Questions tagged [geometric-algebras]
Geometric algebras are Clifford algebras over the real numbers. They are applied in geometry and theoretical physics.
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Do unit bivectors square to -1 in vector spaces of any dimension?
We know that in a 2D vector space (VS) the unit bivector squares to -1. My question is:
Is the geometric product of a unit bivector with itself equal to -1 in any VS, independently of the VS dimension?...
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Can a vector have a wedge product with a scalar? And a geometric product?
One of the necessary requirements of a vector space is to be endowed with the multiplication by a scalar. I am now interested in other operations, the wedge and the geometric product in geometric ...
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Proof of bivector multiplication with reciprocal frame vector from Doran, Lasenby?
Doran and Lasenby (Geometric Algebra for Physicists) introduce the reciprocal frame vector and make the below assertion about multiplication with arbitrary bivectors (page 102, eq 4.104):
$e_i e^i \...
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A proof for bivector decomposition
Let $V$ be an $n$-dimensional vector space over some field $\mathbb F$. I'm interested in the following result:
For every bivector $\alpha\in\bigwedge^2 V$, there exists a linearly independent set $S=...
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Is Clifford product only for scalars already defined?
hi i'm a high school student interested in Clifford algebra. Since it is hard for me to understand several things thus i hope to know as follows:
Is Clifford product only for scalars already defined?
...
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Intersection of *segments* in 3D using projective geometric algebra?
There is a relatively common technique to find the intersection of two segments in 3D, which can be found in page 304 of Graphics Gems
Note that a segment is a compact subset of a line.
The technique ...
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How can one generalize the split-complex algebra to a coordinate space with an arbitrary number of asymptotes?
Given that split-complex numbers generates 2D coordinates with 4 asymptotes (when multiplying) that look as follows:
The arrows along the hyperbola indicate a positive direction for boosting points ...
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Orthogonal transformations and reflections
In the context of geometric algebra, why is it that any orthogonal transformation can be expressed as successive reflections? (Both, a mathematical and an intuitive explanation would be great)
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Representation of angular momentum as bivector
I read about derivation of angular velocity and angular momentum here - https://peeterjoot.com/archives/geometric-algebra/angular_acc.pdf
Peeter Joot wrote: "writing the angular momentum as ...&...
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Inequality involving dot products of sphere points
Given $p_1,p_2,p_3 \in S^2 = \{x \in \mathbb{R}^3 \mid \lVert x \rVert = 1\}$ all distinct, I wish to prove that
\begin{align*}
J &:=(p_1p_2 - 1)^2 + (p_1p_3 - 1)^2 + (p_2p_3 - 1)^2\\
&\qquad -...
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Why does the wedge product produce a scalar and not a graded element?
Reading the definition of the geometric product we get that it satisfies:
$$ a_1 \wedge \cdots \wedge a_r = \frac{1}{r!} \sum_{\sigma \in G_r} \mathrm{sgn}(\sigma)a_{\sigma(1)}\cdots a_{\sigma(r)} $$
...
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Solving for the rotor in quaternion rotation
If we have 2 vectors $v_1,v_2$ which have been rotated into $v'_1,v'_2$ by the following operations:
$v'_1 = e^{θ\hat{n}/2}v_1e^{-θ\hat{n}/2}$
$v'_2 = e^{θ\hat{n}/2}v_2e^{-θ\hat{n}/2}$
Where $\hat{n}$ ...
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Possible error in Doran’s “Geometric Algebra for Physicists”
On page 40 of Geometric Algebra for Physicists, Doran states in equation 2.98 that for a unit vector $n$ and an arbitrary vector $a$ with $a_\perp = n n\wedge a$, that
$$n\cdot a_\perp=\langle n n \...
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How do units work for multivectors? (Clifford/Geometric algebra) [closed]
Suppose we work with $\mathcal{G}(2)$. If the units of $e_{1}$ and $e_{2}$ are of length, it makes sense that the units of $e_{12}$ would be of area. But then what are the units of scalars? And what ...
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Outer product for r vectors in terms of the geometric product
I found a definition in (An introduction to geometric algebra and calculus by Alan Bromborsky) which say
let $a_1,a_2,...,a_r$ be vectors in $\mathbb{R}^n$ and let $\varepsilon_{1...r}^{i_1...i_r}$ be ...
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For what types of objects is the outer product defined in geometric algebra?
I just started to learn geometric algebra from the "Geometric Algebra for Physicists" book. Authors first give definition of outer product of two vectors $a \wedge b$. Then they give ...
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Solving a equation with a 2-form/bivector differential equation
I was thinking about a problem and a bivector differential equation appeared, not sure how to start solving the issue.
I want to find parameterization of the n-sphere, we know the implicit equation of ...
