SLIDE 1
1 / 23
Plane Isometries
2 / 23
- Theorem. Any isometry of R2 is a composition
- f ≤ 3 reflections in lines.
Head to tail compositions Oleg Viro November 27, 2014 1 / 23 - - PowerPoint PPT Presentation
Head to tail compositions Oleg Viro November 27, 2014 1 / 23 - - PowerPoint PPT Presentation
Head to tail compositions Oleg Viro November 27, 2014 1 / 23 Plane Isometries Theorem. Any isometry of R 2 is a composition of 3 reflections in lines. 2 / 23 Plane Isometries Theorem. Any isometry of R 2 is a composition of 3
SLIDE 2
SLIDE 3
Plane Isometries
2 / 23
- Theorem. Any isometry of R2 is a composition
- f ≤ 3 reflections in lines.
- Lemma. A plane isometry is determined by its restriction
to any three non-collinear points.
SLIDE 4
Plane Isometries
2 / 23
- Theorem. Any isometry of R2 is a composition
- f ≤ 3 reflections in lines.
Proof of Theorem. Given an isometry:
A B C f(C) f(B) f(A) f
SLIDE 5
Plane Isometries
2 / 23
- Theorem. Any isometry of R2 is a composition
- f ≤ 3 reflections in lines.
Proof of Theorem.
A B C R1(C) f(C) f(B) f(A) R1(B) R1 = R1(A)
SLIDE 6
Plane Isometries
2 / 23
- Theorem. Any isometry of R2 is a composition
- f ≤ 3 reflections in lines.
Proof of Theorem.
A B C R1(C) f(C) R2 ◦ R1(C) f(B) f(A) R1(B) R2
SLIDE 7
Plane Isometries
2 / 23
- Theorem. Any isometry of R2 is a composition
- f ≤ 3 reflections in lines.
Proof of Theorem.
A B C f(C) R2 ◦ R1(C) f(B) f(A) R3
SLIDE 8
Plane Isometries
2 / 23
- Theorem. Any isometry of R2 is a composition
- f ≤ 3 reflections in lines.
Proof of Theorem.
A B C f(C) f(B) f(A) f
We are done.
SLIDE 9
Compositions of reflections in parallel lines
3 / 23
m l
SLIDE 10
Compositions of reflections in parallel lines
3 / 23
m l
SLIDE 11
Compositions of reflections in parallel lines
3 / 23
m l
SLIDE 12
Compositions of reflections in parallel lines
3 / 23
m l
SLIDE 13
Compositions of reflections in parallel lines
3 / 23
is a translation
m l
SLIDE 14
Compositions of reflections in parallel lines
3 / 23
is a translation
x Rm(x) Rl ◦ Rm(x) m l
SLIDE 15
Compositions of reflections in parallel lines
3 / 23
is a translation
x Rm(x) Rl ◦ Rm(x) m l
The decomposition is not unique:
SLIDE 16
Compositions of reflections in parallel lines
3 / 23
is a translation
x Rm(x) Rl ◦ Rm(x) m l
The decomposition is not unique:
Rl ◦ Rm = Rl′ ◦ Rm′
iff l′, m′ can be obtained from l, m by a translation.
SLIDE 17
Compositions of reflections in parallel lines
3 / 23
is a translation
x Rl′ ◦ Rm′(x) m′ l′ Rm′(x)
The decomposition is not unique:
Rl ◦ Rm = Rl′ ◦ Rm′
iff l′, m′ can be obtained from l, m by a translation.
SLIDE 18
Compositions of reflections in intersecting lines
4 / 23
m l
SLIDE 19
Compositions of reflections in intersecting lines
4 / 23
m l
SLIDE 20
Compositions of reflections in intersecting lines
4 / 23
m l
SLIDE 21
Compositions of reflections in intersecting lines
4 / 23
m l
SLIDE 22
Compositions of reflections in intersecting lines
4 / 23
is a rotation
α α β β x Rm(x) Rl ◦ Rm(x) m l
SLIDE 23
Compositions of reflections in intersecting lines
4 / 23
is a rotation
α α β β x Rm(x) Rl ◦ Rm(x) m l
Decomposition of rotation is not unique:
SLIDE 24
Compositions of reflections in intersecting lines
4 / 23
is a rotation
α α β β x Rm(x) Rl ◦ Rm(x) m l
Decomposition of rotation is not unique:
Rl ◦ Rm = Rl′ ◦ Rm′
iff l′, m′ can be obtained from l, m by a rotation about the intersection point m ∩ l .
