Point-hyperplane frameworks, slider joints, and rigidity preserving transformations

A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in $\mathbb{R}^d$ and those in $\mathbb{S}^d$ is a classical observation by Pogorelov, and further connections among different rigidity models in various different spaces have been extensively studied. In this paper, we shall extend this line of research to include the infinitesimal rigidity of frameworks consisting of points and hyperplanes. This enables us to understand correspondences between point-hyperplane rigidity, classical bar-joint rigidity, and scene analysis. Among other results, we derive a combinatorial characterization of graphs that can be realized as infinitesimally rigid frameworks in the plane with a given set of points collinear. This extends a result by Jackson and Jord\'{a}n, which deals with the case when three points are collinear.


Introduction
Given a collection of objects in a space satisfying particular geometric constraints, a fundamental question is whether the given constraints uniquely determine the whole configuration up to congruence. The rigidity problem for bar-joint frameworks in R d , where the objects are points, the constraints are pairwise distances and only local deformations are considered, is a classical example.
Pogorelov [19,Chapter V] observed that the space of infinitesimal motions of a barjoint framework on a semi-sphere is isomorphic to those of the framework obtained by a central projection to Euclidean space. Since then, connections between various types of rigidity 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 consequences of the fact that infinitesimal rigidity is preserved by projective transformations.
A key essence of the research is its geometric and combinatorial interpretations, which sometimes 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-point distance constraints, point-hyperplane distance constraints, and hyperplane-hyperplane angle constraints. The 2-dimensional point-line version has been considered, for example in [11,18,33], for a possible application to CAD. We will show that the infinitesimal rigidity of a point-hyperplane framework is closely related to that of a bar-joint framework with nongeneric positions for its joints. Understanding the infinitesimal rigidity of such nongeneric bar-joint frameworks is a classical but still challenging problem, and our results give new insight into this problem.
Specifically, in Section 2 we establish a one-to-one correspondence between the space of infinitesimal motions of a point-hyperplane framework and that of a bar-joint framework with a given set of joints in the same hyperplane by extending the correspondence between Euclidean rigidity and spherical rigidity. Combining this with a result by Jackson and Owen [11] for point-line rigidity, we give a combinatorial characterization of a graph that can be realized as an infinitesimally rigid bar-joint framework in the plane with a given set of points collinear. This extends a result by Jackson and Jordán [10], which deals with the 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 of hyperplanes, respectively. The edge set is partitioned into E P P , E LP , E LL according to the bipartition {V P , V L } of the vertex set. Each i ∈ V P is associated with p i ∈ R d while each j ∈ V L is associated with a hyperplane {x ∈ R d : a j , x + r j = 0} for some a j ∈ S d−1 and r j ∈ R. We will see in Section 2.2 that the infinitesimal motions of the framework are given by the solutions of the following system of linear equations inṗ i ,ȧ j ,ṙ j : Now observe that, if V L = ∅ then the system is exactly that of a bar-joint framework on V P in Euclidean space while, if V P = ∅ then the system becomes that of a bar-joint framework on V L in spherical space. Hence the system of point-hyperplane frameworks is a mixture of these two settings. Further detailed restrictions of the system enable us to link various types of rigidity models with point-hyperplane rigidity. In the second part of the paper, the following new results are obtained: • Ifȧ j = 0 (j ∈ V L ), the system models the case when the normal of each hyperplane is 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 the result of Jackson and Owen [11].
When E P P = E LL = ∅, we further point out a connection to the parallel drawing problem from scene analysis, and we derive a combinatorial characterization of graphs G = (V P ∪ V L , E P L ) which can be realized as a fixed-normal rigid pointhyperplane framework in R d using a theorem of Whiteley [29].
• Ifṙ j = 0 (j ∈ V L ), the system can model the case when concurrent hyperplanes can rotate around a common intersection point. We derive a characterization of graphs which can be realized as a rigid point-line framework in the plane in this rigidity model. By using the rigidity transformation established in Section 2, this result is translated to a characterization of the infinitesimal rigidity of bar-joint frameworks in the plane with horizontal slider-joints on a line. Our result allows us to put slider points anywhere on this line.
• Ifȧ j =ṙ j = 0 (j ∈ V L ), the system models the case when each hyperplane is fixed. A combinatorial characterization is derived for d = 2 by first transforming the point-line framework to a bar-joint framework (with nongeneric positions for its joints) and then applying a theorem by Servatius et al [23].
Point-line frameworks in the plane with different types of constraints imposed on the lines 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 sliderjoints 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 problems involving 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 1 , B 2 , B 3 , B 4 ) connected at five slider joints (labeled 1 , 2 , . . . , 5 ). Each slider joint constrains the relative motion between its two incident bodies to be a translation in a direction determined by the orientation 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, as indicated in Figure 1(b). We will see in Section 2.3 that this framework has one degree of freedom.

