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 thepoints 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.

= 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-rigidsub-frame

X Y

If X is rigid then X U Y is not independentbut 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 dimensionsCompare 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

p1 p2 l2

|(p1-p2)|2=d2

p1-p2 p2-p1

(p1.t2-l2)2=d2 t2 -1

p1.t2-l2=0 t2 -1

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.

The usual framework (for points) is a point-line-position framework with |Vl|=0

The direction-length framework is a point-line-position framework withevery point-line edge is a coincidence edgeevery 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!

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)

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.

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 p1 and p2 are in R(0) then geometrically p1 = p2

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-separatedif distinct vertices in id(G) have distinct coordinates.

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.

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.

Proof Method

Need more than Henneberg moves

ph

(Gt,p,l)(G’t,p,l)

Does (G’t,p,l) independent imply ( Gt,p,l) independent ????

Note that the coordinates of ph are fully determined by G’.

First new graph move: Vertex split/merge.

Point vertex pmhas line vertex neighbours l1 and l2 via coincidence edges: Merge vertices l1 and l2

G G’=m(pm,l1,l2)G

f(G’) = f(G)

R

G is independent

m(pm,l1,l2)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 rY(G).

R

G rY(G)

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.

Y

Also need id(m(pm,l1,l2)rY(G)) independent - not always true

pmpm

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.

Point-line-position frameworks give a reasonable representation for some Cad drawings.

Point-line-position frameworks include distance-angle frameworks andallow 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.