SLIDE 25
Compositions of reflections in intersecting lines
4 / 23
is a rotation
x Rl′ ◦ Rm′(x) Rm′(x) m′ l′
Decomposition of rotation is not unique:
Rl ◦ Rm = Rl′ ◦ Rm′
iff l′, m′ can be obtained from l, m by a rotation about the intersection point m ∩ l .
SLIDE 26
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
SLIDE 27
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines.
SLIDE 28
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 ∦ 2 and 3 ∦ 4 .
2 1 3 4
SLIDE 29
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 ∦ 2 and 3 ∦ 4 .
3 4 1 2
SLIDE 30
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 ∦ 2 and 3 ∦ 4 .
3 4 2 1
SLIDE 31
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 ∦ 2 and 3 ∦ 4 .
1 4
SLIDE 32
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let either 1 ∦ 2 or 3 ∦ 4 .
SLIDE 33
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let either 1 ∦ 2 or 3 ∦ 4 .
3 4 2 1
SLIDE 34
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let either 1 ∦ 2 or 3 ∦ 4 .
3 4 2 1
SLIDE 35
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let either 1 ∦ 2 or 3 ∦ 4 .
3 4 1 2
SLIDE 36
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let either 1 ∦ 2 or 3 ∦ 4 .
3 = 2 4 1
SLIDE 37
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let either 1 ∦ 2 or 3 ∦ 4 .
4 1
SLIDE 38
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 2 and 3 4 .
SLIDE 39
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 2 and 3 4 .
1 2 3 4
SLIDE 40
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 2 and 3 4 .
1 4 2 3
SLIDE 41
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 2 and 3 4 .
1 4 2 3
SLIDE 42
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 2 and 3 4 .
4 3 1 2
SLIDE 43
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 2 and 3 4 .
4 3 1 2
SLIDE 44
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 2 and 3 4 .
1 2 3 4
SLIDE 45
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 2 and 3 4 .
1 2 4 3=
SLIDE 46
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Let 1 2 and 3 4 .
1 4
SLIDE 47
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines.
SLIDE 48
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Proof of Theorem. By Lemma, any relation can be reduced to a relation of length ≤ 3 .
SLIDE 49
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Proof of Theorem. By Lemma, any relation can be reduced to a relation of length ≤ 3 . A composition of odd number of reflections reverses orientation and cannot be id .
SLIDE 50
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Proof of Theorem. By Lemma, any relation can be reduced to a relation of length ≤ 3 . A composition of odd number of reflections reverses orientation and cannot be id . A composition of two different reflections is not identity.
SLIDE 51
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Generalization of Lemma. In Rn , a composition of any n + 2 reflections in hyperplanes is a composition of n reflections in hyperplanes.
SLIDE 52
Relations
5 / 23
- Theorem. Any relation among reflections in lines follow from relations
R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′ , where l, m, l′, m′ are as above.
- Lemma. A composition of any 4 reflections in lines can be transformed
by these relations to a composition of 2 reflections in lines. Generalization of Lemma. In Rn , a composition of any n + 2 reflections in hyperplanes is a composition of n reflections in hyperplanes. Generalization of Theorem. Any relation among reflections in hyperplanes of Rn follow from relations R2
l = 1 and Rl ◦ Rm = Rl′ ◦ Rm′.
SLIDE 53
Flips and flippers
6 / 23
Generalize reflections!
SLIDE 54
Flips and flippers
6 / 23
Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set.
SLIDE 55
Flips and flippers
6 / 23
Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip.
SLIDE 56
Flips and flippers
6 / 23
Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x → 2a − x , the reflection of R in a point a .
SLIDE 57
Flips and flippers
6 / 23
Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x → 2a − x , the reflection of R in a point a .
a → 2a − a = a
SLIDE 58
Flips and flippers
6 / 23
Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x → 2a − x , the reflection of R in a point a . Generalization: a symmetry of Rn in a k -subspace.
SLIDE 59
Flips and flippers
6 / 23
Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x → 2a − x , the reflection of R in a point a . Generalization: a symmetry of Rn in a k -subspace. Further examples in hyperbolic spaces, spheres, projective spaces and other symmetric spaces.
