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

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.

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|V p (G)|+2|V l (G)| coordinates in total.

= line=point = dimension= coincidence

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 if the coordinates of the geometries are generic subject to the requirement that the logical constraints are satisfied.

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 XY 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

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 neighbours is 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

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

Equation Rigidity Matrix p 1 p 2 l 2 |(p 1 -p 2 )| 2 =d 2 p 1 -p 2 p 2 -p 1 (p 1.t 2 -l 2 ) 2 =d 2 t 2 -1 p 1.t 2 -l 2 =0 t 2 -1

Definition: A point-line-position framework (G t,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 G t. A point-line-position graph is independent if f(X)=2|V p (X)|+|V l (X)|-|E(X)| ≥ 2+∂(|V l (X)|), where ∂ l (X) = 1 if |V l (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.

The usual framework (for points) is a point-line-position framework with |V l |=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 |V l | algebraically independent real numbers. This is not a good assumption but we hope it is not significant!

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 (G t,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)

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 (G t,p,l) is generic if it is a generic point of the coincidence subspace.

A subgraph R (0) of G (0) is a rigid coincidence subgraph if f(R (0) )=2. Rigid coincidence subgraphs of G play a special role If p 1 and p 2 are in R (0) then geometrically p 1 = p 2

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 (G t,p,l) is well-separated if distinct vertices in id(G) have distinct coordinates.

Theorem: If G is independent and t generic then the framework (G t,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 G t (0) is independent. Consequence: A generic framework vector for (G t,p,l) is well- separated.

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 (G t,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+∂(|V l (X)|)+ ∂(|E d (X)|) and id(G)=G.

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

First new graph move: Vertex split/merge. Point vertex p m has line vertex neighbours l 1 and l 2 via coincidence edges: Merge vertices l 1 and l 2 GG’=m(p m,l 1,l 2 )G f(G’) = f(G) R G is independent m(p m,l 1,l 2 )G is not independent

Second new graph move: If Y is a rigid subgraph of G with f(Y)=2, rearrange the distance edges in Y to generate r Y (G). R G r Y (G) r(G (0) ) =G (0). G and r Y (G) have the same coincidence subspace If (Y t,p,l) and r(Y t,p,l) are both independent: (G t,p,l) is independent if and only if (r Y (G t ),p,l) is independent. Can show: there is r Y such that m(p m,l 1,l 2 )r Y (G) is independent. Y

Also need id(m(p m,l 1,l 2 )r Y (G)) independent - not always true pmpm pmpm Can prove: There is always p m,l 1,l 2 and r Y and r Z such that m(p m,l 1,l 2 )r Y (G) and id(m(p m,l 1,l 2 )r Z r Y (G)) are independent.

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.