and Jord´an, which deals with thecase when three points are collinear. Keywords: infinitesimal rigidity, bar-joint framework, point-hyperplane framework, spher-ical framework, slider constraints
Given a collection of objects in a space satisfying particular geometric constraints, afundamental question is whether the given constraints uniquely determine the whole con-figuration up to congruence. The rigidity problem for bar-joint frameworks in R d , wherethe objects are points, the constraints are pairwise distances and only local deformationsare considered, is a classical example. ∗ York University, Toronto, Canada, [email protected], supported in part by NSERC, Canada † Queen Mary, University of London, UK, [email protected] ‡ Lancaster University, UK, [email protected] § Lancaster University, UK, [email protected], supported by EPSRC Grant EP/M013642/1 ¶ Kyoto University, Kyoto, Japan, and Centrum Wiskunde & Informatica (CWI), Amsterdam, TheNetherlands, supported by JSPS Postdoctoral Fellowships for Research Abroad and JSPS Grant-in-Aidfor Scientific Research(A)(25240004) (cid:107)
York University, Toronto, Canada, [email protected], supported in part by NSERC,Canada a r X i v : . [ m a t h . C O ] M a r ogorelov [19, Chapter V] observed that the space of infinitesimal motions of a bar-joint framework on a semi-sphere is isomorphic to those of the framework obtained by acentral projection to Euclidean space. Since then, connections between various types ofrigidity models in different spaces have been extensively studied, see, e.g., [1, 2, 9, 21, 22,27, 28]. When talking about infinitesimal rigidity, these connections are often just conse-quences of the fact that infinitesimal rigidity is preserved by projective transformations.A key essence of the research is its geometric and combinatorial interpretations, whichsometimes give us unexpected connections between theory and real applications.In this paper we shall extend this line of research to include point-hyperplane rigidity .A point-hyperplane framework consists of points and hyperplanes along with point-pointdistance constraints, point-hyperplane distance constraints, and hyperplane-hyperplaneangle constraints. The 2-dimensional point-line version has been considered, for examplein [11, 18, 33], for a possible application to CAD. We will show that the infinitesimalrigidity of a point-hyperplane framework is closely related to that of a bar-joint frameworkwith nongeneric positions for its joints. Understanding the infinitesimal rigidity of suchnongeneric bar-joint frameworks is a classical but still challenging problem, and our resultsgive new insight into this problem.Specifically, in Section 2 we establish a one-to-one correspondence between the space ofinfinitesimal motions of a point-hyperplane framework and that of a bar-joint frameworkwith a given set of joints in the same hyperplane by extending the correspondence betweenEuclidean rigidity and spherical rigidity. Combining this with a result by Jackson andOwen [11] for point-line rigidity, we give a combinatorial characterization of a graph thatcan be realized as an infinitesimally rigid bar-joint framework in the plane with a givenset of points collinear. This extends a result by Jackson and Jord´an [10], which deals withthe case when three points are collinear.Let us denote the underlying graph of a point-hyperplane framework in R d by G =( V P ∪ V L , E P P ∪ E P L ∪ E LL ), where V P and V L represent the set of points and the set ofhyperplanes, respectively. The edge set is partitioned into E P P , E LP , E LL according tothe bipartition { V P , V L } of the vertex set. Each i ∈ V P is associated with p i ∈ R d whileeach j ∈ V L is associated with a hyperplane { x ∈ R d : (cid:104) a j , x (cid:105) + r j = 0 } for some a j ∈ S d − and r j ∈ R . We will see in Section 2.2 that the infinitesimal motions of the frameworkare given by the solutions of the following system of linear equations in ˙ p i , ˙ a j , ˙ r j : (cid:104) p i − p j , ˙ p i − ˙ p j (cid:105) = 0 ( ij ∈ E P P ) (cid:104) p i , ˙ a j (cid:105) + (cid:104) ˙ p i , a j (cid:105) + ˙ r j = 0 ( ij ∈ E P L ) (cid:104) a i , ˙ a j (cid:105) + (cid:104) ˙ a i , a j (cid:105) = 0 ( ij ∈ E LL ) (cid:104) a i , ˙ a i (cid:105) = 0 ( i ∈ V L ) . Now observe that, if V L = ∅ then the system is exactly that of a bar-joint frameworkon V P in Euclidean space while, if V P = ∅ then the system becomes that of a bar-jointframework on V L in spherical space. Hence the system of point-hyperplane frameworks isa mixture of these two settings. Further detailed restrictions of the system enable us tolink various types of rigidity models with point-hyperplane rigidity. In the second part ofthe paper, the following new results are obtained:2 If ˙ a j = 0 ( j ∈ V L ), the system models the case when the normal of each hyperplaneis fixed. Such a rigidity model was investigated by Owen and Power [17] for d = 2.We show how to derive their combinatorial characterization in the plane from theresult of Jackson and Owen [11].When E P P = E LL = ∅ , we further point out a connection to the parallel draw-ing problem from scene analysis, and we derive a combinatorial characterization ofgraphs G = ( V P ∪ V L , E P L ) which can be realized as a fixed-normal rigid point-hyperplane framework in R d using a theorem of Whiteley [29]. • If ˙ r j = 0 ( j ∈ V L ), the system can model the case when concurrent hyperplanes canrotate around a common intersection point. We derive a characterization of graphswhich can be realized as a rigid point-line framework in the plane in this rigiditymodel. By using the rigidity transformation established in Section 2, this result istranslated to a characterization of the infinitesimal rigidity of bar-joint frameworksin the plane with horizontal slider-joints on a line. Our result allows us to put sliderpoints anywhere on this line. • If ˙ a j = ˙ r j = 0 ( j ∈ V L ), the system models the case when each hyperplane isfixed. A combinatorial characterization is derived for d = 2 by first transformingthe point-line framework to a bar-joint framework (with nongeneric positions for itsjoints) and then applying a theorem by Servatius et al [23].Point-line frameworks in the plane with different types of constraints imposed on thelines may be used to model structures in engineering with various types of slider-joints(e.g. linkages with prismatic joints in mechanical engineering). Indeed, the use of slider-joints in both mechanical and civil engineering provides a key motivation for our work.The following example from [20] illustrates how our results may be applied to problemsinvolving slider-joints in engineering, see also [12, 15, 25]. Consider the ‘sliding pair chain’shown in Figure 1(a) consisting of four rigid bodies (labelled B , B , B , B ) connectedat five slider joints (labeled (cid:96) , (cid:96) , . . . , (cid:96) ). Each slider joint constrains the relative motionbetween its two incident bodies to be a translation in a direction determined by theorientation of the slider joint. We may model this system as a point-line framework,with each body represented by a ‘bar’ i.e. two points joined by a distance constraint, asindicated in Figure 1(b). We will see in Section 2.3 that this framework has one degreeof freedom. In this section we explain how the rigidity of point-hyperplane frameworks is related tothe rigidity of bar-joint frameworks on the sphere or in Euclidean space by using a rigiditypreserving transformation.We use R d to denote d -dimensional Euclidean space equipped with the standard innerproduct (cid:104)· , ·(cid:105) and S d to denote the unit d -dimensional sphere centered at the origin, andconsider S d ⊂ R d +1 . Let e ∈ R d +1 be the vector whose last coordinate is one and the3 (cid:96) (cid:96) (cid:96) (cid:96) B B B B B B (a) B (cid:96) B (cid:96) B (cid:96) B (cid:96) (cid:96) (b)Figure 1: A 4-body sliding pair chain (a) that is modelled as a point-line framework (b).A dashed line between a point and a line indicates a point-line distance constraint, and asolid line between two points indicates a point-point distance constraint.others are equal to zero, and let A d = { x ∈ R d +1 : (cid:104) x, e (cid:105) = 1 } be the hyperplane of R d +1 with e ∈ A d and with normal e . We also use S d> = { x ∈ S d : (cid:104) x, e (cid:105) > } , S d ≥ = { x ∈ S d : (cid:104) x, e (cid:105) ≥ } and put S d< = S d \ S d ≥ . The equator of S d is defined to be S d ≥ \ S d> and is denoted by Q . In the following discussion, the last coordinate in R d +1 will have a special role (as one may expect from the definitions of A d and S d> ). Hencewe sometimes refer to a coordinate of a point in R d +1 as a pair ( x, x (cid:48) ) ∈ R d × R , where x (cid:48) denotes the last coordinate. For example, a point in A d is denoted by ( x,
1) with x ∈ R d . It is a classical fact that there is a one-to-one correspondence between frameworks in R d and those in S d> at the level of infinitesimal motions. Since the transformation betweenthese two spaces is the starting point of our study, we first give a detailed description ofthis transformation.By a framework in a space M we mean a pair ( G, p ) of an undirected finite graph G = ( V, E ) and a map p : V → M . The most widely studied examples are frameworks( G, p ) in R d , where p is a map from V to R d . In this space we are interested in whetherthere is a different framework (up to congruences) in some neighborhood of p satisfyingthe same system of length constraints: (cid:107) p i − p j (cid:107) = const ( ij ∈ E ) . A common strategy to answer this question is to take the derivative of the square of eachlength constraint to get the first-order length constraint, (cid:104) p i − p j , ˙ p i − ˙ p j (cid:105) = 0 ( ij ∈ E ) , (1)and then check the dimension of the solution space with variables ˙ p . We say that ˙ p : V → R d is an infinitesimal motion of ( G, p ) if ˙ p satisfies (1), and ( G, p ) is called infinitesimally igid if the dimension of the space of infinitesimal motions of ( G, p ) is equal to (cid:0) d +12 (cid:1) (assuming that the points p ( V ) affinely span R d ).Less well-known but still widely appearing models of frameworks are those in S d . In S d the spherical distance between two points is determined by their inner product. Hencewe are interested in the solutions to the system of inner product constraints: (cid:104) p i , p j (cid:105) = const ( ij ∈ E ) . Since p i is constrained to be on S d , we also have the extra constraints (cid:104) p i , p i (cid:105) = 1 ( i ∈ V ) . Again, taking the derivative, we can obtain the system of first-order inner product con-straints: (cid:104) p i , ˙ p j (cid:105) + (cid:104) p j , ˙ p i (cid:105) = 0 ( ij ∈ E ) (2) (cid:104) p i , ˙ p i (cid:105) = 0 ( i ∈ V ) . (3)A map ˙ p : V → R d +1 is said to be an infinitesimal motion of ( G, p ) if it satisfies this systemof linear constraints, and the framework (