Rigidity preserving transformations
In this section we explain how the rigidity of point-hyperplane frameworks is related to the rigidity of bar-joint frameworks on the sphere or in Euclidean space by using a rigidity preserving transformation.
We use R d to denote d-dimensional Euclidean space equipped with the standard inner product ·, · and S d to denote the unit d-dimensional sphere centered at the origin, and consider S d ⊂ R d+1 . Let e ∈ R d+1 be the vector whose last coordinate is one and the others are equal to zero, and let A d = {x ∈ R d+1 : x, e = 1} be the hyperplane of R d+1 with e ∈ A d and with normal e. We also use 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 >0 ). Hence we sometimes refer to a coordinate of a point in R d+1 as a pair (x, x ) ∈ R d × R, where x denotes the last coordinate. For example, a point in A d is denoted by (x, 1) with x ∈ R d .

Euclidean space vs spherical space
It is a classical fact that there is a one-to-one correspondence between frameworks in R d and those in S d >0 at the level of infinitesimal motions. Since the transformation between these two spaces is the starting point of our study, we first give a detailed description of this 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 whether there is a different framework (up to congruences) in some neighborhood of p satisfying the same system of length constraints: A common strategy to answer this question is to take the derivative of the square of each length constraint to get the first-order length constraint, and then check the dimension of the solution space with variablesṗ. We say thatṗ : V → R d is an infinitesimal motion of (G, p) ifṗ satisfies (1), and (G, p) is called infinitesimally rigid if the dimension of the space of infinitesimal motions of (G, p) is equal to d+1 2 (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. Hence we are interested in the solutions to the system of inner product constraints: Since p i is constrained to be on S d , we also have the extra constraints Again, taking the derivative, we can obtain the system of first-order inner product constraints: A mapṗ : V → R d+1 is said to be an infinitesimal motion of (G, p) if it satisfies this system of linear constraints, and the framework (G, p) is infinitesimally rigid if the dimension of its space of infinitesimal motions is equal to d+1 2 (assuming the points p(V ) linearly span R d+1 ). For each x ∈ S d , let be the tangent hyperplane at x. Then we may give an equivalent definition for an infinitesimal motion of (G, p) as a map i →ṗ 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 the hyperplane A d in R d+1 . For a framework (G, p) in A d , we define an infinitesimal motion as a map i →ṗ i ∈ T p i A d satisfying (1), where for all x ∈ A d . Then the space of infinitesimal motionsṗ of a framework (G, p) in R d coincides with the space of infinitesimal motionsṗ of the framework (G,p) in A d , when we takep i = (p i , 1) andṗ i = (ṗ i , 0) for all i ∈ V . Hence in the subsequent discussion we may 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 : For each where x, m, x e = m, x follows from the fact that the last coordinate of x ∈ A d is equal to one. Given a framework (G, p) in A d and an infinitesimal motionṗ of (G, p), a simple calculation shows that for all ij ∈ E, and hence ψ(ṗ) := (ψ p i (ṗ 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 invertible and 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]. In the following, we will extend the correspondence between infinitesimally rigid frameworks in R d and S d given in Proposition 2.1 further by allowing points to lie on the equator of the sphere. Note that, in the transformation described above, a point on the equator of S d corresponds to a 'point at infinity' in A d .

