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Yanxi [email protected]
Today’s Theme:
Formalization ---how to represent your application problem in group theory terms?
And Computation ---
how to denote, compute and use
symmetry groups on computers?
Example I: Surface contact of 3D solids
Solids in Contact-MotionLower-pairs
‘Put that cube in the corner !’ (how many different ways?)
1
23
‘Put that cube in the corner with face 1 on top !’(4 different ways)
Complete and unambiguous task specifications can be tedious for symmetrical objects
Each Algebraic Surface as one primitive feature associated with its own coordinates
World coordinates
Solid coordinates
Surface coordinates
l
f
Contacting surfaces
translations
Surface 2
Surface 1
Relative Motion and Contacting Surface Symmetry
Relative Locations of Two Contacting Solids
Solid 1l1
f1
G1 -- symmetry group of surface 1
-- symmetry group of surface 2G2
Surface 2
world
Solid 2 Surface 1
l 2
f2
l1-1
l 2
U
f1 G1
-1f2
σ G2
G1 G2σ
Containing multiple motions
Using group properties ofthe transformations!
Insight:
The contacting surface pair from two different solids coincide, thus has the samesymmetry group which determines their relative motions/locations
Relative locations of solids in terms of their contacting surface symmetry groups
General contact: l1-1
l 2
U
f1 G1
-1f2
σ G2
Relative locations of solids in terms of their contacting surface symmetry groups
l1-1
l 2U
f1
G -1f2Under surface contact:
Relative locations of solids in terms of their contacting surface symmetry groups
Under multiple surface contacts:
l1-1
l 2 f1
-1f2
U (G1
U
G2…)
Relative locations of solids in terms of their contacting surface symmetry groups
Under multiple general contacts:
Relative locations of solids in terms of their contacting surface symmetry groups
Primitive surface features:
finite or infinite?
To guarantee a physical contact, either it is considered as infinitesimal motion from a real contact, or additional constraints are needed
Primitive surface features:
‘hair’ or no ‘hair’?
Symmetries for a Surface:
keeping the orientation invariant or not?
Symmetries for a Surface:
keeping the orientation invariant or not?
Symmetries for an oriented primitive feature F = (S,ρ)
• S is a connected, irreducible and continuous algebraic surface (a point set), bounding a finite solid M
• ρ ⊆ S x unit sphere is a set of pairs• For all s in S and (s,v) in ρ v poins away from
M
Intuitively, now primitive feature F is composed of both ‘skin’ S and ‘hair’ ρ
Symmetries of primitive feature F
• A symmetry of F has to keep two sets of points invariant
• Symmetries of an oriented feature F form a group (proof the four properties of a group)
Consider the symmetry group of the contacting surfaces collectively
DISTINCT
1-congruent
2-congruent
COMPLEMENT Ex. 2:
Ex. 1:
A pair of Surface Features
• distinct• 1-congruent• 2-congruent• complementary
Proposition 2.2.22
Distinct, 1-congruent, 2-congruent and complementary are the ONLY possible relationships between a pair of primitive features
Proposition 2.3.30
If a compound feature F is composed of pairwise distinct primitive features, the symmetry group G of F is the intersection of the symmetry groups of the primitive features
Proposition 2.3.30 ADD the case of 2-congrudent!!!
If a compound feature F is composed of pairwise distinct primitive features, the symmetry group G of F is the intersection of the symmetry groups of the primitive features
Different subgroups of the proper Euclidean Group
The hierarchy of the subgroups of The Euclidean Group
O orthogonal groupSO special orthogonal groupT translation groupD dihedral groupC cyclic group
How to represent this diverse set of subgroups on computers?
How to do group intersection on computers?
Symmetry groups of contacting surfaces can be finite, infinite, discrete, non-discrete and
continuous …
How to represent this diverse set of subgroups on computers?
How to do group intersection on computers?
Symmetry groups of contacting surfaces can be finite, infinite, discrete, non-discrete and
continuous …
Examples of group G versus set S
G SAct upon
Invariant of
Algebraic domain Geometric domain
subgroups
G3
G2
G1 CI-1
CI-3
CI-2
Characteristic invariance
G1 G2 = G3
Geometric operations
Mapping from subgroups to geometric invariants
ff
f
-1
U
Inverse of
Orbits (review)
An orbit of a point x in R3 under group G is G(x) = { g(x)| all g in G}
Group acts transitively on S
If for each x, y in S, there exists a g in G, such that y = g(x).