G, p ) is infinitesimally rigid if the dimension ofits space of infinitesimal motions is equal to (cid:0) d +12 (cid:1) (assuming the points p ( V ) linearly span R d +1 ). For each x ∈ S d , let T x S d = { m ∈ R d +1 | (cid:104) x, m (cid:105) = 0 } be the tangent hyperplane at x . Then we may give an equivalent definition for an in-finitesimal motion of ( G, p ) as a map i (cid:55)→ ˙ p i ∈ T p i S d which satisfies (2) for all i ∈ V .In order to relate the rigidity models in R d and S d , a key step is to identify R d with thehyperplane A d in R d +1 . For a framework ( G, p ) in A d , we define an infinitesimal motionas a map i (cid:55)→ ˙ p i ∈ T p i A d satisfying (1), where T x A d = { m ∈ R d +1 | (cid:104) e , m (cid:105) = 0 } for all x ∈ A d . Then the space of infinitesimal motions ˙ p of a framework ( G, p ) in R d coincides with the space of infinitesimal motions ˙ˆ p of the framework ( G, ˆ p ) in A d , whenwe take ˆ p i = ( p i ,
1) and ˙ˆ p i = ( ˙ p i ,
0) for all i ∈ V . Hence in the subsequent discussion wemay consider the infinitesimal rigidity of frameworks in A d rather than R d .We can now describe the rigidity preserving transformation from S d to A d . Let φ : A d → S d> be the central projection, that is, φ ( x ) = x (cid:107) x (cid:107) ( x ∈ A d ) . (4)For each x ∈ A d , define ψ x : T x A d → T φ ( x ) S d by ψ x ( m ) = m − (cid:104) m, x (cid:105) e (cid:107) x (cid:107) ( m ∈ T x A d ) . ψ x indeed lies in T φ ( x ) S d because (cid:104) φ ( x ) , ψ x ( m ) (cid:105) = (cid:104) x, m − (cid:104) m, x (cid:105) e (cid:105)(cid:107) x (cid:107) = (cid:104) x, m (cid:105) − (cid:104) m, x (cid:105)(cid:107) x (cid:107) = 0where (cid:104) x, (cid:104) m, x (cid:105) e (cid:105) = (cid:104) m, x (cid:105) follows from the fact that the last coordinate of x ∈ A d isequal to one.Given a framework ( G, p ) in A d and an infinitesimal motion ˙ p of ( G, p ), a simplecalculation shows that (cid:104) φ ( p i ) , ψ p j ( ˙ p j ) (cid:105) + (cid:104) φ ( p j ) , ψ p i ( ˙ p i ) (cid:105) = − (cid:104) p i − p j , ˙ p i − ˙ p j (cid:105)(cid:107) p i (cid:107)(cid:107) p j (cid:107) = 0for all ij ∈ E , and hence ψ ( ˙ p ) := ( ψ p i ( ˙ p i )) i ∈ V is an infinitesimal motion of ( G, φ ◦ p ) in S d . Since ψ x is bijective for all x ∈ A d , it is invertible. This implies that ψ is invertibleand this gives us an isomorphism between the spaces of infinitesimal motions of ( G, p )and (
G, φ ◦ p ) (see also Figure 2). In particular, we have the following result discussed in[9, 21, 22]. Proposition 2.1.
A bar-joint framework ( G, p ) is infinitesimally rigid in A d if and onlyif ( G, φ ◦ p ) is infinitesimally rigid in S d> . p i ˙ p i ψ p i ( ˙ p i ) - (cid:104) ˙ p i , p i (cid:105) e φ ( p i )Figure 2: Transfer of infinitesimal motions between A d and S d> .In the following, we will extend the correspondence between infinitesimally rigid frame-works in R d and S d given in Proposition 2.1 further by allowing points to lie on the equatorof the sphere. Note that, in the transformation described above, a point on the equatorof S d corresponds to a ‘point at infinity’ in A d . The frameworks considered in Section 2.1 model a structure consisting of rigid bars andjoints. Such frameworks are usually called bar-joint frameworks . A different kind of frame-work consisting of points and lines in R mutually linked by distance or angle constraints(see Figure 1(b) for example), usually referred to as point-line frameworks , were intro-duced in [18]. A combinatorial characterization for generic rigidity of such frameworkswas recently provided in [11]. We will consider the d -dimensional generalisation of theseframeworks and refer to them as point-hyperplane frameworks . We will use the rigidity6reserving transformation given in Section 2.1 to establish an equivalence (at the level ofinfinitesimal rigidity) between a point-hyperplane framework in R d and a bar-joint frame-work in R d in which a given set of joints lie on the same hyperplane. The idea is to usethis transformation to show that these frameworks are equivalent to a pair of congruentspherical frameworks.Formally, we define a point-hyperplane framework in R d to be a triple ( G, p, (cid:96) ) where G = ( V P ∪ V L , E ) is a point-hyperplane graph , i.e. a graph G in which the vertices havebeen partitioned into two sets V P , V L corresponding to points and hyperplanes, respec-tively, and each edge in E indicates a point-point distance constraint, a point-hyperplanedistance constraint, or a hyperplane-hyperplane angle constraint. Thus the edge set E ispartitioned into three subsets E P P , E
P L , E LL according to the types of end-vertices of eachedge. The point-configuration and the line-configuration are specified by p : V P → R d ,and (cid:96) = ( a, r ) : V L → S d − × R , where the hyperplane associated to each j ∈ V L is givenby { x ∈ R d : (cid:104) x, a j (cid:105) + r j = 0 } . For i ∈ V P and j, k ∈ V L , the distance between the point p i and the hyperplane (cid:96) j is equal to |(cid:104) p i , a j (cid:105) + r j | , and the angle between the two hyperplanes (cid:96) j , (cid:96) k is determined by (cid:104) a j , a k (cid:105) . Hence the system of constraints can be written as (cid:107) p i − p j (cid:107) = const ( ij ∈ E P P ) (5) |(cid:104) p i , a j (cid:105) + r j | = const ( ij ∈ E P L ) (6) (cid:104) a i , a j (cid:105) = const ( ij ∈ E LL ) . (7)Since a j ∈ S d − , we also have the constraint (cid:104) a i , a i (cid:105) = 1 ( i ∈ V L ) . Taking the derivative we get the system of first order constraints (cid:104) p i − p j , ˙ p i − ˙ p j (cid:105) = 0 ( ij ∈ E P P ) (8) (cid:104) p i , ˙ a j (cid:105) + (cid:104) ˙ p i , a j (cid:105) + ˙ r j = 0 ( ij ∈ E P L ) (9) (cid:104) a i , ˙ a j (cid:105) + (cid:104) ˙ a i , a j (cid:105) = 0 ( ij ∈ E LL ) (10) (cid:104) a i , ˙ a i (cid:105) = 0 ( i ∈ V L ) . (11)A map ( ˙ p, ˙ (cid:96) ) is said to be an infinitesimal motion of ( G, p, (cid:96) ) if it satisfies this systemof linear constraints, and (
G, p, (cid:96) ) is infinitesimally rigid if the dimension of the space ofits infinitesimal motions is equal to (cid:0) d +12 (cid:1) , assuming the points p ( V P ) and hyperplanes (cid:96) ( V L ) affinely span R d . In order to use the rigidity preserving transformation from Section 2.1, we will firsttranslate the point-hyperplane framework (
G, p, (cid:96) ) to a point-hyperplane framework ( G, ˆ p, (cid:96) )in affine space A d by taking ˆ p i = ( p i ,
1) for all i ∈ V p . The system of constraints (8)-(11)then becomes: This formulation is slightly different to the formulation for point-line frameworks given in [11], butthey are easily seen to be equivalent. ˆ p i − ˆ p j , ˙ˆ p i − ˙ˆ p j (cid:105) = 0 ( ij ∈ E, i, j ∈ V P ) (12) (cid:104) ˆ p i , ˙ (cid:96) j (cid:105) + (cid:104) ˙ˆ p i , (cid:96) j (cid:105) = 0 ( ij ∈ E, i ∈ V P , j ∈ V L ) (13) (cid:104) a i , ˙ a j (cid:105) + (cid:104) ˙ a i , a j (cid:105) = 0 ( ij ∈ E, i, j ∈ V L ) (14) (cid:104) ˙ˆ p i , e (cid:105) = 0 ( i ∈ V P ) (15) (cid:104) a i , ˙ a i (cid:105) = 0 ( i ∈ V L ) . (16)We now relate this system of linear equations with that for bar-joint frameworks.We first observe that r j does not appear in (13) because ˙ˆ p i ∈ T ˆ p i A d (and hence thelast coordinate of ˙ˆ p i is equal to zero). This implies that the last coordinate of (cid:96) j is notimportant when analyzing the infinitesimal rigidity of ( G, ˆ p, (cid:96) ), and we may always assumethat (cid:96) is a map with (cid:96) : V L → S d − × { } . Under this assumption, we can regard each (cid:96) i as a point on the equator Q of S d by identifying S d − × { } with Q . Hence (16) can bewritten as (cid:104) (cid:96) j , ˙ (cid:96) j (cid:105) = 0, i.e. ˙ (cid:96) j ∈ T (cid:96) j S d for all j ∈ V L , and (14) gives (cid:104) (cid:96) i , ˙ (cid:96) j (cid:105) + (cid:104) ˙ (cid:96) i , (cid:96) j (cid:105) = 0for all ij ∈ E with i, j ∈ V L . We have already seen that (12) can be rewritten as (cid:104) φ (ˆ p i ) , ψ ˆ p j ( ˙ˆ p j ) (cid:105) + (cid:104) φ (ˆ p j ) , ψ ˆ p i ( ˙ˆ p i ) (cid:105) = (cid:104) ˆ p i − ˆ p j , ˙ˆ p i − ˙ˆ p j (cid:105)(cid:107) ˆ p i (cid:107)(cid:107) ˆ p j (cid:107) = 0for all ij ∈ E with i, j ∈ V P . A similar calculation shows that (13) can be rewritten as (cid:104) φ (ˆ p i ) , ˙ (cid:96) j (cid:105) + (cid:104) ψ ˆ p i ( ˙ˆ p i ) , (cid:96) j (cid:105) = (cid:104) ˆ p i , ˙ (cid:96) j (cid:105) + (cid:104) ˙ˆ p i , (cid:96) j (cid:105)(cid:107) ˆ p i (cid:107) = 0for all ij ∈ E with i ∈ V P and j ∈ V L .These equations imply that ( ˙ˆ p, ˙ (cid:96) ) is an infinitesimal motion of ( G, ˆ p, (cid:96) ) if and only if ˙ q is an infinitesimal motion of ( G, q ), where (
G, q ) is the bar-joint framework in S d ≥ givenby q i = (cid:40) φ (ˆ p i ) ( i ∈ V P )( a i ,
0) ( i ∈ V L ) , (17)and ˙ q i ∈ T q i S d is given by ˙ q i = (cid:40) ψ ˆ p i ( ˙ˆ p i ) ( i ∈ V P )˙ (cid:96) i ( i ∈ V L ) . (18)Since each ψ x is bijective and hence invertible, this gives us an isomorphism between thespaces of infinitesimal motions of ( G, ˆ p, (cid:96) ) and ( G, q ). In particular, if we denote the map q given in (17) by φ ◦ (ˆ p, (cid:96) ), then we have the following result. Theorem 2.2.