Point-hyperplane vs bar-joint
The frameworks considered in Section 2.1 model a structure consisting of rigid bars and joints. Such frameworks are usually called bar-joint frameworks. A different kind of framework consisting of points and lines in R 2 mutually linked by distance or angle constraints (see Figure 1(b) for example), usually referred to as point-line frameworks, were introduced in [18]. A combinatorial characterization for generic rigidity of such frameworks was recently provided in [11]. We will consider the d-dimensional generalisation of these frameworks and refer to them as point-hyperplane frameworks. We will use the rigidity preserving transformation given in Section 2.1 to establish an equivalence (at the level of infinitesimal rigidity) between a point-hyperplane framework in R d and a bar-joint framework in R d in which a given set of joints lie on the same hyperplane. The idea is to use this transformation to show that these frameworks are equivalent to a pair of congruent spherical frameworks.
Formally, we define a point-hyperplane framework in R d to be a triple (G, p, ) where G = (V P ∪ V L , E) is a point-hyperplane graph, i.e. a graph G in which the vertices have been partitioned into two sets V P , V L corresponding to points and hyperplanes, respectively, and each edge in E indicates a point-point distance constraint, a point-hyperplane distance constraint, or a hyperplane-hyperplane angle constraint. Thus the edge set E is partitioned into three subsets E P P , E P L , E LL according to the types of end-vertices of each edge. The point-configuration and the line-configuration are specified by p : V P → R d , and = (a, r) : V L → S d−1 × R, where the hyperplane associated to each j ∈ V L is given by {x ∈ R d : x, a j +r j = 0}. For i ∈ V P and j, k ∈ V L , the distance between the point p i and the hyperplane j is equal to | p i , a j +r j |, and the angle between the two hyperplanes j , k is determined by a j , a k . Hence the system of constraints can be written as Since a j ∈ S d−1 , we also have the constraint Taking the derivative we get the system of first order constraints A map (ṗ,˙ ) is said to be an infinitesimal motion of (G, p, ) if it satisfies this system of linear constraints, and (G, p, ) is infinitesimally rigid if the dimension of the space of its infinitesimal motions is equal to d+1 2 , assuming the points p(V P ) and hyperplanes (V L ) affinely span R d . 1 In order to use the rigidity preserving transformation from Section 2.1, we will first translate the point-hyperplane framework (G, p, ) to a point-hyperplane framework (G,p, ) in affine space A d by takingp i = (p i , 1) for all i ∈ V p . The system of constraints (8)- (11) then becomes: 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) becauseṗ i ∈ Tp i A d (and hence the last coordinate ofṗ i is equal to zero). This implies that the last coordinate of j is not important when analyzing the infinitesimal rigidity of (G,p, ), and we may always assume that is a map with : Under this assumption, we can regard each i as a point on the equator Q of S d by identifying S d−1 × {0} with Q. Hence (16) can be written as j ,˙ j = 0, i.e.˙ j ∈ T j S d for all j ∈ V L , and (14) gives We have already seen that (12) can be rewritten as A similar calculation shows that (13) can be rewritten as for all ij ∈ E with i ∈ V P and j ∈ V L . These equations imply that (ṗ,˙ ) is an infinitesimal motion of (G,p, ) if and only ifq is an infinitesimal motion of (G, q), where (G, q) is the bar-joint framework in S d ≥0 given by Since each ψ x is bijective and hence invertible, this gives us an isomorphism between the spaces of infinitesimal motions of (G,p, ) and (G, q). In particular, if we denote the map q given in (17) by φ • (p, ), then we have the following result.
) be the bar-joint framework in S d ≥0 obtained by central projection of eachp i (i ∈ V P ) and by regarding each The corresponding spherical framework (G, φ • (p, )) in S 2 ≥0 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 points in S d <0 then gives (d). Finally in (e) we have a projection to the plane as a bar-joint framework with three collinear points. Figure 3(a), (b). In order to relate (G, φ • (p, )) with a bar-joint framework in A d , we further consider transformations 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 some orthogonal matrix R. Note thatq is an infinitesimal motion of (G, q) if and only if the

The transformation used in Theorem 2.2 is illustrated in
Given a framework (G, q) in S d and I ⊆ V , the inversion ι (with respect to I) is an operator acting on q such that (ι 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 <0 to S >0 so that a framework ( In the following discussion, ι always refers to such an operator. Then a framework (G, q) in S d can be transformed to a framework (G, ι•γ •q) in S >0 by first applying a rotation γ which moves all points off the equator, and then applying ι to flip points to S >0 . For a framework in S >0 we can then use the inverse of φ to transfer it to A d . An important property of this sequence of transformations is that point-hyperplane incidence is preserved, i.e. points in (G, q) lie on a hyperplane in Combining this with Theorem 2.2 we have our main result. (See also Figure 3 for an illustration.) Note that the above transformation is reversible, i.e., from a framework (G,q) in A d with pointsq(X) being on a hyperplane for X ⊂ V , one obtains a point-hyperplane framework (G,p, ) in A d with V L = X and V P = V \ X such that (G,q) is infinitesimally rigid if and only if (G,p, ) is infinitesimally rigid. We can now associate A d with R d to obtain the following result. (a) G can be realised as an infinitesimally rigid bar-joint framework in R d such that the points assigned to X lie on a hyperplane.
(b) G can be realised as an infinitesimally rigid point-hyperplane framework in R d such that each vertex in X is realised as a hyperplane and each vertex in V \ X is realised as a point.

