Steven De Keninck PRO
Mathematical Experimentalist
Graded Symmetry Groups
Plane and Simple.
the universally accepted unwritten axiom
AGACSE2021
Steven De Keninck (enkimute)
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https://slides.com/d/0A4Ehfc/live
AGACSE2021
On Lie groups defining automorphisms that leave invariant fundamental subspaces of geometric algebra.
Dmitry Shirokov
Graded Symmetry Groups
Plane and Simple.
the universally accepted unwritten axiom
AGACSE2021
L’application des mêmes idées de dualité peut s’étendre à la Mécanique. En effet, l’élément primitif des corps auquel on applique d’abord les premiers principes de cette science, est, comme dans la Géométrie ancienne, le point mathématique. Ne sommes-nous pas autorisés à penser, maintenant, qu’en prenant le plan pour l’élément de l’étendue, et non plus le point, on sera conduit à d’autres doctrines, faisant pour ainsi dire une nouvelle science?
- Michel Chasles, 1875
Graded Symmetry Groups
Plane and Simple.
the universally accepted unwritten axiom
AGACSE2021
The application of the same ideas of duality can be extended to Mechanics. Indeed, the primitive element of bodies to which the first principles of this science are applied is, as in ancient Geometry, the mathematical point. Are we not permitted to think, now, that by taking the plane for the basic element, and no longer the point, we shall be led to other doctrines, making, as it were, a new science?
- Michel Chasles, 1875
Graded Symmetry Groups
Plane and Simple.
the universally accepted unwritten axiom
AGACSE2021
The application of the same ideas of duality can be extended to Mechanics. Indeed, the primitive element of bodies to which the first principles of this science are applied is, as in ancient Geometry, the mathematical point. Are we not permitted to think, now, that by taking the plane for the basic element, and no longer the point, we shall be led to other doctrines, making, as it were, a new science?
- Michel Chasles, 1875
1. Symmetry Groups
2. Geometry
3. Rigid Body Dynamics
- classic approach
- k-reflections
- geometric gauges
- Mozzi-Chasles
- coordinate agnostic
- dimension agnostic
- metric agnostic
- Forques
- Demo
Graded Symmetry Groups
Plane and Simple.
the universally accepted unwritten axiom
AGACSE2021
The application of the same ideas of duality can be extended to Mechanics. Indeed, the primitive element of bodies to which the first principles of this science are applied is, as in ancient Geometry, the mathematical point. Are we not permitted to think, now, that by taking the plane for the basic element, and no longer the point, we shall be led to other doctrines, making, as it were, a new science?
- Michel Chasles, 1875
1. Symmetry Groups
2. Geometry
3. Rigid Body Dynamics
- classic approach
- k-reflections
- geometric gauges
- Mozi-Chasles
- coordinate agnostic
- dimension agnostic
- metric agnostic
- Forques
- Demo
All our examples will be using the Euclidean groups in 2 and 3 dimensions.
Why?
- these are non-simple Lie groups
- the natural choice for intuition
Graded Symmetry Groups
Plane and Simple.
the universally accepted unwritten axiom
AGACSE2021
The application of the same ideas of duality can be extended to Mechanics. Indeed, the primitive element of bodies to which the first principles of this science are applied is, as in ancient Geometry, the mathematical point. Are we not permitted to think, now, that by taking the plane for the basic element, and no longer the point, we shall be led to other doctrines, making, as it were, a new science?
- Michel Chasles, 1875
1. Symmetry Groups
2. Geometry
3. Rigid Body Dynamics
- classic approach
- k-reflections
- geometric gauges
- Mozi-Chasles
- coordinate agnostic
- dimension agnostic
- metric agnostic
- Forques
- Demo
All our examples will be using the Euclidean groups in 2 and 3 dimensions.
Why?
- these are non-simple Lie groups
- the natural choice for intuition
a vector space
transformations on a vector space
\begin{bmatrix}
x_1 \\ \vdots \\ x_n
\end{bmatrix}
\begin{bmatrix}
m_{11} & \dots & m_{1n} \\ \vdots & & \vdots \\ m_{n1} & \dots & m_{nn}
\end{bmatrix}
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
4x4 matrix
6x6 matrix
4x4 matrix
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups
a vector space
transformations on a vector space
\begin{bmatrix}
x_1 \\ \vdots \\ x_n
\end{bmatrix}
\begin{bmatrix}
x_{11} & \dots & x_{1n} \\ \vdots & & \vdots \\ x_{n1} & \dots & x_{nn}
\end{bmatrix}
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
4x4 matrix
6x6 matrix
4x4 matrix
In this model, perfectly reasonable starting point.
Also valid, but why would we?