Let ( G, ˆ p, (cid:96) ) be a point-hyperplane framework with G = ( V P ∪ V L , E ) , ˆ p : V P → A d and (cid:96) = ( a, r ) : V L → S d − × R . Let ( G, φ ◦ (ˆ p, (cid:96) )) be the bar-jointframework in S d ≥ obtained by central projection of each ˆ p i ( i ∈ V P ) and by regarding eachhyperplane (cid:96) i = ( a i , r i ) ( i ∈ V L ) as the point ( a i , on the equator of S d . Then ( G, ˆ p, (cid:96) ) isinfinitesimally rigid if and only if ( G, φ ◦ (ˆ p, (cid:96) )) is infinitesimally rigid. v v u u u u (a) u u u u v v v (b) u u u u v v v (c) u u u u v v v (d) u u u u v v v (e)Figure 3: An illustration of the rigidity preserving transformations in Theorems 2.2 and2.3. (a) A point-line framework ( G, ˆ p, (cid:96) ). (b) The corresponding spherical framework( G, φ ◦ (ˆ p, (cid:96) )) in S ≥ with three points on the equator. The spherical framework in (c)arises from (b) by a small rotation to take points off the equator. An inversion of pointsin S d< then gives (d). Finally in (e) we have a projection to the plane as a bar-jointframework with three collinear points. 9he transformation used in Theorem 2.2 is illustrated in Figure 3(a), (b).In order to relate ( G, φ ◦ (ˆ p, (cid:96) )) with a bar-joint framework in A d , we further considertransformations for frameworks in S d introduced in [22]. Given a framework ( G, q ) in S d ,a rotation γ is an operator acting on q such that ( γ ◦ q ) i = Rq i , for all i ∈ V , for someorthogonal matrix R . Note that ˙ q is an infinitesimal motion of ( G, q ) if and only if themap γ ◦ ˙ q defined by ( γ ◦ ˙ q ) i = R ˙ q i ( i ∈ V ) is an infinitesimal motion of ( G, γ ◦ q ). Inparticular, ( G, q ) is infinitesimally rigid if and only if (
G, γ ◦ q ) is infinitesimally rigid.Given a framework ( G, q ) in S d and I ⊆ V , the inversion ι (with respect to I ) is anoperator acting on q such that ( ι ◦ q ) i = − q i if i ∈ I and ( ι ◦ q ) i = q i otherwise. Note that˙ q is an infinitesimal motion of ( G, q ) if and only if ι ◦ ˙ q defined by ( ι ◦ ˙ q ) i = − ˙ q i ( i ∈ I )and ( ι ◦ ˙ q ) i = ˙ q i ( i ∈ V \ I ) is an infinitesimal motion of ( G, ι ◦ q ), which again means that ι preserves infinitesimal rigidity.We shall use an inversion to flip points in S < to S > so that a framework ( G, q ) in S d is transferred to a framework ( G, ι ◦ q ) in S d ≥ . In the following discussion, ι always refersto such an operator. Then a framework ( G, q ) in S d can be transformed to a framework( G, ι ◦ γ ◦ q ) in S > by first applying a rotation γ which moves all points off the equator, andthen applying ι to flip points to S > . For a framework in S > we can then use the inverseof φ to transfer it to A d . An important property of this sequence of transformations isthat point-hyperplane incidence is preserved, i.e. points in ( G, q ) lie on a hyperplane in S d if and only if the corresponding points in ( G, φ − ◦ ι ◦ γ ◦ p ) lie on a hyperplane in A d . Combining this with Theorem 2.2 we have our main result. (See also Figure 3 for anillustration.) Theorem 2.3.
Let ( G, ˆ p, (cid:96) ) be a point-hyperplane framework in A d with G = ( V P ∪ V L , E ) , ˆ p : V P → A d and (cid:96) = ( a, r ) : V L → S d − × R . Let ( G, ˆ q ) be the bar-joint framework in A d with ˆ q = φ − ◦ ι ◦ γ ◦ φ ◦ (ˆ p, (cid:96) ) . Then the points in ˆ q ( V L ) all lie on a hyperplane in A d ,and ( G, ˆ p, (cid:96) ) is infinitesimally rigid if and only if ( G, ˆ q ) is infinitesimally rigid. Note that the above transformation is reversible, i.e., from a framework ( G, ˆ q ) in A d with points ˆ q ( X ) being on a hyperplane for X ⊂ V , one obtains a point-hyperplaneframework ( G, ˆ p, (cid:96) ) in A d with V L = X and V P = V \ X such that ( G, ˆ q ) is infinitesimallyrigid if and only if ( G, ˆ p, (cid:96) ) is infinitesimally rigid. We can now associate A d with R d toobtain the following result. Corollary 2.4.
Let G = ( V, E ) be a graph and X ⊆ V . Then the following are equivalent:(a) G can be realised as an infinitesimally rigid bar-joint framework in R d such that thepoints assigned to X lie on a hyperplane.(b) G can be realised as an infinitesimally rigid point-hyperplane framework in R d suchthat each vertex in X is realised as a hyperplane and each vertex in V \ X is realisedas a point. To see the power of our main theorem, let us consider the case when d = 2. In the plane,Jackson and Owen [11] were able to give a combinatorial characterization of graphs which10an be realised as an infinitesimally rigid point-line framework. Combining this withCorollary 2.4 we immediately obtain the following characterization of graphs which canbe realised as infinitesimally rigid bar-joint frameworks in the plane with a given set ofcollinear points. This theorem extends a result by Jackson and Jord´an [10], where theygive a characterization for the case when three specified points are collinear. We will needthe following notation. Given a graph G = ( V, E ), X ⊆ V and A ⊆ E , let ν X ( A ) be thenumber of vertices of X which are incident to edges in A . Theorem 2.5.
Let G = ( V, E ) be a graph and X ⊆ V . Then the following are equivalent:(a) G can be realised as an infinitesimally rigid bar-joint framework in R such that thepoints assigned to X lie on a line.(b) G can be realised as an infinitesimally rigid point-line framework in R such thateach vertex in X is realised as a line and each vertex in V \ X is realised as a point.(c) G contains a spanning subgraph G (cid:48) = ( V, E (cid:48) ) such that E (cid:48) = 2 | V | − and, for all ∅ (cid:54) = A ⊆ E (cid:48) and all partitions { A , . . . , A s } of A , | A | ≤ s (cid:88) i =1 (2 ν V \ X ( A i ) + ν X ( A i ) −
2) + ν X ( A ) − . We illustrate this result using the underlying graph G = ( V, E ) in Figure 3, taking X = { v , v , v } . We have | E | = 11 = 2 | V | − G (cid:48) described in Theorem 2.5(c) is G (cid:48) = G . However, if we take A = E and A , A and A to be the edge-sets induced by { v , u , u , v } , { v , u , u , v } and { v , v } ,respectively, then { A , A , A } partitions E , ν V \ X ( A ) = ν V \ X ( A ) = 2, ν V \ X ( A ) = 0and ν X ( A ) = ν X ( A ) = ν X ( A ) = 2. Since ν X ( E ) = 3, this gives | E | = 11 > (cid:88) i =1 (2 ν V \ X ( A i ) + ν X ( A i ) −
2) + ν X ( E ) − G cannot be realised as aninfinitesimally rigid point-line framework in R such that each vertex in X is realised asa line and each vertex in V \ X is realised as a point, and G cannot be realised as aninfinitesimally rigid bar-joint framework in R such that the points assigned to X lie ona line. Note however that every generic realisation of G as a bar-joint framework in R isinfinitesimally rigid by Laman’s theorem [13].As a second example, consider the point-line framework in Figure 1(b). The underlyingpoint-line graph is given in Figure 4(a), and is shown as a bar-joint framework withcollinear points. It has | V P | = 8 and | V L | = 5, and hence we have 2 | V P | + 2 | V L | − X to be the set of line vertices V L and A i to be the set of edges incident to thebody B i for i = 1 , , ,
4, then (cid:88) i =1 (2 ν V \ X ( A i ) + ν X ( A i ) −
2) + ν X ( E ) − . A , A , A , A partition E , no subset A of E with | A | = 23 can satisfy Theo-rem 2.5(c). (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) B B B B (a) (cid:96) (cid:96) (cid:96) B (cid:96) (cid:96) B (cid:96) (cid:96) B (cid:96) (cid:96) (cid:96) B (b)Figure 4: A bar-joint framework with five collinear points corresponding to the point-lineframework in Figure 1(b) and a partition of the edge set into A , A , A and A . It is standard to analyze a linear system by using its matrix representation. Here we shallpresent the matrices corresponding to the key linear systems discussed above.Let (
G, p ) be a bar-joint framework in R d with underlying graph G = ( V, E ). The | E | × d | V | matrix corresponding to the linear system given in (1) is the well known rigiditymatrix of ( G, p ), and has the form R ( G, p ) = i j ... { i, j } . . . p i − p j ) 0 . . . p j − p i ) 0 . . . ,where, for each edge { i, j } ∈ E , R ( G, p ) has the row with ( p i − p j ) , . . . , ( p i − p j ) d in thecolumns associated with i , ( p j − p i ) , . . . , ( p j − p i ) d in the columns associated with j , and0 elsewhere [30]. It is clear from the discussion in Section 2.1 that the rigidity matrix R ( G, p ) is the fundamental tool to analyse the infinitesimal rigidity properties of (
G, p ).Similarly, for a spherical framework (
G, p ) with p : V → S d , we can also write downa standard spherical rigidity matrix, as described in [16, 22]. For our purposes it is moreinstructive to consider the ( | E | + | V | ) × ( d + 1) | V | matrix R S ( G, p ) corresponding to thelinear system given in (2) and (3), which can easily be seen to be row-equivalent to the12atrix in [16, 22]: R S ( G, p ) = i j { i, j } . . . p j . . . p i . . . i . . . p i . . . . . . j . . . . . . p j . . . .In Section 2.2 we showed that infinitesimal rigidity can be transferred between a point-hyperplane framework in R d and a bar-joint framework in R d which has a given set ofpoints lying in a hyperplane. We proved this by showing that this transfer of infinitesimalrigidity works for a point-hyperplane framework ( G, p, (cid:96) ) in A d and the spherical frame-work ( G, q ) in S d with q : V → S d ≥ defined as in (17). In the following, we present therigidity matrix R A ( G, p, (cid:96) ) for (