Combinatorial characterization in the plane
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 which can be realised as an infinitesimally rigid point-line framework. Combining this with Corollary 2.4 we immediately obtain the following characterization of graphs which can be realised as infinitesimally rigid bar-joint frameworks in the plane with a given set of collinear points. This theorem extends a result by Jackson and Jordán [10], where they give a characterization for the case when three specified points are collinear. We will need the following notation. Given a graph G = (V, E), X ⊆ V and A ⊆ E, let ν X (A) be the number 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 2 such that the points assigned to X lie on a line.
(b) G can be realised as an infinitesimally rigid point-line framework in R 2 such that each vertex in X is realised as a line and each vertex in V \ X is realised as a point.
We illustrate this result using the underlying graph G = (V, E) in Figure 3, taking We have |E| = 11 = 2|V | − 3 so the only possible choice for the subgraph G described in Theorem 2.5(c) is G = G. However, if we take A = E and A 1 , A 2 and A 3 to be the edge-sets induced by so the condition in Theorem 2.5(c) fails to hold. Hence G cannot be realised as an infinitesimally rigid point-line framework in R 2 such that each vertex in X is realised as a line and each vertex in V \ X is realised as a point, and G cannot be realised as an infinitesimally rigid bar-joint framework in R 2 such that the points assigned to X lie on a line. Note however that every generic realisation of G as a bar-joint framework in R 2 is infinitesimally rigid by Laman's theorem [13].
As a second example, consider the point-line framework in Figure 1(b). The underlying point-line graph is given in Figure 4(a), and is shown as a bar-joint framework with collinear points. It has |V P | = 8 and |V L | = 5, and hence we have 2|V P | + 2|V L | − 3 = 23. If we take X to be the set of line vertices V L and A i to be the set of edges incident to the body B i for i = 1, 2, 3, 4, then Figure 4: A bar-joint framework with five collinear points corresponding to the point-line framework in Figure 1(b) and a partition of the edge set into A 1 , A 2 , A 3 and A 4 .

Rigidity matrices
It is standard to analyze a linear system by using its matrix representation. Here we shall present 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 rigidity matrix of (G, p), and has the form in the columns associated with j, and 0 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 down a standard spherical rigidity matrix, as described in [16,22]. For our purposes it is more instructive to consider the (|E| + |V |) × (d + 1)|V | matrix R S (G, p) corresponding to the linear system given in (2) and (3), which can easily be seen to be row-equivalent to the matrix in [16,22]: In Section 2.2 we showed that infinitesimal rigidity can be transferred between a pointhyperplane framework in R d and a bar-joint framework in R d which has a given set of points lying in a hyperplane. We proved this by showing that this transfer of infinitesimal rigidity works for a point-hyperplane framework (G, p, ) in A d and the spherical frame- (17). In the following, we present the rigidity matrix R A (G, p, ) for (G, p, ) in A d corresponding to the linear system given in (12)-(16).
Let the underlying graph of (G, p, ) be G = (V P ∪ V L , E) and let p : Since r j does not appear in (13), the last coordinate of k in the row corresponding to {i, k} may be assumed to be zero, i.e., k = (a k , 0). (Recall Section 2.2). It then follows from the definition of the maps φ and ψ in Section 2.2 that this matrix is row equivalent to the spherical rigidity matrix R S (G, q), where q : V → S d ≥0 is defined as in (17). Note that the rows corresponding to i and j in R A (G, p, ) guarantee thatṗ i andṗ j lie in T p i A d and T p j A d , respectively, for any infinitesimal motion (ṗ,˙ ) of (G, p, ). By row operations the 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 that the 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.

Connection to scene analysis
In this section we describe a connection between point-hyperplane frameworks and scene analysis.
A d-scene is a pair consisting of a set of points and a set of hyperplanes in R d . By using 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, and each edge in E to a point-hyperplane incidence), a d-scene can be formally defined as a triple (G, p, ) of a bipartite graph G, and maps p : V P → R d and : V L → S d−1 × R, satisfying the incidence condition taking j = (a j , r j ) for all j ∈ V L . Given the hyperplane normals (a j ) j∈V L , we can always construct 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 = − t, a j (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 nontrivial d-scene with a given set of hyperplane normals (a j ) j∈V L . Note that the set of all realisations forms a linear space whose dimension is at least d, with equality if and only if every realisation is trivial. It follows that the existence of a nontrivial realisation can be checked by determining the dimension of the solution space of the following linear system of equations for the variables x : V p → R d and t : E → R: Now let us return to point-hyperplane frameworks, and consider the restricted rigidity model when the normal of each hyperplane is fixed. We can obtain the first order constraints for a fixed-normal point-hyperplane framework (G, p, ) with G = (V P ∪ V L , E) by settingȧ j = 0 in the system (12)- (16). This gives whereṗ andṙ are variables. We say that the (G, p, ) is infinitesimally fixed-normal rigid if the dimension of the space of infinitesimal motions, i.e. the solution space of the system of equations (21) and (22), is equal to d. Note that the system of equations (22) depends only on the normals (a j ) j∈V L . This implies that the infinitesimal fixed-normal rigidity of (G, p, ) depends only on the normals (a j ) j∈V L when (G, p, ) 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 by identifying x withṗ and t withṙ. This means that, for any bipartite graph G = (V P ∪ V L , E) and any fixed set of hyperplane normals (a j ) j∈V L , every realisation of G as a dscene 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 is infinitesimally fixed-normal rigid.
Whiteley [29] gave a combinatorial characterization of generic d-scenes which have only trivial realisations. By the above discussion, this gives a combinatorial characterization of the infinitesimal fixed-normal rigidity of naturally bipartite point-hyperplane frameworks with generic hyperplane normals, i.e., the set of entries in (a j ) j∈V L is algebraically independent over Q. (c) Every realisation of G as a point-hyperplane framework in R d with generic hyperplane normals is infinitesimally fixed-normal rigid.
Proof. The equivalence of (a), (b) and (c) follows from the above discussion. The equivalence of (a) and (d) follows from [29,Theorem 4.1]. The equivalence of (d) and (e) follows from a result of Edmonds [4] on matroids induced by submodular functions.
Note that the problem of characterising fixed normal rigidity of generic point-hyperplane frameworks in R d which are not naturally bipartite is at least as difficult as that of characterising the rigidity of generic bar-joint frameworks in R d , which is notoriously difficult when d ≥ 3. We will solve the fixed normal rigidity problem when d = 2 in the next section.