SLIDE 60
Flips and flippers
6 / 23
Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x → 2a − x , the reflection of R in a point a . Generalization: a symmetry of Rn in a k -subspace. Further examples in hyperbolic spaces, spheres, projective spaces and other symmetric spaces. Correspondence Flipper S ←
→ Flip in S is
the shortest connection between simple static geometric objects - flippers - and isometries.
SLIDE 61
Symmetry about a point
7 / 23
is a flip.
SLIDE 62
Symmetry about a point
7 / 23
is a flip. Composition of flips in points
B A
SLIDE 63
Symmetry about a point
7 / 23
is a flip. Composition of flips in points
B A
SLIDE 64
Symmetry about a point
7 / 23
is a flip. Composition of flips in points
B A
SLIDE 65
Symmetry about a point
7 / 23
is a flip. Composition of flips in points
B A
SLIDE 66
Symmetry about a point
7 / 23
is a flip. Composition of flips in points is a translation:
RB(RA(X)) RA(X) X B A
SLIDE 67
Symmetry about a point
7 / 23
is a flip. Composition of flips in points is a translation:
RB(RA(X)) RA(X) X B A
− → AB = 1
2
− − − − − − − − − → X RB(RA(X)
SLIDE 68
Symmetry about a point
7 / 23
is a flip. Composition of flips in points is a translation:
RB(RA(X)) RA(X) X B A
− → AB = 1
2
− − − − − − − − − → X RB(RA(X) − → AB is half the arrow representing RB ◦ RA .
SLIDE 69
Head to tail
8 / 23
Compare the head to tail addition −
→ AB + − − → BC = − → AC
to (RC ◦ RB) ◦ (RB ◦ RA) = RC ◦ R2
B ◦ RA = RC ◦ RA .
SLIDE 70
Flip-flop decomposition
9 / 23
Which isometries are compositions of two flips?
SLIDE 71
Flip-flop decomposition
9 / 23
Which isometries are compositions of two flips? Any isometry of Rn .
SLIDE 72
Flip-flop decomposition
9 / 23
Which isometries are compositions of two flips? Any isometry of Rn . Djokoviˇ c Theorem. Any isometry of a non-degenerate inner product space over any field can be presented as a composition of two involutions isomteries. ( Product of two involutions, Arch. Math. 18 (1967), 582-584.)
SLIDE 73
Flip-flop decomposition
9 / 23
Which isometries are compositions of two flips? Any isometry of Rn . Djokoviˇ c Theorem. Any isometry of a non-degenerate inner product space over any field can be presented as a composition of two involutions isomteries. ( Product of two involutions, Arch. Math. 18 (1967), 582-584.)
- Corollary. Any isometry of an affine space with a non-degenerate
bilinear form can be presented as a composition of two flips.
SLIDE 74
Flip-flop decomposition
9 / 23
Which isometries are compositions of two flips? Any isometry of Rn . Djokoviˇ c Theorem. Any isometry of a non-degenerate inner product space over any field can be presented as a composition of two involutions isomteries. ( Product of two involutions, Arch. Math. 18 (1967), 582-584.)
- Corollary. Any isometry of an affine space with a non-degenerate
bilinear form can be presented as a composition of two flips.
- Corollary. Any isometry of a hyperbolic space, sphere, projective space,
- etc. is a composition of two flips.
A flip-flop decomposition.
SLIDE 75
Biflippers
10 / 23
An ordered pair of flippers (A, B) is a biflipper, an analogue for an arrow representing a translation.
SLIDE 76
Biflippers
10 / 23
An ordered pair of flippers (A, B) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow.
SLIDE 77
Biflippers
10 / 23
An ordered pair of flippers (A, B) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow.
A B
SLIDE 78
Biflippers
10 / 23
An ordered pair of flippers (A, B) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow.
A B
If A ∩ B = ∅ ,
A B
SLIDE 79
Biflippers
10 / 23
An ordered pair of flippers (A, B) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow.
A B
If A ∩ B = ∅ ,
A B
SLIDE 80
Biflippers
10 / 23
An ordered pair of flippers (A, B) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow.
A B
If A ∩ B = ∅ ,
A B
To what extent are the representations non-unique?
SLIDE 81
Biflippers
10 / 23
An ordered pair of flippers (A, B) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow.
A B
If A ∩ B = ∅ ,
A B
To what extent are the representations non-unique? Equivalence relation:
(A, B) ∼ (A′, B′)
if
RB ◦ RA = RB′ ◦ RA′ .
SLIDE 82
Biflippers
10 / 23
An ordered pair of flippers (A, B) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow.