Stabilizer Subgroup
For x in X, if the set Gx={g in G | g(x) = x)} is a group, then G is called the stabilizer subgroup of G at x
Fixed Point Set of a Group
FG= { x in X | for all g in G, g(x) = x }
Euclidean Group (review)The set of all the isometries in 3D Euclidean space with the composition of isometries as the binary operation.If we restrict the set of isometries to those that preserve handedness, thus exluding reflections, the remaining set with the same composition forms the proper Euclidean group.
Group Conjugation (review)
G1, G2 are subgroups of G, we call G1 is a conjugate to G2 iff for some g in G, such that G1 = g * G2 * g-1
Pole Set
A pole of a rotation group R = tRct-1 is a point p on the unit sphere centered at the origin which is left fixed by some rotation of the group Rc other than the identity.
Translation Subgroups T
G = TR
Translational Invariant of T (translation subgroup): The T orbit of the origin
Rotation Subgroups R
G = TR
Characteristic Invariants of R (rotation subgroup):
{ Fixed Point set, Pole-set }
Group Fixed-point Set Pole-set
Group Fixed-point Set Pole-set
TR Subgroups of the Proper Euclidean Group
Translational Invariant: The T orbit of the origin
Rotational Invariant:
(fixed-point-set, pole-set) p90 on Yanxi’s book manuscript
Note: the fixed-point-set of R is translated by the T subgroup of the TR groups.
Symmetry group denotation on computers
<a,b,c><p,inf>
<-p,inf>
Example: Symmetry group of a non-directed plane G = TR
Translational invariance:T orbit of the origin
Rotational invariance:fixed point set + pole set
aX + bY + cZ = 0{(p,0),(-p,0),<circle,2>}
<circle,2>
Symmetry group denotation on computers
<a,b,c><p,inf>
<-p,inf>
Example: Symmetry group of a directed plane G = TR
Translational invariance:T orbit of the origin
Rotational invariance:fixed point set + pole set
aX + bY + cZ = 0{(p,0),(-p,0)}
Symmetry group denotation on computers
<x,y,z>
<p,inf>
<-p,inf>
Example: Symmetry group of a cylindrical surface G = TR
Translational invariance:T orbit of the origin
Rotational invariance:fixed point set + pole set
{(p,0),(-p,0),<circle,2>}
<circle,2>
<a,b,c>
<a,b,c,x,y,z>
Group Intersection
This is an O(n ) algorithm, where n is the number of countable poles. 2
translationrotation
22
2
2
2
2 G= Gplane Gcyl
= T D2x21
U
002
2
0
0
What about Crystallographic Groups?
Continuous Discrete Translation Symmetry
Symmetry groups of contacting surfaces: a computational representation
S2 S3
S4S1 Solid level
Surface level1 ... n
G1 Gn
1 ... n
1 ...n
1 ... n
Input: Solid 1 fits solid 2, and solid 2 fits solid 3, ...
ACIS(PADL2)
Disassembly-motion analysis
Possible AssemblyConfigurations
FeasibleAssemblyConfigurations
PartiallyOrdered AssemblyMotionSets
Mating-feature inference
Spatial constraint satisfaction
attach symmetry groupto each surface feature
S1
S2
S4
S3
S5
A gearbox example
S1
S2
S3
S4
S5
S1 S2
S3S4
S5
S1
S5
S4S3
S2 E(3)
cyl
SO(2)
mesh
mesh
meshfit
fitfit
fit
Precedence diagram
Contributions• Formalization of general solid contacts in
terms of their surface symmetry groups• development of a geometric TR subgroup
denotation and an efficient group intersection algorithm treating both continuous and discrete symmetries uniformly
• application of symmetry groups to assembly task specification from CAD models, bridging between symbolic reasoning and numerical computation
Future Research Issues
• Computation of symmetry group of a surface (algebraic, parametric, splined, etc.)
• Computation of general solid contact other than surface contact (require group product)
• Symmetry guided sensory and manipulation planning and execution of complicated (dis)assembly tasks
Ingredients of using group theory in real world problems
• The problem itself presents some type of regularity (e.g. rotation, translation symmetries). Usually not perfect.
• There exists a benefit to unveil this regularity• Find the right group structure• Express the relation (mapping) between the
symmetry groups and the real data• Compute the mapping between the
symmetry groups and the data