Combinatorial characterizations of constrained pointline frameworks in the plane
Bar-joint frameworks with pinned vertices can be understood by deleting the corresponding columns from the rigidity matrix. In this section we investigate analogous constrained point-line frameworks with either fixed lines, fixed normals or fixed intercepts.

Fixed-line rigidity
We begin with the fixed-line rigidity of point-hyperplane frameworks in R d . In this rigidity model, each line is fixed and hence has no velocity. More formally, given a pointhyperplane framework (G, p, ), we are interested in the following system: obtained by settingȧ j = 0 andṙ j = 0 in the system (8)- (11). We say that (G, p, ) 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, ) in R d to a bar-joint framework (G, q) in R d in such a way that infinitesimal rigidity is preserved. 2 From the isomorphism between the spaces of infinitesimal motions of (G, p, ) and (G, q) (given in the proof of Theorem 2.3), it is easy to see that˙ i = 0 if and only if the correspondingq i = 0 for i ∈ V L . This implies that (G, p, ) is infinitesimally fixed-line rigid if and only if (G, q) is an infinitesimally rigid bar-joint framework under the constraint that the vertices in V L are pinned.
The rigidity of pinned bar-joint frameworks is a classical concept, and in R 2 several combinatorial characterizations are known. Here we should be careful since, as shown in Theorem 2.3, the points in q(V L ) all lie on a line, and hence (G, q) may not be a generic barjoint 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 frameworks in R 2 in which the assumption of genericity is not required for the positions of the pinned vertices. This gives us the following characterization of infinitesimal fixed-line rigidity.

Fixed-normal rigidity
We introduced the fixed-normal rigidity of a point-hyperplane framework (G, p, ) in Section 3 and observed that the infinitesimal motions of (G, p, ) which preserve the normals 2 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 ) top i = p i 1 , applying the transformation to (G,p, ) to obtain (G,q), and then projectingq i = of the hyperplanes are determined by the system of equations Owen and Power [17] had previously used a recursive construction to characterise the fixed-normal rigidity of generic point-line frameworks in R 2 . We will show that their result can be deduced from Theorem 2.5. (b) G + T can be realized as an infinitesimally rigid bar-joint framework in R 2 such that the points assigned to V L are collinear; 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 satisfies the conditions of Theorem 2.5(c) with V \X = V P and X = V L . Since |E| = 2|V P |+|V L |−2, Thus G + T satisfies the condition of Theorem 2.5(c). Hence G + T also satisfies Theorem 2.5(a) so (b) holds Finally we suppose that (b) holds. Then G + T can be realised as an infinitesimally rigid point-line framework (G + T, p, ). This implies that the dimension of the solution space of the system (8)- (11) for (G + T, p, ) is equal to three. Choose a special vertex i * ∈ V L , and add the extra constraint to the system (8)- (11), where x ⊥ denotes the π/2 clockwise rotation of a vector x ∈ R 2 . Since the system (8)-(11) contains a rotation in its solution space, adding the extra equation (25) decreases the dimension of the solution space by one. Note that, in the system (8)-(11) for (G + T, p, ), each edge in T gives the following constraint: A simple inductive argument, starting from i * , implies that (25), (26), and (11) hold if and only ifȧ Since the combination of (12)- (16) with (27) is equivalent to the system (23)-(24) for (G, p, ), we conclude that the dimension of the solution space of the latter system is equal to two. In other words, (G, p, ) admits only trivial infinitesimal motions as a fixed-normal point-line framework and (a) holds.
An example illustrating Theorem 4.2 is shown in Figure 6.

