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