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups
a vector space
transformations on a vector space
\begin{bmatrix}
x_1 \\ \vdots \\ x_n
\end{bmatrix}
\begin{bmatrix}
x_{11} & \dots & x_{1n} \\ \vdots & & \vdots \\ x_{n1} & \dots & x_{nn}
\end{bmatrix}
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
4x4 matrix
6x6 matrix
4x4 matrix
a graded vector space
x \in \mathbb R_{p,q,r}
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
versors/rotors
R \in \mathbb R_{p,q,r}^+
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups
a vector space
transformations on a vector space
\begin{bmatrix}
x_1 \\ \vdots \\ x_n
\end{bmatrix}
\begin{bmatrix}
x_{11} & \dots & x_{1n} \\ \vdots & & \vdots \\ x_{n1} & \dots & x_{nn}
\end{bmatrix}
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
4x4 matrix
6x6 matrix
4x4 matrix
a graded vector space
x \in \mathbb R_{p,q,r}
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
versors/rotors
R \in \mathbb R_{p,q,r}^+
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | k-reflections
a graded vector space
x \in \mathbb R_{p,q,r}
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
versors/rotors
R \in \mathbb R_{p,q,r}^+
start from Hamilton's observation that all isometries can be generated by composing reflections. Then stick with it.
\(E(2)\) - The Euclidean Group of the plane
| # reflections | Isometry |
|---|---|
| 0 | identity |
| 1 | reflection |
| 2 | rotation/translation |
| 3 | glide reflection |
| 4 ? | → 2 |
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | k-reflections
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
start from Hamilton's observation that all isometries can be generated by composing reflections. Then stick with it.
\(E(2)\) - The Euclidean Group of the plane
| # reflections | Isometry |
|---|---|
| 0 | identity |
| 1 | reflection |
| 2 | rotation/translation |
| 3 | glide reflection |
| 4 ? | → 2 |
| Invariant |
|---|
| plane |
| line |
| point / ∞point |
| line + ∞point |
| → 2 |
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | k-reflections
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
start from Hamilton's observation that all isometries can be generated by composing reflections. Then stick with it.
\(E(2)\) - The Euclidean Group of the plane
| # reflections | Isometry |
|---|---|
| 0 | identity |
| 1 | reflection |
| 2 | rotation/translation |
| 3 | glide reflection |
| Invariant |
|---|
| plane |
| line |
| point / ∞point |
| line + ∞point |
| GA |
|---|
| scalar |
| vector |
| bivector |
| trivector |
Using Chasles' hyperplanes as vectors in a GA gives us exactly the structure we need.
the reflections give us everything.
Geometric Algebra Sandwich
\( -a b a^{-1}\)
Group Theory Conjugation
\( a b a^{-1}\)
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | k-reflections
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
\(E(2)\) - The Euclidean Group of the plane
| # reflections | Isometry |
|---|---|
| 0 | identity |
| 1 | reflection |
| 2 | rotation/translation |
| 3 | glide reflection |
| Invariant |
|---|
| plane |
| line |
| point / ∞point |
| line + ∞point |
Geometric Algebra Sandwich
\( -a b a^{-1}\)
| GA |
|---|
| scalar |
| vector |
| bivector |
| trivector |
Using Chasles' hyperplanes as vectors in a GA gives us exactly the structure we need.
the reflections give us everything.
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | geometric gauges
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | geometric gauges
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | geometric gauges
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozzi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozzi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
The Mozzi-Chasles Theorem :
Every rigid body motion can be decomposed into a rotation around followed or preceded by a a translation along a single line.
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
The Mozzi-Chasles Theorem :
Every rigid body motion can be decomposed into a rotation around followed or preceded by a a translation along a single line.
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
The Mozzi-Chasles Theorem :
Every quadreflection can be decomposed into a rotation around followed or preceded by a a translation along a single line.
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
The Mozzi-Chasles Theorem :
Every quadreflection can be decomposed into a commuting rotation and translation
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
The Mozzi-Chasles Theorem :
Every quadreflection can be decomposed into commuting bireflections
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
Our final theorem :
Every k-reflection can be decomposed into \(\lceil \frac k 2 \rceil\) commuting parts.
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
Our final theorem :
Every k-reflection can be decomposed into \(\lceil \frac k 2 \rceil\) commuting parts.
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
Our final version :
Every k-reflection can be decomposed into \(\lceil \frac k 2 \rceil\) commuting parts.
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
Martin's talk follows up on this and goes into all the algebraic details!
Today, lets envision how this could make a difference in applications and intuition.
2. Geometry | coordinate, dimension and metric agnostic
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
Our final version :
Every k-reflection can be decomposed into \(\lceil \frac k 2 \rceil\) commuting parts.
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
Martin's talk follows up on this and goes into all the algebraic details!
Today, lets envision how this could make a difference in applications and intuition.
