carnegie mellon school of computer science | - yanxi...
TRANSCRIPT
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Yanxi [email protected]
RI and CALDSchool of Computer Science, Carnegie Mellon University
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Today’s Theme:Different Types of Symmetries
and Symmetry Groups
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What is a symmetry?
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What is a symmetry of S ⊆R3?
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DefinitionIf g is a distance preserving transformation in n-dimensional Euclidean space R , and S is a subset of R , then g is a symmetry of S iff g(S) = S, and S is (setwise) invariant under action g.
An example:a square plate in R(2)
Reflection axis
4-fold rotations
nn
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DefinitionA group G is a set of elements with a binaryoperation * defined on the set that satisfy:
• * is associative a*(b*c) = (a*b)*c• G is closed, for any a,b in G, a*b also in G• any element in G has an inverse• there is an identity element id in G, id*a = a
a square plate in R(2)
G = symmetries of asquare
* = transformation composition
An example:
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Symmetry Group
All symmetries of a subset S of Euclidean space R have a group structure G, and G is called the symmetry group of S.
n
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Examples of group G versus set S
G SAct upon
Invariant of
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The Promise:… beyond absolute regularity there are nuances
of decreasing regularities, of reductive invariance: that there is an order to so-called disorder.
Ordering Disorder after K.L. Wolfby W.S. Huff (an architect)
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The PerilsTyger! Tyger! burning bright,In the forest of the night,What immortal hand or eyeCould frame thy fearful symmetry?
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William Blake 1794
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What are different types of symmetries?
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Symmetry or Near-Symmetry Patterns are ubiquitous
Symmetry or Near-Symmetry Patterns are ubiquitous
a square shape in R2
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Symmetry or Near-Symmetry Patterns are ubiquitous
Symmetry or Near-Symmetry Patterns are ubiquitous
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Symmetries in 3D
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Hierarchy of Symmetries
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Which one is more symmetrical?
or
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How to organize these different types of
symmetries?
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Types of Symmetry Groups G: for all g in G, g(S) =S
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• point group: All symmetries g in group G leave at least one point in R3 invariant • Space group: symmetry groups in R3
that contains a non-trivial translation symmetry
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Euclidean Group
All symmetries of R3 for the symmetry group of the 3D Euclidean space and is called the Euclidean Group
Examples of elements in the Euclidean Group are:Rotations
TranslationsReflections
Glide-reflections…
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Proper Euclidean Group
All handedness-preserving isometries form the proper Euclidean group that
excludes reflections
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Group relations: Subgroup
Definition:Let (G,*) be a group and let H be subset of G . Then H is a subgroup of G defined under the same operation if H is a group by itself (with respect to * ), that is:• H is closed under the operation. • There exists an identity element e such that for all h, h*e = e*h = h• Let h in H, then there exists an inverse h-1 such that h-1*h = e = h*h-1.
The subgroup is denoted likewise (H, *).We denote H being a subgroup of G by writing H G .
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Group Relations: Group Conjugate
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
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Orbits
An orbit of a point x in R3 under group G is G(x) = { g(x)| all g in G}
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Types of Symmetry Groups G of S
• G is finite• G is infinite
• G is discrete (Definition 1.1.10)• G is continuous
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Definitions of Different Types of Subgroups of the Euclidean Group
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Discrete Point Groups asSubgroups of theEuclidean Group
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The hierarchy of the subgroups of The Euclidean Group
O orthogonal groupSO special orthogonal groupT translation groupD dihedral groupC cyclic group
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S(2), S(3), SO(2), SO(3), T1-3
???continuous
T1C2 …???Non-discrete
Crystallographic groups
Cn, D2n, G of regular solids
discrete
infinitefinite
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An Exploration of Hierarchies of Symmetry Groups
• Surface Contact/Motion among solids (robotics assembly planning)
• Periodic Patterns (visual)• Papercut Patterns (fold-then-cut)
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Example I: Symmetry in Contact Motions
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Solids in Contact-MotionLower-pairs
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Insight:
The contacting surface pair from two different solids coincide, thus has the samesymmetry group which determines their relative motions/locations
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‘Put that cube in the corner !’ (how many different ways?)
1
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‘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
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Constructive Solid Geometry (CSG) Representation
Different types of surfaces associated with different types of groups
The concept of “conjugated symmetry groups”are used here
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Algebraic Surface feature and its coordinates
World coordinates
Solid coordinates
Surface coordinates
l
f
Contacting surfaces
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Example II: Symmetry in Visual Form
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p2
t1
t2
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p6
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Hilbert’s 18th Problem
Question:
In n-dimensional euclidean space is there … only a finite number of essentially different kinds of (symmetry)groups of motions with a fundamental region?
Answer:
Yes! (Bieberbach and Frobenius, published 1910-1912)
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11 1g m1 12 mg 1m mm
I II III IV V VI VII
Examples of Seven Frieze Patterns and their symmetry groups
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Hierarchy of Frieze Groups
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Examples of 17 Wallpaper Patterns and Their Symmetry Groups
From a web page by:David Joyce, Clark Univ.
p1 p2 pm pg cm
pmm pmg pgg cmm p4
p4m p4g p3 p31m p3m1
p6 p6m
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lattice units of the 17 wallpaper groups
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p1
p2
p3
p4
p6
pg
pmcm
pgg
pmgpmm
cmm
p4gp4m
p3m1
p31m
p6m
Subgroup Relationship Among the 17 Wallpaper Groups(Coxeter)
P1 camp P2 camp
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Where does the symmetry group of an affinely deformed pattern migrate to?
p2
cmmp4m
cmm
cmm
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Examples of Symmetry Group Migrationcmcm p1
p3m1
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Potential Symmetry= largest Euclidean subgroup in AGA-1An inherent, affine-invariant, non-face-value
property of a patternp6mcmmp2
p4mcmmpmmp2
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Example III: Reflection Symmetry for Folding
A Symmetry Group Formalization of Papercut Patterns
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SIGGRPHA 2005 Technical Sketch Slides
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Symmetry Groups are not just decorative but (computationally and physically) functional
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Digital Papercutting
Yanxi Liu and James Hays Carnegie Mellon University
Ying-Qing Xu and Heung-Yeung ShumMicrosoft Research Asia
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Motivation
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Fold and Cut Example
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Contribution• We present a general computational
formalization of this particular art form for the first time.
• We combine the analysis and synthesis of a papercut pattern into one automatic process, as well as the generation of a physically feasible folding plan.
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Related work
• Computational geometry– Demaine, Erik., Demaine, Martin, and Lubiw, Anna.
Folding and cutting paper. In JCDCG’98.– Demaine, Erik., Demaine, Martin, and Lubiw, Anna.
Folding and One Straight Cut Suffice SODA'99.
• Papercraft– Mitani, J., and Suziki, H. Making papercraft toys from
meshes using strip-based approximate unfolding. SIGGRAPH 2004.
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Formal treatment of papercutting
P S1 S2 S3
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Characterization by Symmetry Groups
Frieze Group Dihedral Group
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Dihedral group examples
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Dihedral group examples
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Dihedral group examples
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Dihedral group examples
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Dihedral group examples
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Dihedral group examples
Fundamental Regions
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Dihedral group examples
Fundamental Regions
3 folds 3 folds 4 folds 4 folds 4 folds 4 folds
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Dihedral group examples
Fundamental Regions
3 folds 3 folds 4 folds 4 folds 4 folds 4 folds