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Taking the geometric derivative of $e^{\frac{1}{2}\left[\gamma_0 \vec{k} \vec{x} \gamma^0 - \vec{x} \gamma^0 \gamma_0 \vec{k}\right]}$
As in the title: I'm looking to take the geometric derivative with respect to $\vec{x}$ of the exponential of the commutator of $\gamma_0 \vec{k}$ and $\vec{x} \gamma^0$:
$$e^{\frac{1}{2}\left[\...
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Do all bivectors simplify to 2-blades in seven-dimensional space?
The wedge product of two vectors $\vec{v}, \vec{w}\in\mathbb{R}^{n}$ can be defined as an anti-symmetrized tensor product. In three dimensions, there is a correspondence between the wedge product of ...
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Geometric product of 2 r-blades
In some lecture notes the geometric product of a $r$-blade $A_r$ and an $s$-blade $B_s$ is given as:
$A_r B_s = \langle A_r B_s\rangle_{|r-s|} + \langle A_r B_s\rangle_{|r-s|+2} +... + \langle A_r B_s\...
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For an even multivector $A$, if the map $X\mapsto AX\tilde A$ preserves grade, must $A$ be a product of vectors?
We're working in a Clifford algebra over a non-degenerate $n$-dimensional vector space $V$, and considering various properties a multivector $A$ could have:
(0) $A$ is invertible.
(1) $\tilde AA$ is a ...
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Is there a field with infinite elements and $2^k$-th roots of unity as integers or integer vectors?
Let $F$ be a field with infinite elements such that for any integer $k$, there exists a $2^k$-th root of unity, denoted by $N$, with the property that $N$ is either an integer or can be represented as ...
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How do you show that $\textsf f(e_1) \wedge \textsf f(e_2) I^{-1} = \det (\textsf f) \bar{\textsf f}^{-1} (e_3)$ in geometric algebra in 3D?
I would like to show that
$\textsf f(e_1) \wedge \textsf f(e_2) I^{-1} = \det (\textsf f) \bar{\textsf f}^{-1} (e_3)$
in geometric algebra in 3 dimensions, as in Geometric Algebra for Physicists (...
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Can Fast Fourier Transform (FFT) be implemented using Clifford Algebra over GF(3)?
Background. Fast Fourier Transform (FFT) is an algorithm used to quickly calculate the discrete Fourier transform (DFT) of a sequence. It is widely used in signal processing, image analysis, and data ...
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How do you prove this determinant-related identity in geometric algebra?
I would like to show that
$$
\begin{align}
F_{\alpha i} F_{\beta j} \cdots F_{\gamma k} e_\alpha \wedge e_\beta \cdots \wedge e_\gamma I^\dagger
&= e_{\alpha} \cdot F(e_i) e_\beta \cdot F(e_j) \...
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Geometric algebra sweeping surface/change of basis
Someone has asked a similar question before but the answer was not that useful.
I want to make a sweeping surface with PGA, i.e. to transport a curve along a parametric curve.
With linear algebra one ...
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In Geometric Algebra, does $A\tilde A=0$ imply $\tilde AA=0$?
Reversion in Geometric Algebra is defined by the property $(AB)^\sim=\tilde B\tilde A$ (for any two multivectors $A$ and $B$), along with linearity, and that it leaves scalars and vectors unchanged. ...
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Help with intuitively understanding why the space of bivectors from 4D Euclidean space is 6 dimensional
The wedge product of two 4D Euclidean space vectors is of dimension 6. I can understand mathematically that the six basis vectors are linearly independent, from the math itself. But I fail to see ...
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How to extract weight, direction and support point from a flat in Conformal Geometric Algebra? [closed]
I've been studying Geometric Algebra and I'm specially interested in the Conformal Model. The main reference I'm using is Leo Dorst (Geometric Algebra for Computer Science).
It's not clear to me how I ...
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Can we often only consider homogeneous elements of exterior or tensor algebras because their products preserve homogeneity?
Background: I'm just an innocent physicist with very little formal training in algebra. So I think of things in very naive, concrete (as opposed to abstract), and non-rigorous ways. I probably won't ...
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Does a vector space need a quadratic form in order to define a wedge product?
Geometric (Clifford) algebras require the vector space to be endowed with a quadratic form in order to define the geometric product.
Meanwhile, an exterior algebra has the wedge product as its ...
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Identifying if 2 motors are "compatible." (no candy wrapping)
Preamble (Trying to describe candywrapping)
Consider a translation and rotation encoded as a motor in $Cl(3, 0, 1)$, also known as PGA.
All rotations in 3D can be considered to be a planar rotation up ...
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Converting quaternion to motor causes candy wrapping
I have an animated model that I am trying to animate using PGA motors.
The animation mostly works but I am noticing that some of my motors seem to have an angle that is exactly 180 degrees off from ...
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Is Projective Geometric Algebra a strict subset of Conformal Geometric Algebra?