2. Geometry | coordinate, dimension and metric agnostic
this is not a vector.
ceci n'est pas un vecteur
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
Martin's talk follows up on this and goes into all the algebraic details!
Today, lets envision how this could make a difference in applications and intuition.
2. Geometry | coordinate, dimension and metric agnostic
this is a vector.
ceci n'est pas un vecteur
how many dimensions ?
does it have an orientation ?
Graded Symmetry Groups
AGACSE2021
1. Symmetry Groups | Mozi-Chasles Theorem
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
Martin's talk follows up on this and goes into all the algebraic details!
Today, lets envision how this could make a difference in applications and intuition.
2. Geometry | coordinate, dimension and metric agnostic
this is a vector.
ceci n'est pas un vecteur
\( x^2 = \lVert x \rVert^2 \)
only its length is well defined.
Graded Symmetry Groups
AGACSE2021
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
2. Geometry | coordinate, dimension and metric agnostic
this is a vector.
\( x^2 = \lVert x \rVert^2 \)
When a second non colinear vector is added
- we can measure two lengths
- we can measure one angle
- we know our space is >= 2D
Graded Symmetry Groups
AGACSE2021
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
2. Geometry | coordinate, dimension and metric agnostic
this is a vector.
\( x^2 = \lVert x \rVert^2 \)
When a second non colinear vector is added
- we can measure two lengths
- we can measure one angle
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Graded Symmetry Groups
AGACSE2021
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
2. Geometry | coordinate, dimension and metric agnostic
this is a vector.
\( x^2 = \lVert x \rVert^2 \)
When a second non colinear vector is added
- we can measure two lengths
- we can measure one angle
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.
Graded Symmetry Groups
AGACSE2021
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
2. Geometry | coordinate, dimension and metric agnostic
this is a vector.
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.
This works for spaces of arbitrary (constant) curvature.
Graded Symmetry Groups
AGACSE2021
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
2. Geometry | coordinate, dimension and metric agnostic
this is a vector.
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.
This works for spaces of arbitrary (constant) curvature.
Graded Symmetry Groups
AGACSE2021
a 4D homogeneous point
a 6D Plucker line
a 4D homogeneous plane
any \(k\), even and odd
any # dimensions,
any metric
closed form exp and log. \(e^{a+b} = e^ae^b\) iff \(ab = ba\)
2. Geometry | coordinate, dimension and metric agnostic
this is a vector.
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.
This works for spaces of arbitrary (constant) curvature.
Graded Symmetry Groups
AGACSE2021
2. Geometry | coordinate, dimension and metric agnostic
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.
This works for spaces of arbitrary (constant) curvature.
Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.
Graded Symmetry Groups
AGACSE2021
2. Geometry | coordinate, dimension and metric agnostic
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.
However, formulating a problem independent of the dimensionality of the space does require some extra care...
Graded Symmetry Groups
AGACSE2021
2. Geometry | coordinate, dimension and metric agnostic
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.
However, formulating a problem independent of the dimensionality of the space does require some extra care...
robot arm
floor plan
robot arm
Both 2D drawings have points and lines. What happens if we go from two to three dimensions?
Graded Symmetry Groups
AGACSE2021
2. Geometry | coordinate, dimension and metric agnostic
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.
However, formulating a problem independent of the dimensionality of the space does require some extra care...
robot arm
floor plan
robot arm
Both 2D drawings have points and lines. What happens if we go from two to three dimensions?
When going to 3D, the lines on the left stay lines. Those on the right become planes .. working dimension independent means embracing duality!
Graded Symmetry Groups
AGACSE2021
2. Geometry | coordinate, dimension and metric agnostic
- we know our space is >= 2D
We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'
Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.
However, formulating a problem independent of the dimensionality of the space does require some extra care...
robot arm
floor plan
robot arm
When going to 3D, the lines on the left stay lines. Those on the right become planes .. working dimension independent means embracing duality!
points and lines defined dually
points and lines defined directly
Graded Symmetry Groups
AGACSE2021
2. Geometry | Forques !
However, formulating a problem independent of the dimensionality of the space does require some extra care...
robot arm
floor plan
robot arm
When going to 3D, the lines on the left stay lines. Those on the right become planes .. working dimension independent means embracing duality!
points and lines defined dually
points and lines defined directly
Graded Symmetry Groups
AGACSE2021
2. Geometry | Forques !
In the body frame :
push at origin = translation
push at infinity = rotation
Force and Torque are both lines.
in any # dimensions.
Graded Symmetry Groups
AGACSE2021
2. Geometry | Forques !
In the body frame :
push at origin = translation
push at infinity = rotation
Force and Torque are both lines.
in any # dimensions.
nobody in their right mind does live programming during a talk.
Unfortunately for you all, that leaves me qualified.
Graded Symmetry Groups
By Steven De Keninck
Graded Symmetry Groups
- 672
