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2D Body-and-cad Rigidity
Linnea LaMon1
Carnegie Mellon University
July 29, 2012
1Supported by NSF grant DMS-0849637
Body-and-cad Structures
bodiescoincidence, angular, distance constraints
Figure: Line-line distance constraint between two bodies.
L. LaMon 2D Body-and-cad Rigidity
Rigidity
Flexible
Figure: Line-line distance constraint.
Rigid
Figure: Point-point distance,point-line coincidence, line-lineperpendicular constraints.
L. LaMon 2D Body-and-cad Rigidity
Motivation
geometric-based CAD software
click to startL. LaMon 2D Body-and-cad Rigidity
Approach
Algebraic Rigidity
Infinitesimal Rigidity
Combinatorial Rigidity
L. LaMon 2D Body-and-cad Rigidity
2D Motions
rotations
translations
(0 −11 0
)(10
)+
(11
)=
(12
)
L. LaMon 2D Body-and-cad Rigidity
2D Motions
(0 −11 0
)(10
)+
(11
)=
(12
)∈ R2
0 −1 11 0 10 0 1
101
=
121
∈ R3
In general, T =
cos θ − sin θ τxsin θ cos θ τy
0 0 1
L. LaMon 2D Body-and-cad Rigidity
Body-and-cad Constraints
CoincidenceDistanceAngular: only between pairs of lines
Figure: Point-point distance, point-line coincidence, line-lineperpendicular constraints.
L. LaMon 2D Body-and-cad Rigidity
Basic Angular Constraint
Line-line fixed angular
cosβ = di · djcosβ = Tidi · Tjdj
Figure: Two bodies, i and j , constrained by a fixed angle β.
L. LaMon 2D Body-and-cad Rigidity
Basic Blind Constraint
Point-line coincidence
p = a + λdTj(p, 1) = Ti (a, 1) + λTi (d, 0)
Figure: A point p on body j is constrained to the line determined bypoint a and direction d on body i .
L. LaMon 2D Body-and-cad Rigidity
Approach
Algebraic Rigidity
Infinitesimal Rigidity
Combinatorial Rigidity
L. LaMon 2D Body-and-cad Rigidity
Infinitesimal Rigidity
Analyze localbehavior
Exterior algebra
Rigidity matrix
(pi ,1, 1) ∨ (d̂1, 0) −((pi ,1, 1) ∨ (d̂1, 0))
(pi ,2, 1) ∨ (d̂j , 0) −((pi ,2, 1) ∨ (d̂j , 0))(di , 0) ∨ (dj , 0) −((di , 0) ∨ (dj , 0))
Figure: Rigid. Point-point distance, point-line coincidence, line-lineperpendicular constraints.
L. LaMon 2D Body-and-cad Rigidity
Approach
Algebraic Rigidity
Infinitesimal Rigidity
Combinatorial Rigidity
L. LaMon 2D Body-and-cad Rigidity
Summary
2D motions, algebraic rigidity
instantaneous motions, infinitesimal rigidity, rigidity matrix
combinatorial rigidity (Sidman-Lee-St. John, 2012)
L. LaMon 2D Body-and-cad Rigidity