G, p, (cid:96) ) in A d corresponding to the linear system given in(12)-(16).Let the underlying graph of ( G, p, (cid:96) ) be G = ( V P ∪ V L , E ) and let p : V P → A d and (cid:96) : V L → S d − × R with (cid:96) i = ( a i , r i ). Further, let i, j ∈ V P and i (cid:48) , j (cid:48) ∈ V L . Then R A ( G, p, (cid:96) )is an ( | E | + | V | ) × ( d + 1) | V | matrix of the following form: i j k l { i, j } . . . p i − p j ) 0 . . . p j − p i ) 0 . . . . . . . . . { i, k } . . . a k ,
0) 0 . . . . . . p i . . . . . . { k, l } . . . . . . . . . a l ,
0) 0 . . . a k ,
0) 0 . . . i . . . e . . . . . . . . . . . . j . . . . . . e . . . . . . . . . k . . . . . . . . . a k ,
0) 0 . . . . . . l . . . . . . . . . . . . a l ,
0) 0 . . . .Since r j does not appear in (13), the last coordinate of (cid:96) k in the row corresponding to { i, k } may be assumed to be zero, i.e., (cid:96) k = ( a k , φ and ψ in Section 2.2 that this matrix is row equivalentto the spherical rigidity matrix R S ( G, q ), where q : V → S d ≥ is defined as in (17). Notethat the rows corresponding to i and j in R A ( G, p, (cid:96) ) guarantee that ˙ p i and ˙ p j lie in T p i A d and T p j A d , respectively, for any infinitesimal motion ( ˙ p, ˙ (cid:96) ) of ( G, p, (cid:96) ). By row operations13he entries e in those rows are converted to q i = φ ( p i ) and q j = φ ( p j ) in R S ( G, q ), so thatthe infinitesimal velocity vectors of (
G, q ) at q i and q j lie in T q i S d and T q j S d , respectively. In this section we describe a connection between point-hyperplane frameworks and sceneanalysis.A d -scene is a pair consisting of a set of points and a set of hyperplanes in R d . Byusing a bipartite graph G = ( V P ∪ V L , E ) to represent the point-hyperplane incidences(where each vertex in V P corresponds to a point, each vertex in V L to a hyperplane, andeach edge in E to a point-hyperplane incidence), a d -scene can be formally defined as atriple ( G, p, (cid:96) ) of a bipartite graph G , and maps p : V P → R d and (cid:96) : V L → S d − × R ,satisfying the incidence condition (cid:104) p i , a j (cid:105) + r j = 0 ( ij ∈ E, i ∈ V P , j ∈ V L ) , (19)taking (cid:96) j = ( a j , r j ) for all j ∈ V L . Given the hyperplane normals ( a j ) j ∈ V L , we can alwaysconstruct a d -scene with these normals by choosing the points P i to be coincident, i.e.putting p i = t ( i ∈ V P ) and r j = −(cid:104) t, a j (cid:105) ( j ∈ V L ) for some t ∈ R d . We will call such a d -scene trivial .In the realisation problem (see [30] for example) we are asked whether there is a non-trivial d -scene with a given set of hyperplane normals ( a j ) j ∈ V L . Note that the set of allrealisations forms a linear space whose dimension is at least d , with equality if and only ifevery realisation is trivial. It follows that the existence of a nontrivial realisation can bechecked by determining the dimension of the solution space of the following linear systemof equations for the variables x : V p → R d and t : E → R : (cid:104) x i , a j (cid:105) + t j = 0 ( ij ∈ E, i ∈ V P , j ∈ V L ) . (20)Now let us return to point-hyperplane frameworks, and consider the restricted rigiditymodel when the normal of each hyperplane is fixed. We can obtain the first order con-straints for a fixed-normal point-hyperplane framework ( G, p, (cid:96) ) with G = ( V P ∪ V L , E )by setting ˙ a j = 0 in the system (12)-(16). This gives (cid:104) p i − p j , ˙ p i − ˙ p j (cid:105) = 0 ( ij ∈ E P P ) (21) (cid:104) ˙ p i , a j (cid:105) + ˙ r j = 0 ( ij ∈ E P L ) , (22)where ˙ p and ˙ r are variables. We say that the ( G, p, (cid:96) ) is infinitesimally fixed-normal rigid if the dimension of the space of infinitesimal motions, i.e. the solution space of the systemof equations (21) and (22), is equal to d . Note that the system of equations (22) dependsonly on the normals ( a j ) j ∈ V L . This implies that the infinitesimal fixed-normal rigidityof ( G, p, (cid:96) ) depends only on the normals ( a j ) j ∈ V L when ( G, p, (cid:96) ) is naturally bipartite i.e.when all constraints are point-hyperplane distance constraints.Now observe that (20) and (22) represent exactly the same system of equations byidentifying x with ˙ p and t with ˙ r . This means that, for any bipartite graph G = ( V P ∪ L , E ) and any fixed set of hyperplane normals ( a j ) j ∈ V L , every realisation of G as a d -scene with hyperplane normals ( a j ) j ∈ V L is trivial if and only if every realisation of G as a naturally bipartite point-hyperplane framework with hyperplane normals ( a j ) j ∈ V L isinfinitesimally fixed-normal rigid.Whiteley [29] gave a combinatorial characterization of generic d -scenes which have onlytrivial realisations. By the above discussion, this gives a combinatorial characterizationof the infinitesimal fixed-normal rigidity of naturally bipartite point-hyperplane frame-works with generic hyperplane normals, i.e., the set of entries in ( a j ) j ∈ V L is algebraicallyindependent over Q . Theorem 3.1.
Let G = ( V P ∪ V L , E ) be a bipartite graph. Then the following are equiv-alent.(a) The dimension of the solution space of system (20) is equal to d for some ( a j ) j ∈ V L .(b) Every realisation of G as a d -scene with generic hyperplane normals is trivial.(c) Every realisation of G as a point-hyperplane framework in R d with generic hyper-plane normals is infinitesimally fixed-normal rigid.(d) G contains a spanning subgraph G (cid:48) = ( V P ∪ V L , E (cid:48) ) with | E (cid:48) | = d | V P | + | V L | − d suchthat | A | ≤ d ν V P ( A ) + ν V L ( A ) − d for all ∅ (cid:54) = A ⊆ E (cid:48) .(e) For any partition { A , . . . , A s } of E, s (cid:88) i =1 ( d ν V P ( A i ) + ν V L ( A i ) − d ) ≥ d | V P | + | V L | − d. Proof.
The equivalence of (a), (b) and (c) follows from the above discussion. The equiva-lence of (a) and (d) follows from [29, Theorem 4.1]. The equivalence of (d) and (e) followsfrom a result of Edmonds [4] on matroids induced by submodular functions.Note that the problem of characterising fixed normal rigidity of generic point-hyperplaneframeworks in R d which are not naturally bipartite is at least as difficult as that of char-acterising the rigidity of generic bar-joint frameworks in R d , which is notoriously difficultwhen d ≥
3. We will solve the fixed normal rigidity problem when d = 2 in the nextsection. Bar-joint frameworks with pinned vertices can be understood by deleting the correspond-ing columns from the rigidity matrix. In this section we investigate analogous constrainedpoint-line frameworks with either fixed lines, fixed normals or fixed intercepts.15 .1 Fixed-line rigidity
We begin with the fixed-line rigidity of point-hyperplane frameworks in R d . In this rigid-ity model, each line is fixed and hence has no velocity. More formally, given a point-hyperplane framework ( G, p, (cid:96) ), we are interested in the following system: (cid:104) p i − p j , ˙ p i − ˙ p j (cid:105) = 0 ( ij ∈ E P P ) (cid:104) ˙ p i , a j (cid:105) = 0 ( ij ∈ E P L )obtained by setting ˙ a j = 0 and ˙ r j = 0 in the system (8)-(11). We say that ( G, p, (cid:96) ) is infinitesimally fixed-line rigid if this system has no nonzero solution.By the results of Section 2, we know how to convert a point-hyperplane framework(
G, p, (cid:96) ) in R d to a bar-joint framework ( G, q ) in R d in such a way that infinitesimalrigidity is preserved. From the isomorphism between the spaces of infinitesimal motionsof (
G, p, (cid:96) ) and (
G, q ) (given in the proof of Theorem 2.3), it is easy to see that ˙ (cid:96) i = 0 ifand only if the corresponding ˙ q i = 0 for i ∈ V L . This implies that ( G, p, (cid:96) ) is infinitesimallyfixed-line rigid if and only if (
G, q ) is an infinitesimally rigid bar-joint framework underthe constraint that the vertices in V L are pinned.The rigidity of pinned bar-joint frameworks is a classical concept, and in R severalcombinatorial characterizations are known. Here we should be careful since, as shown inTheorem 2.3, the points in q ( V L ) all lie on a line, and hence ( G, q ) may not be a generic bar-joint framework. Fortunately, Servatius et al. [23, Theorem 4] (see also [12, Theorem 7.5])already gave a characterization of the infinitesimal rigidity of pinned bar-joint frameworksin R in which the assumption of genericity is not required for the positions of the pinnedvertices. This gives us the following characterization of infinitesimal fixed-line rigidity. Theorem 4.1.
Let G = ( V P ∪ V L , E ) be a point-line graph and let a i ∈ S for each i ∈ V L .Then G can be realised as a minimally infinitesimally fixed-line rigid point-line frameworkin R such that each i ∈ V L is realised as the line with normal a i if and only if | E | = 2 | V P | and | F | ≤ ν V P ( F ) − { , a ( F ) } for all nonempty F ⊆ E , where a ( F ) := dim (cid:104) a i : i ∈ V L ( F ) (cid:105) . An example illustrating Theorem 4.1 is shown in Figure 5.
