Independence Conditions for Point-Line-Position Frameworks

1. Independence Conditions for Point-Line-Position Frameworks John Owen and Steve Power

3. A drawing has geometries - points, lines, circles........ A drawing has dimensional constraints – distance, radius, angle...... Usually between one or two geometries A drawing has logical constraints - coincident, tangent, parallel, concentric...... A drawing is fully-defined when the geometries are completely determined (locally) by the constraints (dimensional and logical). A drawing is well-dimensioned when the value of any dimensional constraint can be changed (by a small amount) and the drawing can still be realised consistently with the constraints. A drawing defines a constraint graph G and a framework.

4. Often circles can be replaced by their centre point. We have a point-line framework. We can denote a point-line framework by (G,p,l) where G gives the graph, p gives the coordinates of all the points and l gives the position coordinates and direction (slope) coordinates of all the lines. There are 2|Vp(G)|+2|Vl(G)| coordinates in total.

5. = line =point = dimension = coincidence

6. A drawing is fully-dimensioned if its framework is rigid A drawing is well-dimensioned if the bars in the framework which represent dimensional constraints are independent i.e. their values can be varied independently. A drawing is well-dimensioned if its generic framework is independent. A drawing or framework is generic ifthe coordinates of the geometries are generic subject to the requirement that the logical constraints are satisfied.

7. There is a problem with lines An angle constraint (between two lines) is unchanged by a translation of either line. An angular constraint between two lines can be induced by a non-rigid sub-frame If X is rigid then X U Y is not independent but X U Y is not rigid. Same problem as double banana for points in 3D. Angle constraints may not be evident X Y

8. Work around solution Assume that all lines are connected in a tree of angle dimensions Compare with all hinges present for points in 3D. In fact it is enough that every line with more than two neighboursis in this tree – this is often a good approximation (for example it works for the design above, but not for the triangle) This is equivalent to assuming that a line has only a positional freedom and that the direction (slope) of the line is fixed. This gives rise to a point-line-position framework

9. Definition: A point-line-position graph G is a graph in which there are: Vertices which are labelled as points or lines Edges between two point vertices which are labelled as distance edges Edges between a point vertex and a line vertex which are labelled distance or coincidence There are no edges between two lines

10. Equation Rigidity Matrix p1 p2 l2 |(p1-p2)|2=d2 p1-p2 p2-p1 (p1.t2-l2)2=d2 t2 -1 p1.t2-l2=0 t2 -1

11. Definition: A point-line-position framework (Gt,p,l) is a point-line-position graph, an assignment t for the line directions and an assignment (p,l) for the point and line positions which satisfy the coincidence equations in Gt. A point-line-position graph is independent if f(X)=2|Vp(X)|+|Vl(X)|-|E(X)| ≥ 2+∂(|Vl(X)|), where ∂l(X) = 1 if |Vl(X)|=0 else ∂l(X) = 0, for every subgraph X with |E(X)| ≥1. A point-line-position framework is independent if its Rigidity Matrix has linearly independent rows.

12. The usual framework (for points) is a point-line-position framework with |Vl|=0 The direction-length framework is a point-line-position framework with every point-line edge is a coincidence edge every line vertex is degree two – no three points are collinear Many CAD drawings can be described by a point-line-position framework (after a bit of manipulation). We will also mostly assume that the line directions t are generic i.e. determined by a set of |Vl| algebraically independent real numbers. This is not a good assumption but we hope it is not significant!

13. Some Results for Point-Line-Position Frameworks Theorem 1. If there are no coincidence constraints then (p,l) may be simply generic (algebraically independent) and (Gt,p,l) is independent for generic (p,l) and generic t if and only if G is independent. The proof is quite straightforward. It can be done using only the usual Henneberg moves (vertex addition and edge splitting with link addition)

14. Now with distance constraints and coincidence constraints . G(0) is the subgraph of G with the same vertices as G but only the coincidence edges. If G is independent then G(0) and(G(0)t,p,l) are independent. The equations determined by G(0) and t are all linear because t is considered as fixed. They are also homogeneous. The framework vectors (p,l) which satisfy these linear equations lie in a subspace of R(2Vp+Vl) with dimension f(G(0)).We call this the coincidence subspace. The coincidence subspace is determined by G(0)t. A framework vector (p,l) for the framework (Gt,p,l) is generic if it is a generic point of the coincidence subspace.

15. A subgraph R(0) of G(0) is a rigid coincidence subgraphif f(R(0))=2. Rigid coincidence subgraphs of Gplay a special role If p1 and p2 are in R(0) then geometrically p1 = p2

16. Define a new graph id(G) by merging all point vertices which are in the same rigid coincidence subgraph G Can easily prove id(id(G)) = id(G). A framework vector (p,l) for the framework (Gt,p,l) is well-separated if distinct vertices in id(G) have distinct coordinates.

17. Theorem: If G is independent and t generic then the framework (Gt,p,l) has a framework vector (p,l) that is well-separated. Proof: Add a projected distance edge between a pair of points in G(0) which are not the same vertex in id(G). This system of linear equations has a solution because the framework Gt(0) is independent. Consequence: A generic framework vector for (Gt,p,l) is well-separated.

18. Main Theorem: G is a point-line-position graph and t a set of generic directions (slopes) for the lines. Then G and id(G) are both independent (as point-line-position graphs) if and only if (Gt,p,l) is independent for a generic framework vector (p,l) Note: Could simply forbid rigid coincidence subgraphs with 2 or more point vertices. Then f(X) ≥ 2+∂(|Vl(X)|)+ ∂(|Ed(X)|) and id(G)=G.

19. Proof Method Need more than Henneberg moves ph (G’t,p,l) (Gt,p,l) Does (G’t,p,l) independent imply ( Gt,p,l) independent ???? Note that the coordinates of ph are fully determined by G’.

20. First new graph move: Vertex split/merge. Point vertex pmhas line vertex neighbours l1 and l2 via coincidence edges: Merge vertices l1 and l2 f(G’) = f(G) G G’=m(pm,l1,l2)G G is independent m(pm,l1,l2)G is not independent R

21. Second new graph move: If Y is a rigid subgraph of G with f(Y)=2, rearrange the distance edges in Y to generate rY(G). G rY(G) Y R r(G(0)) =G(0) . G and rY(G) have the same coincidence subspace If (Yt,p,l) and r(Yt,p,l) are both independent: (Gt,p,l) is independent if and only if (rY(Gt),p,l) is independent. Can show: there is rY such that m(pm,l1,l2)rY(G) is independent.

22. Also need id(m(pm,l1,l2)rY(G)) independent - not always true pm pm Can prove: There is always pm,l1,l2 and rY and rZ such that m(pm,l1,l2)rY(G) and id(m(pm,l1,l2)rZrY(G)) are independent.

23. Point-line-position frameworks give a reasonable representation for some Cad drawings. Point-line-position frameworks include distance-angle frameworks and allow points to be constrained collinear. We have a combinatorial (matroid) description for generic rigidity. There is a pebble game to determine generic rigidity, circuits and rigid components.