Geometric Algebra
Table of Contents
- Vectorspace Geometric Algebra, VGA
1. Definition
- \(\mathrm{Cl}_{n,0}(\mathbb{R})\) with the quadratic form \(g\) satisfies \(g(e_i, e_i) = 1\).
1.1. Grade
- The grade of the graded subalgebra.
1.1.1. Grade Projection
- \(\langle A\rangle_r\): Take the grade \(r\) part.
1.2. k-Vector
- Multivector consisting solely of grade \(k\) multivectors.
1.3. k-Blade
- Exterior product of \(k\) vectors.
2. Structures
2.1. Geometric Product
- The product in Clifford algebra.
- From \[ ab = \frac{1}{2}(ab+ba) + \frac{1}{2}(ab-ba), \]
- It follows that: \[ ab = a\cdot b + a\wedge b \]
2.2. Exterior Product
- Meet
2.2.1. Of Vectors
- \[ a\wedge b := \frac{1}{2}(ab-ba) \]
2.2.2. Canonical Extension
- It is extended to exactly match the structure of the exterior algebra
- It is defined to satisfy for each graded algebra:
\[
\begin{align*} 1\wedge a_i &= a_i\wedge 1 = a_i\\ a_1\wedge a_2\wedge \cdots \wedge a_r &= \frac{1}{r!}\sum_{\sigma\in S_r}\mathrm{sgn}(\sigma) a_{\sigma(1)}a_{\sigma(2)}\cdots a_{a\sigma(r)} \end{align*}\]
- for all vectors \(a_i\).
- And extended by liearity.
- Equivalently, \[ C\wedge D := \sum_{r,s}\langle \langle C\rangle_r\langle D\rangle_s\rangle_{r+s}. \]
2.2.3. Alternative Extensions
- Commutator Product
- \[ C\times D := \frac{1}{2}(CD-DC) \]
2.3. Regressive Product
- Join
- Dual of the exterior product
- \[ C\vee D := (CI^{-1})\wedge (CI^{-1}))I \]
- with \(I\) being the unit pseudoscalar of the algebra.
2.4. Inner Product
2.4.1. Of Vectors
- Vector Inner product \(a\cdot b := g(a,b)\) satisfies
- \[
\frac{1}{2}((a+b)^2 - a^2 - b^2) = a\cdot b
\]
- The .
2.4.2. Extensions
- Left Contraction
- \[ C\mathbin{\rfloor} D := \sum_{r,s}\langle \langle C\rangle_r\langle D\rangle_s\rangle_{s-r} \]
- Identity for any vector \(a\) and multivector \(B\): \[ aB = a\rfloor B + a\wedge B. \]
- Right Contraction
- \[ C\mathbin{\lfloor} D := \sum_{r,s}\langle \langle C\rangle_r\langle D\rangle_s\rangle_{r-s} \]
- Scalar Product
- \[ C\ast D := \sum_{r,s}\langle\langle C\rangle_r\langle D\rangle_s\rangle_0 \]
- See ((6684e628-7453-41e8-837d-0a1a7a60d8d5))
- (Fat) Dot Product
- \[ C\mathbin{\bullet} D := \sum_{r,s}\langle \langle C\rangle_r\langle D\rangle_s\rangle_{|s-r|} \]
2.5. Dual Basis
- Reciprocal Basis, Reciprocal Frame
- If the quadratic form is nondegenerate, then \(V^*\) is naturally identified with \(V\).
- The dual basis can be constructed \[ e^i = (-1)^{i-1}(e_1\cdots \check{e_i}\cdots e_n)I^{-1} \]
- where \(I\) is the pseudoscalar and \(\check{e_i}\) denotes that \(i\)th basis vector is omitted from the product.
- Note that when the change of basis happens the basis and dual basis transforms differently, which makes it a ((655459b9-8584-48ba-82a3-a521a935e24b)).
3. 2-Dimensional
- \(\mathcal{G_2}\)
3.1. Polar Coordinates
- The basis vectors can be represented as
\[
\begin{align*} \hat{\mathbf{r}} &= \mathbf{e}_1e^{i\theta} \\ \hat{\bm{\theta}} &= \mathbf{e}_2e^{i\theta} \end{align*}\]
- where \(i\) is the pseudoscalar \(\mathbf{e_1}\mathbf{e}_2\).
- Note that \(\hat{\mathbf{r}}i = \hat{\bm{\theta}}\).
- Circular velocity and acceleration with geometric algebra - YouTube
4. 3-Dimensional
- \(\mathcal{G}_3\), \(\mathcal{G}(3,0,0)\)
4.1. Triple Vector Product
\[
\begin{align*} a\cdot (b\wedge c) &= a\cdot (ib\times c) = \frac{1}{2}(aib\times c - ib\times ca)\\ &=\frac{i}{2}(ab\times c - b\times c a) = ia\wedge(b\times c)\\ &= -a\times (b\times c) \end{align*}\]
- Note \(ai = ia\).
\[
\begin{align*} (a\wedge b)\cdot c &= \frac{1}{2}(ia\times b c - cia\times b)\\ &= i (a\times b)\wedge c = -(a\times b)\times c \end{align*}\]
5. Dimensionality
- This happens because Geometric algebra is doing algebra on geometry.
- Geometric Dimension
- Algebraic Dimension