We introduced the fixed-normal rigidity of a point-hyperplane framework (
G, p, (cid:96) ) in Sec-tion 3 and observed that the infinitesimal motions of (
G, p, (cid:96) ) which preserve the normals Although the transformation was presented in affine space A d , it can be extended to Euclidean space R d by simply first lifting p i ( i ∈ V P ) to ˆ p i = (cid:18) p i (cid:19) , applying the transformation to ( G, ˆ p, (cid:96) ) to obtain( G, ˆ q ), and then projecting ˆ q i = (cid:18) q i (cid:19) ( i ∈ V ) to q i . v u u u u (a) v v u u u u (b) (c) (d)Figure 5: A minimally infinitesimally fixed-line rigid point-line framework (a) and itsassociated point-line graph (b). The corresponding framework on the sphere (c) and itsprojection to the plane (d).of the hyperplanes are determined by the system of equations (cid:104) p i − p j , ˙ p i − ˙ p j (cid:105) = 0 ( ij ∈ E P P ) (23) (cid:104) ˙ p i , a j (cid:105) + ˙ r j = 0 ( ij ∈ E P L ) . (24)Owen and Power [17] had previously used a recursive construction to characterise thefixed-normal rigidity of generic point-line frameworks in R . We will show that theirresult can be deduced from Theorem 2.5. Theorem 4.2.
Let G = ( V P ∪ V L , E ) be a point-line graph with | V P | ≥ and | V L | ≥ and T be the edge set of a tree with vertex set V L . Then the following statements areequivalent:(a) G can be realised as a point-line framework in R which is minimally infinitesimallyfixed-normal rigid;(b) G + T can be realized as an infinitesimally rigid bar-joint framework in R such thatthe points assigned to V L are collinear;(c) | E | = 2 | V P | + | V L | − , | F | ≤ ν V P ( F ) − for all ∅ (cid:54) = F ⊆ E with ν V L ( F ) = 0 , and | F | ≤ ν V P ( F ) + ν V L ( F ) − for all ∅ (cid:54) = F ⊆ E .Proof. It is straightforward to show that (a) implies (c).Suppose that G satisfies (c). We will show that G + T satisfies (b) by showing it satisfiesthe conditions of Theorem 2.5(c) with V \ X = V P and X = V L . Since | E | = 2 | V P | + | V L |− G + T has 2 | V | − A ⊆ E ∪ T , let A = { A , A , . . . , A s } bea partition of A and let A (cid:48) = { A i ∈ A : A i \ T (cid:54) = ∅} . Then (cid:88) A i ∈A (2 ν V P ( A i ) + ν V L ( A i ) −
2) + ν V L ( A ) − ≥ (cid:88) A i ∈A (cid:48) (2 ν V P ( A i \ T ) + ν V L ( A i \ T ) −
2) + | A ∩ T |≥ (cid:88) A i ∈A (cid:48) | A i \ T | + | A ∩ T | = | A | . Thus G + T satisfies the condition of Theorem 2.5(c). Hence G + T also satisfies Theorem2.5(a) so (b) holds 17inally we suppose that (b) holds. Then G + T can be realised as an infinitesimallyrigid point-line framework ( G + T, p, (cid:96) ). This implies that the dimension of the solutionspace of the system (8)-(11) for ( G + T, p, (cid:96) ) is equal to three. Choose a special vertex i ∗ ∈ V L , and add the extra constraint (cid:104) a ⊥ i ∗ , ˙ a i ∗ (cid:105) = 0 (25)to the system (8)-(11), where x ⊥ denotes the π/ x ∈ R . Since the system (8)-(11) contains a rotation in its solution space, adding the extraequation (25) decreases the dimension of the solution space by one.Note that, in the system (8)-(11) for ( G + T, p, (cid:96) ), each edge in T gives the followingconstraint: (cid:104) a i , ˙ a j (cid:105) + (cid:104) ˙ a i , a j (cid:105) = 0 ( ij ∈ T ) . (26)A simple inductive argument, starting from i ∗ , implies that (25), (26), and (11) hold ifand only if ˙ a j = 0 ( j ∈ V L ) . (27)Since the combination of (12)-(16) with (27) is equivalent to the system (23)-(24) for( G, p, (cid:96) ), we conclude that the dimension of the solution space of the latter system isequal to two. In other words, (
G, p, (cid:96) ) admits only trivial infinitesimal motions as afixed-normal point-line framework and (a) holds.An example illustrating Theorem 4.2 is shown in Figure 6. u u u v v v (a) u u u v v v (b) (c) (d)Figure 6: An infinitesimally flexible fixed-normal point-line framework (a) and its as-sociated point-line graph (b). The corresponding framework on the sphere (c) and itsprojection to the plane (d) in which the line-vertics are collinear and constraints thatthe lines have fixed normals are modeled by adding a tree of grey edges between thethree line-vertices. The point-line framework is fixed-normal flexible since it has only sixconstraints and 2 | V P | + | V L | − We now consider point-line frameworks in which each line is allowed to rotate about somefixed point but cannot translate. Such a framework will have at most one trivial motion,18nd this will exist only when each of the lines are allowed to rotate about the same point.We will focus on the special case when all of the lines are concurrent and are allowedto rotate about their common point of intersection. We will refer to such a point-lineframework as a line-concurrent framework .Given a line-concurrent framework (
G, p, (cid:96) ), we may always assume that the commonintersection point of the lines is the origin, i.e., r j = 0 for all j ∈ V L , and hence the fixed-intercept constraint implies that ˙ r j = 0 for all j ∈ V L . Substituting ˙ r j = 0 into (8)-(11),we deduce that the infinitesimal motions are determined by the following system: (cid:104) p i − p j , ˙ p i − ˙ p j (cid:105) = 0 ( ij ∈ E P P ) (28) (cid:104) p i , ˙ a j (cid:105) + (cid:104) ˙ p i , a j (cid:105) = 0 ( ij ∈ E P L ) (29) (cid:104) a i , ˙ a j (cid:105) + (cid:104) ˙ a i , a j (cid:105) = 0 ( ij ∈ E LL ) (30) (cid:104) a i , ˙ a i (cid:105) = 0 ( i ∈ V L ) . (31)We say that ( G, p, (cid:96) ) is infinitesimally fixed-intercept rigid if the above system admits onlythe trivial infinitesimal motion.Our theorem gives a characterization of infinitesimal fixed-intercept rigidity even in thecase when the normals of the lines are specified as input without assuming genericity. (Wewill see below that allowing arbitrary normals gives potential applications to engineering.)
Theorem 4.3.
Let G = ( V P ∪ V L , E ) be a point-line graph with | V L | ≥ and let a i ∈ S for each i ∈ V L . Suppose that each line has a distinct normal. Then G can be realised asa minimally infinitesimally fixed-intercept rigid line-concurrent framework such that each i ∈ V L is realised as the line with normal a i if and only if | E | = 2 | V P | + | V L | − and | F | ≤ ν V P ( F ) + ν V L ( F ) − { , | V L ( F ) |} for all nonempty F ⊆ E . We will in fact prove a stronger statement, in which lines are allowed to have thesame normal (as in the setting of Theorem 4.1). To state the result we need the followingnotation. For a point-line graph G = ( V P ∪ V L , E ), let G P be the graph on V P obtainedfrom G by removing V L and regarding each edge ij in E P L with i ∈ V P as a loop at i .Similarly, let G L be the graph on V L obtained from G by removing V P and regardingeach edge ij in E P L with j ∈ V L as a loop at j . For an edge set F of G , let G [ F ] bethe subgraph of G induced by F . Also for a graph H , let C ( H ) be the set of connectedcomponents in H . Theorem 4.4.
Let G = ( V P ∪ V L , E ) be a point-line graph with | V L | ≥ and let a i ∈ S for each i ∈ V L . Then G can be realised as a minimally infinitesimally fixed-intercept rigidline-concurrent framework such that each i ∈ V L is realised as the line with normal a i ifand only if • | E | = 2 | V P | + | V L | − , • a i (cid:54) = a j for each ij ∈ E LL , and | F | ≤ ν V P ( F ) + ν V L ( F ) − − (cid:80) H ∈ C (( G [ F ]) P ) (2 − dim (cid:104) a j : ij ∈ F ∩ E P L , i ∈ V ( H ) (cid:105) ) for all nonempty F ⊆ E . Consider the point-line graph G shown in Figure 7(a). Two different realisationsas a line-concurrent point-line framework are shown in (d) and (e). The framework in(d) has two lines with the same normal. We can use Theorem 4.4 to show that it isnot infinitesimally fixed intercept rigid by taking F to be the edge-set of sugraph of G shown in (b). Then G [ F ] P is as shown in (c) and the right hand side of the inequality ofTheorem 4.4 is 2 · − − | F | = 8. On the other hand therealisaton shown in (e) is infinitesimally fixed intercept rigid. In particular if we evaluatethe right hand side of the inequality of Theorem 4.3 for F , we obtain 2 · − G, p, (cid:96) ) to a bar-joint framework (
G, q ). Note that a line-concurrentpoint-line framework (
G, p, (cid:96) ) will be mapped to a bar-joint framework (
G, q ) such thatall the points in q ( V L ) lie on a line, say a horizontal line. If the rotation on the sphereis done such that the north pole is mapped to a point on the equator (so that the northpole is finally mapped to a point at infinity after the projection to the plane), then in theisomorphism between the spaces of infinitesimal motions of ( G, p, (cid:96) ) and (
G, q ), we havethat ˙ r j = 0 if and only if ˙ q j is in the horizontal direction. In other words each point q ( v )for v ∈ V L can only slide along the horizontal line. Therefore, the question about thefixed-intercept rigidity of ( G, p, (cid:96) ) can be rephrased as the rigidity question of bar-jointframeworks with horizontal slider joints on the ground. This transformation is illustratedin Figure 7(d),(e),(f),(g) and Figure 8(a),(b).More formally, a bar-joint framework with horizontal slider joints is a tuple (
G, X, p )of a graph G , X ⊆ V ( G ), and p : V → R , where X will represent a set of slider joints. Aninfinitesimal motion ˙ p of ( G, X, p ) is an infinitesimal motion of (
G, p ) with ˙ p ( v ) · (cid:18) (cid:19) = 0for all v ∈ X , and ( G, X, p ) is said to be infinitesimally rigid if horizontal translationsare the only possible infinitesimal motions of (
G, X, p ). By the rigidity transformationexplained above, Theorem 4.4 can be restated as follows.
Theorem 4.5.