My understanding is that CGA has an additional basis vector $e_{\infty}$, so it would seem to me that if that is the only difference then PGA fits entirely within CGA.
However I don't know if the ...
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How do you show that the projection operator for a diffeomorphism is $\textsf P^\prime = \textsf {fP}\textsf{f}^{-1}?$
The pseudoscalar $I(x)$ defines a projection operator that projects an arbitrary multivector onto the component that is intrinsic to the manifold,
\begin{equation}
\textsf P(A(x)) =
\begin{cases}
A_r(...
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What is the relationship between Clifford algebra and differential forms?
Counting the basis vectors, Clifford algebra has 1 scalar, 4 vectors, 6 bivectors, 4 trivectors, 1 pseudoscalar
Differential forms have 1 scalar, 4 one-forms, 6 two-forms, 4 three-forms, 1 4-form (the ...
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Find an orthogonal vector with a specific property in a 2n-dimensional Euclidean space
Suppose $\mathbb{v}_1\in\mathbb{R}^{2n}$ and $\mathbb{v}_2\in\mathbb{R}^{2n}$ are two vectors. Consider a hyperplane $P$ in $\mathbb{R}^{2n}$ space that is perpendicular to $(\mathbf{v_1}-\mathbf{v_2})...
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Adding blades of different grades (computationally)
I ask a previous question which got a really thorough answer.
On the comments it is mentioned that in a geometric algebra you can add blades of different grades.
The goal for me is to understand how ...
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How do you show that this wedge derivative is equal to a certain derivative, $\dot\partial \wedge P(\dot{a}(x)) = P(\partial \wedge a)$?
I would like to show that $\dot\partial \wedge P(\dot{a}) = P(\partial \wedge a)$, where $P$ is a linear function $P(a(x)) = a(x)\cdot I(x)I^{-1}(x)$. The dot is used to indicate that the derivative ...
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Gauge Theory Gravity: transformation of vector potential $A'(x)$
In Gravity, Gauge Theories and Geometric Algebra by A. Lasenby, C. Doran and S. Gull develop a theory for gravity based on the use of a position-gauge field and a rotation-gauge field. The former is ...
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Point at infinity in geometric algebra
In Homogeneous Coordinates, the point at infinity is represented by a vector pointing to "the horizon".
Is there an equivalent representation of the Point at infinity in geometric algebra (...
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Is the definition $a\cdot b= 0.5(ab+ba)$ in geometric algebra justified, or is it mostly arbitrary?
According to one of the basic axioms of geometric algebra, the square of a vector with itself is a scalar. For two vectors $a$ and $b$, this results in $ab+ba = (a+b)^2-a^2-b^2$. Therefore $ab+ba$ ...
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Geometric Algebra: How to prove that the Grade Projection Operator is well defined
The Grade Projection Operator $<\cdot>_r$ is widely used in Geometric Algebra to prove numerous relations and results. I'm looking for a proof that Grade Projection is a well-defined operation.
...
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What would an infinite dimensional Clifford algebra look like in terms of its generators?
In Infinite dimensional Clifford algebras?, the answers spoke on how to construct a Clifford algebra in infinite dimensions. What I want to know is: what does $Cl^{p,q,r}(\Bbb R)$ look like when any ...
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Apparent inconsistency in geometric product associativity
With a little bit of work, I have proven to myself that the geometric product between three vectors is associative ($a$, $b$, and $c$ are 1-vectors):
$$\begin{aligned}(ab)c &= a(bc) \\ &= (b \...
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Am I interpreting the wedge product correctly?
On "The Inner Products of Geometric Algebra" (Leo Dorst), page 39, equation 2.3, the outer product between multivector $A_r$ and $B_s$, of grades $r$, $s$, is defined as:
$$A_r \wedge B_s = \...
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Is the Inner Product on an Exterior Algebra Invariant under Rotors?
Take two k-vectors $x = \bigwedge_{i = 1}^k \vec{x}_i$, $y = \bigwedge_{i = 1}^k \vec{y}_i$ in $\Lambda^k(\mathbb{R}^n)$. Define the inner product between them as:
$$ \langle x, y \rangle = \det(\...
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Missing steps+intuition for geometric algebra manipulations
I'm working my way through Doran and Lasenby's Geometric Algebra for Physicists, but I've run into some trouble in chapter three with a couple of derivations where I can't follow the steps.
The first ...
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Clifford Algebra for 2D spaces
I am studying Clifford and geometric algebra. Specifically, decomposing the line element by finding the matrices that satisfy the necessary algebraic relations. I know that for 4D spacetime, the ...
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How do I take the divergence of a multi-vector?
Say I have a multi-vector of $\mathbb{G}(2,\mathbb{R})$:
$$
\mathbf{u}=a+xe_0+ye_1+be_0e_1
$$
How do I take the divergence? How do I even define it for multi-vector in general?
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