A B
If A ∩ B = ∅ ,
A B
To what extent are the representations non-unique? Equivalence relation:
(A, B) ∼ (A′, B′)
if
RB ◦ RA = RB′ ◦ RA′ .
- Problem. Find an explicit description for the equivalence.
SLIDE 83
Biflippers for a rotation
11 / 23
SLIDE 84
Biflippers for a rotation
11 / 23
an ordered pair of lines.
α α β β x Rm(x) Rl ◦ Rm(x) m l
SLIDE 85
Biflippers for a rotation
11 / 23
an ordered pair of lines. The lines intersect at the center of rotation.
α α β β x Rm(x) Rl ◦ Rm(x) m l
SLIDE 86
Biflippers for a rotation
11 / 23
an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle.
α α β β x Rm(x) Rl ◦ Rm(x) m l
SLIDE 87
Biflippers for a rotation
11 / 23
an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc.
SLIDE 88
Biflippers for a rotation
11 / 23
an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc. Equivalent biflippers are obtained by rotations about the vertex.
SLIDE 89
Biflippers for a rotation
11 / 23
an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc. Equivalent biflippers are obtained by rotations about the vertex.
SLIDE 90
Biflippers for a rotation
11 / 23
an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc. Equivalent biflippers are obtained by rotations about the vertex.
SLIDE 91
Biflippers for a rotation
11 / 23
an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc. Equivalent biflippers are obtained by rotations about the vertex.
SLIDE 92
Head to tail for rotations
12 / 23
SLIDE 93
Head to tail for rotations
12 / 23
Given two rotations, present them by biflippers.
A B l1 m1 m2 n2
SLIDE 94
Head to tail for rotations
12 / 23
Given two rotations, present them by biflippers. By rotating the biflippers, make the second line in the first biflipper to coincide with the first line in the second, so that the biflippers are (l, m) and (m, n) .
A B l1 m1 m2 n2
SLIDE 95
Head to tail for rotations
12 / 23
Given two rotations, present them by biflippers. By rotating the biflippers, make the second line in the first biflipper to coincide with the first line in the second, so that the biflippers are (l, m) and (m, n) .
A B l n m
SLIDE 96
Head to tail for rotations
12 / 23
Given two rotations, present them by biflippers. By rotating the biflippers, make the second line in the first biflipper to coincide with the first line in the second, so that the biflippers are (l, m) and (m, n) . Erase m and draw an oriented arc from l to n , i.e., form the ordered angle (l, n) .
A B l n m
SLIDE 97
Head to tail for rotations
12 / 23
Given two rotations, present them by biflippers. By rotating the biflippers, make the second line in the first biflipper to coincide with the first line in the second, so that the biflippers are (l, m) and (m, n) . Erase m and draw an oriented arc from l to n , i.e., form the ordered angle (l, n) .
l n C
SLIDE 98
Composing reflections in line and point
13 / 23
m O
SLIDE 99
Composing reflections in line and point
13 / 23
m O
SLIDE 100
Composing reflections in line and point
13 / 23
m O
SLIDE 101
Composing reflections in line and point
13 / 23
m O
SLIDE 102
Composing reflections in line and point
13 / 23
m O
This is a glide reflection!
SLIDE 103
Composing reflections in line and point
13 / 23
Indeed!
m O
This is a glide reflection!
SLIDE 104
Composing reflections in line and point
13 / 23
Indeed!
x Rm(x) RO(Rm(x)) m O
This is a glide reflection!
SLIDE 105
Composing reflections in line and point
13 / 23
Indeed!
x Rm(x) RO(Rm(x)) m O l Rl(Rm(x))
This is a glide reflection!
SLIDE 106
Biflippers for a glide reflection
14 / 23
x Rm(x) RO(Rm(x)) m O
SLIDE 107
Biflippers for a glide reflection
14 / 23
x Rm(x) RO(Rm(x)) m O
SLIDE 108
Biflippers for a glide reflection
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m O
SLIDE 109
Biflippers for a glide reflection
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m O
SLIDE 110
Biflippers for a glide reflection
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x Rm(x) RO(Rm(x)) m O l Rl(Rm(x))
SLIDE 111
Biflippers for a glide reflection
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x Rm(x) RO(Rm(x)) m O l Rl(Rm(x)) O′ RO′(x) = Rl(RO′(x))
SLIDE 112
Biflippers for a glide reflection
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x l O′ RO′(x) Rl(RO′(x))
SLIDE 113
Biflippers for a glide reflection
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l O′
SLIDE 114
Biflippers for a glide reflection
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m O
SLIDE 115
Biflippers for a glide reflection
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m O
A biflipper for a glide reflection may glide along itself.