Let G = ( V P ∪ V L , E ) be a point-line graph with | V L | ≥ and let x i ∈ R for each i ∈ V L . Then G can be realised as a minimally infinitesimally rigid bar-jointframework in R with V L as a set of horizontal slider joints such that the coordinate of i ∈ V L is (cid:18) x i (cid:19) if and only if u u v v v u (a) u u u u v v (b) u u u u (c) u u u u v v v (d) u u u v v v u (e) u u u u v v v (f) v v v u u u u (g)Figure 7: (a) G = ( V P ∪ V L , E ), where the left side is V P and the right side is V L .(b) an edge set F violating the count in Theorem 4.4 when two normals coincide as in(d). (c) G [ F ] P . (d) A line-concurrent realization such that the two lines u , u have thesame normal. (e) A generic line-concurrent realization. (f)(g) Bar-joint frameworks withhorizontal slider joints corresponding to (d)(e).21 | E | = 2 | V P | + | V L | − , • x i (cid:54) = x j for each ij ∈ E LL , and • | F | ≤ ν V P ( F ) + ν V L ( F ) − − (cid:80) H ∈ C (( G [ F ]) P ) max { , − |{ x j : ij ∈ F ∩ E P L , i ∈ V ( H ) }|} ) for all nonempty F ⊆ E . Note that Theorem 4.5 has no restriction on the coordinates of the slider joints. Thisis a much stronger statement than previous results [25, 12], where a certain genericity isassumed for the coordinates of slider joints. Such bar-joint frameworks with horizontalsliders frequently appear in the structural engineering literature, where sliders are oftenlocated on the horizontal ground.(a) (b) )Figure 8: (a) shows the spherical framework corresponding to the point-line framework inFigure 7(e), in which we have added a new point-vertex joined to the three line-verticesby grey edges to model the constraints that the lines must rotate about a fixed point. Werotate this framework to move the line-vertices off the equator and the new point-vertexonto the equator to obtain the spherical framework in (b). This projects to the bar-jointframework in Figure 7(g). The three line-vertices project to three collinear joints andthe new point-vertex and its incident grey edges correspond to the constraints that thecollinear joints are forced to move along the line.
Let G = ( V, E ) be a graph which may contain loops and let d be a positive integer. Weassign a copy of R d to each vertex and let ( R d ) V be the direct sum of those spaces overall vertices. For x ∈ ( R d ) V , let x ( i ) ∈ R d be the restriction of x to the space assigned to i ∈ V . Consider the incidence matrix of an oriented G , that is, the ( | E | × | V | )-matrixin which the entries in row e = ij with i < j are 1 in column i , − j and 0elsewhere, and the entries in a row corresponding to a loop at i are 1 in column i and 0elsewhere. This matrix gives a linear representation of a variant of the cycle matroid of22 . This matroid has rank equal to | V | − (cid:88) H ∈ C ( G ) λ ( H ) , where λ ( H ) := 1 if H has no loop and λ ( H ) = 0 otherwise. We can obtain a linearrepresentation of this matroid by assigning a one-dimensional vector space A e to each e ∈ E , where A e = { x ∈ R V : x ( i ) + x ( j ) = 0 , x ( k ) = 0 ∀ k ∈ V \ { i, j }} ( e = ij is not a loop) A e = { x ∈ R V : x ( k ) = 0 ∀ k ∈ V \ { i }} ( e is a loop at i )with each edge e ∈ E . Then dim (cid:104) A e : e ∈ E (cid:105) = | V | − (cid:80) H ∈ C ( G ) λ ( H ).Next we take the direct sum of d copies of A e , which gives a d -dimensional vectorspace A de for each edge e : A de = { x ∈ ( R d ) V : x ( i ) + x ( j ) = 0 , x ( k ) = 0 ∀ k ∈ V \ { i, j }} ( e = ij is not a loop) A de = { x ∈ ( R d ) V : x ( k ) = 0 ∀ k ∈ V \ { i }} ( e is a loop at i ) . Clearly dim (cid:104) A de : e ∈ E (cid:105) = d | V | − (cid:88) H ∈ C ( G ) dλ ( H ) . (32)We now establish a variant of equation (32). Suppose that a d -dimensional vector a e is assigned to each loop e . We then assign a vector space B e to e by putting B e = A de ( e = ij is not a loop) B e = { x ∈ ( R d ) V : x ( i ) ∈ (cid:104) a e (cid:105) , x ( k ) = 0 ∀ k ∈ V \ { i }} ( e is a loop at i )where (cid:104) a e (cid:105) denotes the span of a e . Let Loop( H ) be the set of loops in a graph H . Lemma 4.6. dim (cid:104) B e : e ∈ E (cid:105) = d | V | − (cid:88) H ∈ C ( G ) ( d − dim (cid:104) a e : e ∈ Loop( H ) (cid:105) ) . (33) Proof.
This is implicit in [12], but we give a direct proof since the claim is easy. A vector y ∈ ( R d ) V is in the orthogonal complement of (cid:104) B e : e ∈ E \ Loop( G ) (cid:105) if and only if y ( i ) = y ( j ) for every H ∈ C ( G ) and every i, j ∈ V ( H ). Such vectors form a d | C ( G ) | -dimensional space. Among those vectors, a vector y is in the orthogonal complement of (cid:104) B e : e ∈ E (cid:105) if and only if y ( i ) is in the orthogonal complement of (cid:104) a e : e ∈ Loop( H ) (cid:105) forevery H ∈ C ( G ) and every i ∈ V ( H ). Thus the orthogonal complement of (cid:104) B e : e ∈ E (cid:105) has dimension equal to (cid:80) H ∈ C ( G ) ( d − dim (cid:104) a e : e ∈ Loop( H ) (cid:105) ).23nother type of subspace which we will associate to each edge is C e = { x ∈ R V : x ( k ) = 0 ∀ k ∈ V \ { i, j }} ( e = ij is not a loop) C e = { x ∈ R V : x ( k ) = 0 ∀ k ∈ V \ { i }} ( e is a loop at i ) . These subspaces give a linear representation of the bicircular matroid of G , and we havedim (cid:104) C e : e ∈ E (cid:105) = | V ( E ) | . (34)We will also need the following result of Lov´asz [14] which gives a geometric interpretationof the so-called Dilworth truncation of a matroid. We say that a hyperplane H intersectsa family U of linear subspaces transversally if dim H ∩ U = dim U − U ∈ U . Lemma 4.7.
Let E be a finite set and U = { U e : e ∈ E } be a family of linear subspacesof R d . Then there exists a hyperplane H which intersects U transversally and is such that dim (cid:104) U e ∩ H : e ∈ E (cid:105) = min (cid:40) k (cid:88) i =1 (dim (cid:104) U e : e ∈ E i (cid:105) − (cid:41) , (35) where the minimum is taken over all partitions { E , E , . . . , E k } of E . Proof of Theorem 4.4.
Let (cid:96) j = ( a j ,
0) for each j ∈ V L and let R ( G, p, (cid:96) ) be the rigiditymatrix of a framework (
G, p, (cid:96) ) representing the system (28)–(31). This is a ( | V L | + | E | ) × (2 | V P | + 2 | V L | )-matrix whose rows are of one of the following four types: i j k lij ∈ E P P . . . p i − p j . . . p j − p i . . . . . . . . . . . . . . .jk ∈ E P L . . . . . . . . . a k . . . p j . . . . . . . . .kl ∈ E LL . . . . . . . . . . . . . . . a l . . . a k . . .l ∈ V L . . . . . . . . . . . . . . . . . . . . . a l . . . where i, j denote vertices in V P while k, l denote vertices in V L (and unspecified entriesare equal to zero). Since the set of the row vectors of R ( G, p, (cid:96) ) indexed by the verticesin V L is linearly independent, R ( G, p, (cid:96) ) is row-independent if and only if the projectionsof the remaining row vectors of R ( G, p, (cid:96) ) onto the orthogonal complement of the spacespanned by the row vectors indexed by V L form a linearly independent set. In other words, R ( G, p, (cid:96) ) is row-independent if and only if the following | E | × | V P | + | V L | )-matrix isrow-independent: i j k lij ∈ E P P . . . p i − p j . . . p j − p i . . . . . . . . . . . . . . .jk ∈ E P L . . . . . . . . . a k . . . (cid:104) p j , a ⊥ k (cid:105) a ⊥ k . . . . . . . . .kl ∈ E LL . . . . . . . . . . . . . . . (cid:104) a l , a ⊥ k (cid:105) a ⊥ k . . . (cid:104) a k , a ⊥ l (cid:105) a ⊥ l . . . a k = a l for k, l ∈ V L with kl ∈ E LL , then the corresponding row in theabove matrix becomes zero, and hence a k (cid:54) = a l is necessary for ( G, p, (cid:96) ) to be minimallyinfinitesimally rigid. Thus in the following discussion we assume a k (cid:54) = a l for all kl ∈ E LL .By taking a suitable linear combination of the two columns indexed by each k ∈ V L to convert one of these columns to a zero column and then deleting this zero column,and using the fact that (cid:104) a l , a ⊥ k (cid:105) = −(cid:104) a k , a ⊥ l (cid:105) for all pairs k, l ∈ V L , we may deducethat R ( G, p, (cid:96) ) is row-independent if and only if the following | E | × (2 | V P | + | V L | )-matrix R (cid:48) ( G, p, (cid:96) ) is row-independent: i j k lij ∈ E P P . . . p i − p j . . . p j − p i . . . . . . . . . . . . . . .jk ∈ E P L . . . . . . . . . a k . . . (cid:104) p j , a ⊥ k (cid:105) . . . . . . . . .kl ∈ E LL . . . . . . . . . . . . . . . . . . − . . . (36)We will show that there is an injective map p : V P → R such that R (cid:48) ( G, p, (cid:96) ) isrow-independent if and only if | F | ≤ ν V P ( F ) + ν V L ( F ) − − (cid:88) H ∈ C (( G [ F ]) P ) (2 − dim (cid:104) a j : ij ∈ F ∩ E P L , i ∈ V ( H ) (cid:105) )for all nonempty F ⊆ E , implying the theorem.To this end, we define the following linear subspace U Pe in ( R ) V P for each e ∈ E : U Pe = { x ∈ ( R ) V P : x ( i ) + x ( j ) = 0 , x ( k ) = 0 ∀ k ∈ V P \ { i, j }} ( ik ∈ E P P ) U Pe = { x ∈ ( R ) V P : x ( i ) ∈ (cid:104) a j (cid:105) , x ( k ) = 0 ∀ k ∈ V P \ { i }} ( ij ∈ E P L , j ∈ V L ) U Pe = { } ( ij ∈ E LL )Note that the linear subspaces are in the form of B e given in Section 4.3.1 with theunderlying graph G P . Moreover, for H ∈ C ( G P ), there is a correspondence between aloop in H and an edge ij ∈ E P L with i ∈ V ( H ). Therefore Lemma 4.6 givesdim (cid:104) U Pe : e ∈ E (cid:105) = 2 ν V P ( E ) − (cid:88) H ∈ C (( G [ E ]) P ) (2 − dim (cid:104) a j : ij ∈ E P L , i ∈ V ( H ) (cid:105) ) . (37)For each e ∈ E , we also define the following linear subspace U Le in R V L : U Le = { } ( ik ∈ E P P ) U Le = { x ∈ R V L : x ( k ) = 0 ∀ k ∈ V L \ { j }} ( ij ∈ E P L , j ∈ V L ) U Le = { x ∈ R V L : x ( k ) = 0 ∀ k ∈ V L \ { i, j }} ( ij ∈ E LL )Note that the linear subspaces are in the form of C e given in Section 4.3.1 with theunderlying graph G L obtained from G by removing V P and regarding each edge ij in E P L with j ∈ V L as a loop at j . Hence by (34)dim (cid:104) U Le : e ∈ E (cid:105) = ν V L ( E ) . (38)25ow we consider the direct sum of ( R ) V P and R V L , and let U e be the direct sum of U Pe and U Le for each edge e . Combining (37) and (38),dim (cid:104) U e : e ∈ E (cid:105) = 2 ν V P ( E ) + ν V L ( E ) − (cid:88) H ∈ C (( G [ F ]) P ) (2 − dim (cid:104) a j : ij ∈ E P L , i ∈ V ( H )) . (39)By Lemma 4.7, there is a hyperplane H in ( R ) V P × R V L intersecting { U e : e ∈ E } transversally and satisfying (35). Denote a normal vector of H by s ∈ ( R ) V P × R V L .Since the hyperplane H intersects { U e : e ∈ E } transversally, we may assume s ( i ) (cid:54) = s ( j )for i, j ∈ V P with i (cid:54) = j and s ( j ) (cid:54) = 0 for k ∈ V L (since a small perturbation of s will notchange the property (35).)We define p : V P → R by p ( i ) = s ( i ) ⊥ and show that dim (cid:104) U e ∩ H : e ∈ E (cid:105) is equalto the rank of R (cid:48) ( G, p, (cid:96) ) given in (36). We will use the following claim, which directlyfollows from the definition of U e and the fact that x ∈ U e ∩ H if and only if x ∈ U e and (cid:104) x, s (cid:105) = 0. Claim 1.