Fixed-intercept rigidity
We now consider point-line frameworks in which each line is allowed to rotate about some fixed point but cannot translate. Such a framework will have at most one trivial motion, and 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 allowed to rotate about their common point of intersection. We will refer to such a point-line framework as a line-concurrent framework. Given a line-concurrent framework (G, p, ), we may always assume that the common intersection point of the lines is the origin, i.e., r j = 0 for all j ∈ V L , and hence the fixedintercept constraint implies thatṙ j = 0 for all j ∈ V L . Substitutingṙ j = 0 into (8)-(11), we deduce that the infinitesimal motions are determined by the following system: We say that (G, p, ) is infinitesimally fixed-intercept rigid if the above system admits only the trivial infinitesimal motion. Our theorem gives a characterization of infinitesimal fixed-intercept rigidity even in the case when the normals of the lines are specified as input without assuming genericity. (We will see below that allowing arbitrary normals gives potential applications to engineering.) We will in fact prove a stronger statement, in which lines are allowed to have the same normal (as in the setting of Theorem 4.1). To state the result we need the following notation. For a point-line graph G = (V P ∪ V L , E), let G P be the graph on V P obtained from 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 regarding each 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 ] be the subgraph of G induced by F . Also for a graph H, let C(H) be the set of connected components in H. • a i = a j for each ij ∈ E LL , and for all nonempty F ⊆ E.
Consider the point-line graph G shown in Figure 7(a). Two different realisations as 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 is not 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 of Theorem 4.4 is 2 · 4 + 2 − 1 − 2 = 7, which is greater than |F | = 8. On the other hand the realisaton shown in (e) is infinitesimally fixed intercept rigid. In particular if we evaluate the right hand side of the inequality of Theorem 4.3 for F , we obtain 2 · 4 + 2 − 1 = 9 so the inequality holds.
We will see in the next section that the generic version of Theorem 4.3 can be deduced from Theorem 2.5. However there seems to be no such reduction in the nongeneric case so we provide a direct proof. We first give several tools from matroid theory in Section 4. Before proceeding to the proof, we describe a consequence of Theorem 4.4 for barjoint frameworks. Consider, again, the transformation given in Section 2, which converts a point-line framework (G, p, ) to a bar-joint framework (G, q). Note that a line-concurrent point-line framework (G, p, ) will be mapped to a bar-joint framework (G, q) such that all the points in q(V L ) lie on a line, say a horizontal line. If the rotation on the sphere is done such that the north pole is mapped to a point on the equator (so that the north pole is finally mapped to a point at infinity after the projection to the plane), then in the isomorphism between the spaces of infinitesimal motions of (G, p, ) and (G, q), we have thatṙ j = 0 if and only ifq 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 the fixed-intercept rigidity of (G, p, ) can be rephrased as the rigidity question of bar-joint frameworks with horizontal slider joints on the ground. This transformation is illustrated in Figure 7 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 2 , where X will represent a set of slider joints. An infinitesimal motionṗ of (G, X, p) is an infinitesimal motion of (G, p) withṗ(v) · 0 1 = 0 for all v ∈ X, and (G, X, p) is said to be infinitesimally rigid if horizontal translations are the only possible infinitesimal motions of (G, X, p). By the rigidity transformation explained 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 | ≥ 2 and let x i ∈ R 1 for each i ∈ V L . Then G can be realised as a minimally infinitesimally rigid bar-joint framework in R 2 with V L as a set of horizontal slider joints such that the coordinate of • |E| = 2|V P | + |V L | − 1, • x i = x j for each ij ∈ E LL , and Note that Theorem 4.5 has no restriction on the coordinates of the slider joints. This is a much stronger statement than previous results [25,12], where a certain genericity is assumed for the coordinates of slider joints. Such bar-joint frameworks with horizontal sliders frequently appear in the structural engineering literature, where sliders are often located on the horizontal ground. . This projects to the bar-joint framework in Figure 7(g). The three line-vertices project to three collinear joints and the new point-vertex and its incident grey edges correspond to the constraints that the collinear joints are forced to move along the line.

Matroid preliminaries
Let G = (V, E) be a graph which may contain loops and let d be a positive integer. We assign a copy of R d to each vertex and let (R d ) V be the direct sum of those spaces over all 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 |)-matrix in which the entries in row e = ij with i < j are 1 in column i, −1 in column j and 0 elsewhere, and the entries in a row corresponding to a loop at i are 1 in column i and 0 elsewhere. This matrix gives a linear representation of a variant of the cycle matroid of G. This matroid has rank equal to where λ(H) := 1 if H has no loop and λ(H) = 0 otherwise. We can obtain a linear representation of this matroid by assigning a one-dimensional vector space A e to each e ∈ E, where with each edge e ∈ E. Then dim A e : e ∈ E = |V | − H∈C(G) λ(H).
Next we take the direct sum of d copies of A e , which gives a d-dimensional vector space A d e for each edge e: 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 Proof. This is implicit in [12], but we give a direct proof since the claim is easy. A vector Another type of subspace which we will associate to each edge is a loop at i).
These subspaces give a linear representation of the bicircular matroid of G, and we have dim C e : e ∈ E = |V (E)|.
We will also need the following result of Lovász [14] which gives a geometric interpretation of the so-called Dilworth truncation of a matroid. We say that a hyperplane H intersects a family U of linear subspaces transversally if dim H ∩ U = dim U − 1 for every U ∈ U.
Lemma 4.7. Let E be a finite set and U = {U e : e ∈ E} be a family of linear subspaces of R d . Then there exists a hyperplane H which intersects U transversally and is such that where the minimum is taken over all partitions {E 1 , E 2 , . . . , E k } of E.