SLIDE 116
Biflippers for a glide reflection
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x Rm(x) RO(Rm(x)) m O
A biflipper for a glide reflection may glide along itself.
SLIDE 117
Biflippers for a glide reflection
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x RO(Rm(x)) O Rm(x) m
A biflipper for a glide reflection may glide along itself.
SLIDE 118
Head to tail for glide reflections
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SLIDE 119
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers.
SLIDE 120
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers. The head in the first biflipper and tail in the second one should NOT be lines.
n2 O2 O1 l1
SLIDE 121
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers. The head in the first biflipper and tail in the second one should NOT be lines. By gliding the biflippers, make the head of the first biflipper coinciding with the tail of the second. so that the biflippers are −
→ lO and − → On .
n2 O2 O1 l1
SLIDE 122
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers. The head in the first biflipper and tail in the second one should NOT be lines. By gliding the biflippers, make the head of the first biflipper coinciding with the tail of the second. so that the biflippers are −
→ lO and − → On .
O l n
SLIDE 123
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers. The head in the first biflipper and tail in the second one should NOT be lines. By gliding the biflippers, make the head of the first biflipper coinciding with the tail of the second. so that the biflippers are −
→ lO and − → On .
Draw an oriented arc from l to n
O l n
SLIDE 124
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers. The head in the first biflipper and tail in the second one should NOT be lines. By gliding the biflippers, make the head of the first biflipper coinciding with the tail of the second. so that the biflippers are −
→ lO and − → On .
Draw an oriented arc from l to n
O l n
SLIDE 125
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers. The head in the first biflipper and tail in the second one should NOT be lines. By gliding the biflippers, make the head of the first biflipper coinciding with the tail of the second. so that the biflippers are −
→ lO and − → On .
Draw an oriented arc from l to n and erase O .
O l n
SLIDE 126
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers. The head in the first biflipper and tail in the second one should NOT be lines. By gliding the biflippers, make the head of the first biflipper coinciding with the tail of the second. so that the biflippers are −
→ lO and − → On .
Draw an oriented arc from l to n and erase O .
l n
SLIDE 127
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers. The head in the first biflipper and tail in the second one should NOT be lines. By gliding the biflippers, make the head of the first biflipper coinciding with the tail of the second. so that the biflippers are −
→ lO and − → On .
Draw an oriented arc from l to n and erase O .
l n
This is a rotation!
SLIDE 128
Head to tail for glide reflections
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Given two glide reflections, present them by biflippers. The head in the first biflipper and tail in the second one should NOT be lines. By gliding the biflippers, make the head of the first biflipper coinciding with the tail of the second. so that the biflippers are −
→ lO and − → On .
Draw an oriented arc from l to n and erase O .
l n
- Exercise. Find head to tail rules for rotation ◦ glide reflection.
SLIDE 129
In the 3-space. Rotation
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m l
SLIDE 130
In the 3-space. Rotation
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m l
SLIDE 131
In the 3-space. Rotation
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m l
SLIDE 132
In the 3-space. Rotation
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m m l l
SLIDE 133
In the 3-space. Rotation
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m m l l
SLIDE 134
In the 3-space. Rotation
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Rm(x) Rl ◦ Rm(x) m m α α β β x l α α β β x l α α β β x l m m
SLIDE 135
In the 3-space. Rotation
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Rm(x) Rl ◦ Rm(x) m m α α β β x l α α β β x l α α β β x l m m
Everything like on the plane.
SLIDE 136
In the 3-space. Rotation
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l m
A biflipper formed by two intersecting lines defines a rotation of the 3-space about the axis ⊥ to the plane of the lines.
SLIDE 137
In the 3-space. Rotation
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l m
A biflipper formed by two intersecting lines defines a rotation of the 3-space about the axis ⊥ to the plane of the lines.