A vector x ∈ U e lies in H if and only if: • for e = ij ∈ E P P , x ( i ) = − x ( j ) and x ( i ) is proportional to p ( i ) − p ( j ) ; • for e = ij ∈ E P L wth j ∈ V L , x ( i ) x ( j ) = s ( j ) a j (cid:104) p i ,a ⊥ j (cid:105) ; • for e = ij ∈ E LL , x ( i ) x ( j ) = − s ( j ) s ( i ) . Since each U e ∩ H is one-dimensional, Claim 1 implies that (cid:104) U e ∩ H : e ∈ E (cid:105) is equalto the row space of the | E | × (2 | V P | + | V L | )-matrix having the following form: i j k lij ∈ E P P . . . p i − p j . . . p j − p i . . . . . . . . . . . . . . .jk ∈ E P L . . . . . . . . . a k . . . (cid:104) p j , a ⊥ k (cid:105) /s ( k ) . . . . . . . . .kl ∈ E LL . . . . . . . . . . . . . . . /s ( k ) . . . − /s ( l ) . . . . By scaling each column indexed by a vertex in V L , this matrix is transformed to R (cid:48) ( G, p, (cid:96) )(as defined in (36)). In other words,rank R (cid:48) ( G, p, (cid:96) ) = dim (cid:104) U e ∩ H : e ∈ E (cid:105) . (40)By (35), (39), and (40), we get rank R (cid:48) ( G, p, (cid:96) ) = min (cid:8)(cid:80) F ∈E f ( F ) (cid:9) where f ( F ) = 2 ν V P ( F ) + ν V L ( F ) − − (cid:88) H ∈ C (( G [ F ]) P ) (2 − dim (cid:104) a j : ij ∈ F ∩ E P L , i ∈ V ( H ) (cid:105) )and the minimum is taken over all partitions E of E into nonempty subsets. The function f : 2 E → Z is submodular, nondecreasing and non-negative, since it determines thedimension of (cid:104){ U e : e ∈ F }(cid:105) by (39). Hence f induces the row matroid of R (cid:48) ( G, p, (cid:96) ) by[4]. This implies that rank R (cid:48) ( G, p, (cid:96) ) = | E | if and only if | F | ≤ f ( F ) for all nonempty F ⊆ E . 26 .3.3 Proof of Theorem 4.3 Proof of Theorem 4.3.
By Theorem 4.4 it suffices to prove that the two combinatorialconditions in Theorem 4.3 and Theorem 4.4 are equivalent under the assumption that thenormals are distinct.For each edge set F and each H ∈ C ( G [ F ] P ), recall that V ( H ) is a subset of V ( G ). Welet F ( H ) be the set of edges in F incident to V ( H ) in G . Then the counts of Theorem 4.3and Theorem 4.4 can be written as | F | ≤ ν V P ( F ) + ν V L ( F ) − − max { , − | V L ( F ) |} (41) | F | ≤ ν V P ( F ) + ν V L ( F ) − − (cid:88) H ∈ C ( G [ F ] P ) max { , − | V L ( F ( H )) |} , (42)respectively, where the count of Theorem 4.4 is simplified to (42) due to the assumptionthat the normals are distinct.Since F ( H ) ⊆ F , max { , − | V L ( F ( H )) |} ≥ max { , − | V L ( F ) |} for each H ∈ C ( G [ F ] P ). Thus (42) implies (41) if C ( G [ F ] P ) (cid:54) = ∅ . If C ( G [ F ] P ) = ∅ , then F ⊆ E LL holds, and hence | V L ( F ) | ≥
2. Thus the right hand side of (41) and (42) coincide. Hence,(42) always implies (41).To complete the proof, we show that F satisfies (42) if each nonempty subset of F satisfies (41). Let H , . . . , H k be all the components in C ( G [ F ] P ) with | V L ( F ( H i )) | ≤ F (cid:48) = F \ (cid:83) ki =1 F ( H i ). Then by (41) we have | F ( H i ) | ≤ ν V P ( F ( H i )) + ν V L ( F ( H i )) − − max { , − | V L ( F ( H i )) |}| F (cid:48) | ≤ ν V P ( F (cid:48) ) + ν V L ( F (cid:48) ) − . Since H i ∈ C ( G [ F ] P ), V ( F ( H i )) ∩ V ( F \ F ( H i )) ⊆ V L holds, implying ν V P ( F (cid:48) ) + k (cid:88) i =1 ν V P ( F ( H i )) = ν V P ( F ) . Moreover, by | V L ( F ( H i )) | ≤ ≤ i ≤ k , | V ( F ( H i )) ∩ V ( F \ F ( H i )) | ≤ ν V L ( F (cid:48) ) + k (cid:88) i =1 ν V L ( F ( H i )) ≤ ν V L ( F ) + k. | F | = | F (cid:48) | + k (cid:88) i =1 | F i |≤ ν V P ( F (cid:48) ) + ν V L ( F (cid:48) )) − k (cid:88) i =1 (2 ν V P ( F ( H i )) + ν V L ( F ( H i )) − − max { , − | V L ( F ( H i )) |} ) ≤ ν V P ( F ) + ν V L ( F ) − k (cid:88) i =1 max { , − | V L ( F ( H i )) |} = 2 ν V P ( F ) + ν V L ( F ) − (cid:88) H ∈ C ( G [ F ] P ) max { , − | V L ( F ( H i )) |} , and F satisfies (41). This completes the proof. A natural question is how to generalise the results of Sections 4.1-4.3 to the case when thelines have a mixture of constraints. That is, some lines are completely fixed, some lineshave fixed normals so can translate but not rotate, some lines can rotate about a fixedpoint but cannot translate, and some are unconstrained. We will extend the constructionused in the proof of Theorem 3.1 to show that generic instances of this mixed constraintproblem can be transformed to the unconstrained problem and then solved using Theorem2.3.Suppose we have a point-line graph G which has various types of line vertices i.e. a set V FL of fixed lines, a set V NL of lines with fixed normals, a collection R of pairwise disjointsets of lines with a fixed centre of rotation, and unconstrained lines. A realisation of G in R is a framework ( G, p, (cid:96) ) together with a map c : R → R , where c ( S ) is the centreof rotation for all lines in S for each S ∈ R . We say that the constrained framework( G, p, (cid:96), c ) is generic if the set of coordinates { p i , a j , c S : i ∈ V P , j ∈ V L , S ∈ R} arealgebraically independent over Q .We first consider the case when | V FL | + |R| ≥ | V FL | + |R| + | V NL | ≥
2. (In thiscase no rotation or translation of R will satisfy the constraints on the lines of any genericrealisation of G .) We construct an unconstrained point-line graph G (cid:48) by first adding alarge rigid point-line graph K to G . We then choose a line-vertex v in K and add anedge from v to each v ∈ V NL . This corresponds to the operation of adding the ‘tree ofgrey edges’ joining the (fixed-normal) line-vertices in Figure 6. For each set S ∈ R , wechoose a distinct point-vertex u S in K and add an edge from u S to each vertex of S . Thiscorresponds to the operation of adding a new point-vertex joined by ‘grey edges’ to eachof the (fixed-intercept) line-vertices in Figure 8. Finally we join each v ∈ V FL to v and apoint-vertex of K . This construction is illustrated in Figure 9.Let ( G, p, (cid:96), c ) and ( G (cid:48) , p (cid:48) , (cid:96) (cid:48) ) be generic realisations of G and G (cid:48) in R so that p ( u ) = p (cid:48) ( u ) for all u ∈ V P , (cid:96) ( v ) = (cid:96) (cid:48) ( v ) for all v ∈ V L , and c ( S ) = p (cid:48) ( u S ) for all S ∈ R .28 v v v v v v v v v v v v v v v v u S u S Figure 9: A constrained point-line graph G with eight constrained line vertices: v and v have fixed normals; v is fixed; S = { v , v , v } and S = { v , v } have fixed centresof rotation. We transform G to an unconstrained point-line graph G (cid:48) by adding the rigidgraph K with two point-vertices, u S and u S , and one line-vertex v .Then ( G, p, (cid:96), c ) has a non-zero infinitesimal motion if and only if ( G (cid:48) , p (cid:48) , (cid:96) (cid:48) ) has a non-zero infinitesimal motion which keeps K fixed. Hence ( G, p, (cid:96), c ) is infinitesimally rigidas a constrained point-line framework if and only if ( G (cid:48) , p (cid:48) , (cid:96) (cid:48) ) is infinitesimally rigid asan unconstrained point-line framework. Thus we may determine whether ( G, p, (cid:96), c ) isinfinitesimally rigid by applying Theorem 2.3 to ( G (cid:48) , p (cid:48) , (cid:96) (cid:48) ). Note that our definition ofthe genericity of ( G, p, (cid:96), c ) is independent of the choice of r j for j ∈ V L . (This makessense because the structure of the rigidity matrix given in Section 2.4 implies that therank of R A ( G, p (cid:48) , (cid:96) (cid:48) ) will be maximised for any realisation such that the coordinates of p i , a j , c S are algebraically independent.) This means that we are free to choose the valuesof the r j such that, for each S ∈ R , each of the lines (cid:96) ( v ) with v ∈ S passes through thethe point c ( S ), so we can take the lines in each S to be concurrent as in Section 4.3 if wewish.Similar, but simpler constructions, can be used when | V FL | + |R| + | V NL | ≤ | V FL | + |R| = 0. The combinatorial condition in Theorem 2.5(c), Theorem 3.1(d),(e) and Theorem 4.1can all be checked in polynomial time, see [11], [8, 11, 26] and [23], respectively. For thecondition in Theorem 4.3, as shown in the proof, the right hand side of the count conditiondefines a submodular function f , and hence one can decide whether the condition issatisfied in polynomial time by a general submodular function minimization