Proof of Theorem 4.4
Proof of Theorem 4.4. Let j = (a j , 0) for each j ∈ V L and let R(G, p, ) be the rigidity matrix of a framework (G, p, ) 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:

 
Note that, if a k = a l for k, l ∈ V L with kl ∈ E LL , then the corresponding row in the above matrix becomes zero, and hence a k = a l is necessary for (G, p, ) to be minimally infinitesimally rigid. Thus in the following discussion we assume a k = 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 a l , a ⊥ k = − a k , a ⊥ l for all pairs k, l ∈ V L , we may deduce that R(G, p, ) is row-independent if and only if the following |E| × (2|V P | + |V L |)-matrix R (G, p, ) is row-independent: We will show that there is an injective map p : for all nonempty F ⊆ E, implying the theorem.
To this end, we define the following linear subspace U P e in (R 2 ) V P for each e ∈ E: Note that the linear subspaces are in the form of B e given in Section 4.3.1 with the underlying graph G P . Moreover, for H ∈ C(G P ), there is a correspondence between a loop in H and an edge ij ∈ E P L with i ∈ V (H). Therefore Lemma 4.6 gives For each e ∈ E, we also define the following linear subspace U L e in R V L : Note that the linear subspaces are in the form of C e given in Section 4.3.1 with the underlying 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 U L e : e ∈ E = ν V L (E).
Now we consider the direct sum of (R 2 ) V P and R V L , and let U e be the direct sum of U P e and U L e for each edge e. Combining (37) and (38), (2 − dim a j : ij ∈ E P L , i ∈ V (H)).
(39) By Lemma 4.7, there is a hyperplane H in (R 2 ) V P × R V L intersecting {U e : e ∈ E} transversally and satisfying (35). Denote a normal vector of H by s ∈ (R 2 ) V P × R V L . Since the hyperplane H intersects {U e : e ∈ E} transversally, we may assume s(i) = s(j) for i, j ∈ V P with i = j and s(j) = 0 for k ∈ V L (since a small perturbation of s will not change the property (35).) We define p : V P → R 2 by p(i) = s(i) ⊥ and show that dim U e ∩ H : e ∈ E is equal to the rank of R (G, p, ) given in (36). We will use the following claim, which directly follows from the definition of U e and the fact that x ∈ U e ∩ H if and only if x ∈ U e and x, s = 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); Since each U e ∩ H is one-dimensional, Claim 1 implies that U e ∩ H : e ∈ E is equal to the row space of the |E| × (2|V P | + |V L |)-matrix having the following form: By scaling each column indexed by a vertex in V L , this matrix is transformed to R (G, p, ) (as defined in (36)). In other words, By (35), (39), and (40), we get rank R (G, p, ) = min respectively, where the count of Theorem 4.4 is simplified to (42) due to the assumption that the normals are distinct. Since Therefore, and F satisfies (41). This completes the proof.