SLIDE 138
Rotations of 2-sphere
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l b a B −B A α 2α
SLIDE 139
Rotations of 2-sphere
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Biflippers:
SLIDE 140
Rotations of 2-sphere
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Biflippers: Head to tail for rotations:
SLIDE 141
Rotations of 2-sphere
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Biflippers: Head to tail for rotations:
SLIDE 142
Rotations of 2-sphere
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Biflippers: Head to tail for rotations:
SLIDE 143
Rotations of 2-sphere
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Biflippers: Head to tail for rotations:
SLIDE 144
Rotations of 2-sphere
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Biflippers: Head to tail for rotations: Biflipper vs. angular displacement vector vs. unit quaternion.
SLIDE 145
Rotations of 2-sphere
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Biflippers: Head to tail for rotations: Biflipper vs. angular displacement vector vs. unit quaternion. The rotation encoded by bilipper −
→ wv is defined by quaternion vw = v × w − v · w .
SLIDE 146
Parade of biflippers
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On line:
translation reflections in points the identity
SLIDE 147
Parade of biflippers
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On plane:
translations rotation glide reflections reflections
SLIDE 148
Parade of biflippers
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On plane:
translations rotation glide reflections reflections
On sphere:
rotations reflections rotary reflections
SLIDE 149
Parade of biflippers
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On plane:
translations rotation glide reflections reflections
On sphere:
rotations reflections rotary reflections
On the hyperbolic plane:
reflections translation glide reflections rotation parallel motion
SLIDE 150
Biflippers in the 3-space
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rotations translations central symmetries symmetries about a line (half−turns) reflections glide symmetries about a line glide reflections screw motion rotary reflections
SLIDE 151
In hyperbolic 3-space
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translation rotation parallel motion screw motion
parallel reflections glide reflections rotary reflections
SLIDE 152
Screw displacement
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m l
SLIDE 153
Screw displacement
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m l
SLIDE 154
Screw displacement
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m m l m
SLIDE 155
Screw displacement
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m l m
SLIDE 156
Screw displacement
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m l
SLIDE 157
Screw displacement
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Rm(x) Rl′ ◦ Rm(x) m α α β β x α α β β l α α β β m l′
SLIDE 158
Screw displacement
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Rm(x) Rl′ ◦ Rm(x) m α α β β x α α β β l α α β β m l′
A biflipper presenting a screw displacement is an arrow with two perpendicular lines at the end points skew to each other.
SLIDE 159
Screw displacement
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m l
SLIDE 160
Head to tail for screws
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Given two screw displacement, present them by biflippers.
SLIDE 161
Head to tail for screws
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Given two screw displacement, present them by biflippers. Find the common perpendicular for the axes of the biflippers.
SLIDE 162
Head to tail for screws
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Given two screw displacement, present them by biflippers. Find the common perpendicular for the axes of the biflippers. By gliding the biflippers along their axes and rotating about the axes, make the arrowhead of the first biflipper coinciding with the tail of the second biflipper.
SLIDE 163
Head to tail for screws
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Given two screw displacement, present them by biflippers. Find the common perpendicular for the axes of the biflippers. By gliding the biflippers along their axes and rotating about the axes, make the arrowhead of the first biflipper coinciding with the tail of the second biflipper.
l n m
SLIDE 164
Head to tail for screws
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Given two screw displacement, present them by biflippers. Find the common perpendicular for the axes of the biflippers. Find common perpendicular for the tail of the first biflipper and head of the second biflipper. Draw an arrow along it connecting the flippers which are left.
l n m
SLIDE 165
Head to tail for screws
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Given two screw displacement, present them by biflippers. Find the common perpendicular for the axes of the biflippers. Find common perpendicular for the tail of the first biflipper and head of the second biflipper. Draw an arrow along it connecting the flippers which are left.
l n m
SLIDE 166
Head to tail for screws
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Given two screw displacement, present them by biflippers. Find the common perpendicular for the axes of the biflippers. Find common perpendicular for the tail of the first biflipper and head of the second biflipper. Draw an arrow along it connecting the flippers which are left. Erase old arrows and their common flippers.
l n m
SLIDE 167
Head to tail for screws
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Given two screw displacement, present them by biflippers. Find the common perpendicular for the axes of the biflippers. Find common perpendicular for the tail of the first biflipper and head of the second biflipper. Draw an arrow along it connecting the flippers which are left. Erase old arrows and their common flippers.
n m
SLIDE 168
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SLIDE 169
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SLIDE 170
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SLIDE 171
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SLIDE 172
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SLIDE 173
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SLIDE 174
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Thank you for your attention!
SLIDE 175
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Thank you for your attention!
SLIDE 176
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Thank you for your attention!
SLIDE 177
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