algorithm.Currently we do not have a specialized efficient algorithm for this count.29e have obtained characterizations of fixed-line rigidity and fixed-intercept rigidityin Theorem 4.1 and Theorem 4.4 for a point-line framework with arbitrary normals forits lines. An interesting open problem is to derive an analogous result for fixed-normalrigidity. An important special case is the problem of characterizing fixed-normal rigidityfor point-line frameworks in which the lines have been partitioned into parallel classes withgeneric normals (this was posed by Bill Jackson and John Owen at the rigidity workshopin Banff in 2015). In view of the relationship between fixed normal rigidity and sceneanalysis described in Section 3, this problem is challenging even when the underlyinggraph is naturally bipartite (as it is equivalent to understanding when an arbitrary 2-scene has only trivial realisations). We have constructed examples of (nongeneric) 2-dimensional naturally bipartite point-line frameworks with distinct line-normals whichsatisfy the count condition of Theorem 3.1 but are not fixed-normal rigid.In [9, 22] the transfer of rigidity results between Euclidean spaces and spherical spaceswas extended to Minkowski spaces and hyperbolic spaces (spheres in Minkowski space).Preliminary versions of these infinitesimal rigidity transfers appear in [21]. Applying thesetransfers, Theorem 2.5 transfers to frameworks with a designated coplanar subframeworkin Minkowski space and in hyperbolic space. However there are a number of further inter-esting questions in this direction such as extending our other results on point-hyperplaneframeworks with various constraints on the hyperplanes. Since Minkowski space has thefull space of translations for hyperplanes, it is a natural setting to extend the results ofthis paper.In the setting of body-bar frameworks, including the specific setting of Body CADframeworks, there are preliminary results [7, 12] which include (nongeneric) pinned andslider-joints, and point-line, as well as point-plane, distance constraints. These resultsand the results in this paper, can be refined and extended to explore body-bar-pointframeworks - which do occur in built linkages in 2D and 3D. Some of these extensionsand further related results will appear in the thesis [5].Previous work on direction-length frameworks [24, 31] can be viewed as (i) combina-torially special fixed-normal point-line frameworks, with exactly two points joined to eachline; and (ii) all point-line distances set as 0 i.e. they are point-line incidences . With allthis special geometry, the combinatorial characterization in [24] has one more conditionthan the characterization for general point-line frameworks in Theorem 4.2. This addedcondition comes from the fact that subgraphs with no point-point distance constraints canbe dilated. An extension in which fixed normal lines are allowed to contain an arbitrarynumber of points is given in [17]. Many other extensions are open for exploration. Theseconnections also suggest that prior results on parallel-drawings and mixed frameworks inhigher dimensions, again with two vertices per line, such as [3, 32], can be generalized tocombinatorially special fixed-normal point-hyperplane frameworks in higher dimensions. Acknowledgements
We thank Banff International Research Station for hosting the 2015 workshop on ‘Ad-vances in Combinatorial and Geometric Rigidity’ during which this work was started. We30lso thank HIM (Bonn) and ICMS (Edinburgh) for hosting further rigidity workshops inOctober 2015 and May 2016, respectively, and we thank DIMACS (Rutgers) for hostingthe workshop on ‘Distance Geometry’ in July 2016. The discussions we had during theseworkshops played a crucial role in the writing of this paper.
References [1] R. Connelly and W. Whiteley,
Global rigidity: the effect of coning , Discrete Com-put. Geom., 43:4 (2010) 717–735.[2] H. Crapo and W Whiteley,
Statics of frameworks and motions of panel structures, aprojective geometric introduction , Topol. Struct. 6 (1982), 43–82.[3] H. Crapo and W Whiteley, , Preprint 1994, wiki.math.yorku.ca/index.php/Resources_in_Rigidity_Theory [4] J. Edmonds,
Submodular functions, matroids, and certain polyhedra , in Combina-torial Structures and their Applications, eds. R. Guy, H. Hanani, N. Sauer, and J.Sch¨onheim, Gordon and Breach, New York, 1970, 69-87.[5] Y. Eftekhari,
Geometry of point-hyperplane and spherical frameworks , Ph.D. Thesis,York University, to appear, Spring 2017.[6] J. Graver, B. Servatius and H. Servatius,
Combinatorial rigidity , Graduate Studiesin Math., AMS, 1993.[7] K. Haller, A. Lee-St. John, M. Sitharam, I. Streinu, and N. White,
Body-and-cadgeometric constraint systems , Comput. Geom. 45(2012) 385-405.[8] H. Imai,
On combinatorial structure of line drawings of polyhedra , Discrete Appl.Math. 10 (1985), 79-92.[9] I. Izmestiev:
Projective background of the infinitesimal rigidity of frameworks ,Geom. Dedicata, 140(2009) 183–203.[10] B. Jackson and T. Jord´an,
Rigid two-dimensional frameworks with three collinearpoints , Graphs and Combinatorics, 21:4 (2005) 427–444.[11] B. Jackson and J. Owen,
A characterisation of the generic rigidity of 2-dimensionalpoint-line frameworks , J. Comb. Theory: Series B, 119 (2016) 96–121.[12] N. Katoh and S. Tanigawa,
Rooted-tree decompositions with matroid constraints andthe infinitesimal rigidity of frameworks with boundaries , SIAM J. Discrete Math.,27:1, (2013), 155–185.[13] G. Laman,
On graphs and rigidity of plane skeletal structures , J. Engrg. Math. 4(1970), 331–340. 3114] L. Lov´asz,
Flats in matroids and geometric graphs , in Combinatorial Surverys (Proc.Sixth British Combinatorial Conf., Royal Hollaway College, 1977), Academic Press,London 1977 45–86.[15] D. Myska,
Machines and mechanisms: applied kinematic analysis , Prentice Hall,2012.[16] A. Nixon, J. Owen and S. Power,
Rigidity of frameworks supported on surfaces , SIAMJ. Discrete Math., 26:4 (2012) 1733–1757.[17] J. Owen and S. Power,
Independence conditions for point-line frameworks , unpub-lished manuscript.[18] J. Owen,
Algebraic solution for geometry from dimensional constraints , Proc. the firstACM Symposium on Foundations in Solid Modeling and CAD/CAM applications,(1991), 397–407.[19] A.V. Pogorelov,
Extrinsic geometry of convex surfaces , Translation of the 1969 edi-tion, Translations of Mathematical Monographs 35, AMS, 1973.[20] N. Rosenauer and A.H. Willis,
Kinematics of Mechanisms , Dover Publications, NewYork 1967.[21] F. Saliola and W. Whiteley,
Some notes on the equivalence of first-order rigidity invarious geometries , Preprint (2007) arXiv:0709.3354.[22] B. Schulze and W. Whiteley,
Coning, symmetry and spherical frameworks , DiscreteComput. Geom., 48 (2012) 622–657.[23] B. Servatius, O. Shai, and W. Whiteley,
Combinatorial characterization of the Assurgraphs from engineering , European Journal of Combinatorics 31, 2010, 1091–1104.[24] B. Servatius, and W. Whiteley,
Constraining plane configurations in computer-aideddesign: Combinatorics of Directions and Lengths , SIAM J. Discrete Math. Vol. 12,1999, pp. 136–153.[25] I. Streinu and L. Theran,
Slider-pinning Rigidity: a Maxwell-Laman-type Theorem ,Discrete Comput. Geom., 44 (2007), 812–837.[26] K. Sugihara, An algebraic and combinatorial approach to the analysis of line drawingsof polyhedra, Discrete Appl. Math. 9 (1984), 77-104.[27] W. Whiteley,
Rigidity and polarity I: statics of sheet structures , Geom. Dedicata, 22(1987) 329–362.[28] W. Whiteley,
Rigidity and polarity II: weaving lines and tensegrity frameworks ,Geom. Dedicata, 23 (1989) 75–79. 3229] W. Whiteley,
A matroid on hypergraphs, with applications to scene analysis andgeometry , Discrete Comput. Geom., 4 (1989), 75–95.[30] W. Whiteley,
Some matroids from discrete applied geometry , in Matroid Theory, J.Bonin, J. Oxley and B. Servatius (eds.), AMS Contemporary Mathematics, 1996,171–311.[31] W. Whiteley,
Geometry of Direction and Length Frameworks , Preprint 1996, wiki.math.yorku.ca/index.php/Resources_in_Rigidity_Theory [32] W. Whiteley,
Parallel Drawings in 3-Space , Preprint 1996, wiki.math.yorku.ca/index.php/Resources_in_Rigidity_Theory [33] W. Whiteley,