Mixed constraints
A natural question is how to generalise the results of Sections 4.1-4.3 to the case when the lines have a mixture of constraints. That is, some lines are completely fixed, some lines have fixed normals so can translate but not rotate, some lines can rotate about a fixed point but cannot translate, and some are unconstrained. We will extend the construction used in the proof of Theorem 3.1 to show that generic instances of this mixed constraint problem can be transformed to the unconstrained problem and then solved using Theorem 2.3. Suppose we have a point-line graph G which has various types of line vertices i.e. a set V F L of fixed lines, a set V N L of lines with fixed normals, a collection R of pairwise disjoint sets of lines with a fixed centre of rotation, and unconstrained lines. A realisation of G in R 2 is a framework (G, p, ) together with a map c : R → R 2 , where c(S) is the centre of rotation for all lines in S for each S ∈ R. We say that the constrained framework (G, p, , c) is generic if the set of coordinates {p i , a j , c S : i ∈ V P , j ∈ V L , S ∈ R} are algebraically independent over Q.
We first consider the case when |V F L | + |R| ≥ 1 and |V F L | + |R| + |V N L | ≥ 2. (In this case no rotation or translation of R 2 will satisfy the constraints on the lines of any generic realisation of G.) We construct an unconstrained point-line graph G by first adding a large rigid point-line graph K to G. We then choose a line-vertex v 0 in K and add an edge from v 0 to each v ∈ V N L . This corresponds to the operation of adding the 'tree of grey edges' joining the (fixed-normal) line-vertices in Figure 6. For each set S ∈ R, we choose a distinct point-vertex u S in K and add an edge from u S to each vertex of S. This corresponds to the operation of adding a new point-vertex joined by 'grey edges' to each of the (fixed-intercept) line-vertices in Figure 8. Finally we join each v ∈ V F L to v 0 and a point-vertex of K. This construction is illustrated in Figure 9.
Let (G, p, , c) and (G , p , ) be generic realisations of G and G in R 2 so that p(u) = p (u) for all u ∈ V P , (v) = (v) for all v ∈ V L , and c(S) = p (u S ) for all S ∈ R. Then (G, p, , c) has a non-zero infinitesimal motion if and only if (G , p , ) has a nonzero infinitesimal motion which keeps K fixed. Hence (G, p, , c) is infinitesimally rigid as a constrained point-line framework if and only if (G , p , ) is infinitesimally rigid as an unconstrained point-line framework. Thus we may determine whether (G, p, , c) is infinitesimally rigid by applying Theorem 2.3 to (G , p , ). Note that our definition of the genericity of (G, p, , c) is independent of the choice of r j for j ∈ V L . (This makes sense because the structure of the rigidity matrix given in Section 2.4 implies that the rank of R A (G, p , ) 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 values of the r j such that, for each S ∈ R, each of the lines (v) with v ∈ S passes through the the point c(S), so we can take the lines in each S to be concurrent as in Section 4.3 if we wish.

Further remarks and open problems
The combinatorial condition in Theorem 2.5(c), Theorem 3.1(d),(e) and Theorem 4.1 can all be checked in polynomial time, see [11], [8,11,26] and [23], respectively. For the condition in Theorem 4.3, as shown in the proof, the right hand side of the count condition defines a submodular function f , and hence one can decide whether the condition is satisfied in polynomial time by a general submodular function minimization algorithm. Currently we do not have a specialized efficient algorithm for this count.
We have obtained characterizations of fixed-line rigidity and fixed-intercept rigidity in Theorem 4.1 and Theorem 4.4 for a point-line framework with arbitrary normals for its lines. An interesting open problem is to derive an analogous result for fixed-normal rigidity. An important special case is the problem of characterizing fixed-normal rigidity for point-line frameworks in which the lines have been partitioned into parallel classes with generic normals (this was posed by Bill Jackson and John Owen at the rigidity workshop in Banff in 2015). In view of the relationship between fixed normal rigidity and scene analysis described in Section 3, this problem is challenging even when the underlying graph is naturally bipartite (as it is equivalent to understanding when an arbitrary 2scene has only trivial realisations). We have constructed examples of (nongeneric) 2dimensional naturally bipartite point-line frameworks with distinct line-normals which satisfy 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 spaces was extended to Minkowski spaces and hyperbolic spaces (spheres in Minkowski space). Preliminary versions of these infinitesimal rigidity transfers appear in [21]. Applying these transfers, Theorem 2.5 transfers to frameworks with a designated coplanar subframework in Minkowski space and in hyperbolic space. However there are a number of further interesting questions in this direction such as extending our other results on point-hyperplane frameworks with various constraints on the hyperplanes. Since Minkowski space has the full space of translations for hyperplanes, it is a natural setting to extend the results of this paper.
In the setting of body-bar frameworks, including the specific setting of Body CAD frameworks, there are preliminary results [7,12] which include (nongeneric) pinned and slider-joints, and point-line, as well as point-plane, distance constraints. These results and the results in this paper, can be refined and extended to explore body-bar-point frameworks -which do occur in built linkages in 2D and 3D. Some of these extensions and further related results will appear in the thesis [5].
Previous work on direction-length frameworks [24,31] can be viewed as (i) combinatorially special fixed-normal point-line frameworks, with exactly two points joined to each line; and (ii) all point-line distances set as 0 i.e. they are point-line incidences. With all this special geometry, the combinatorial characterization in [24] has one more condition than the characterization for general point-line frameworks in Theorem 4.2. This added condition comes from the fact that subgraphs with no point-point distance constraints can be dilated. An extension in which fixed normal lines are allowed to contain an arbitrary number of points is given in [17]. Many other extensions are open for exploration. These connections also suggest that prior results on parallel-drawings and mixed frameworks in higher dimensions, again with two vertices per line, such as [3,32], can be generalized to combinatorially special fixed-normal point-hyperplane frameworks in higher dimensions. also thank HIM (Bonn) and ICMS (Edinburgh) for hosting further rigidity workshops in October 2015 and May 2016, respectively, and we thank DIMACS (Rutgers) for hosting the workshop on 'Distance Geometry' in July 2016. The discussions we had during these workshops played a crucial role in the writing of this paper.