# geometric algebra

notes, definitions, and references for geometric algebra (GA), including conformal geometric algebra (CGA).
see also the software implementation [sph-ga](https://github.com/sph-mn/sph-ga).

# overview
geometric algebra unifies inner, exterior, and geometric products into a single algebraic system. it supports scalars, vectors, and higher-grade elements as first-class entities.
applications: geometry, physics, computer graphics, robotics, computational algebra.

# resources

## introductory
* [from zero to geo](https://www.youtube.com/watch?v=2hBWCCAiCzQ\&list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy)
* [geometric algebra for computer science](http://www.geometricalgebra.net/)  -  explanations and software considerations
* [mathoma ga series](https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K)
* [wild linear algebra](https://www.youtube.com/watch?v=yAb12PWrhV0\&list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp)  -  background linear algebra for GA

## tools
* [versor.js](https://github.com/weshoke/versor.js)  -  javascript port of libvsr
* [gaviewer](http://www.geometricalgebra.net/gaviewer_download.html) and [gaviewer tutorial](https://geometricalgebra.org/downloads/gaviewerexercises.pdf)
* [sph-ga](https://github.com/sph-mn/sph-ga)
* [libvsr](https://github.com/wolftype/versor)  -  c++ template-based implementation (fast, fixed-metric)

## reference lists
* [ga explorer resources](https://ga-explorer.netlify.com/index.php/ga-online-resources)
* [bleyer.org ga resources](http://bleyer.org/dw/doku.php?id=geometric_algebra)
* [clifford algebra: a visual introduction](https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction)
* [clifford/geometric algebra on euclideanspace](http://www.euclideanspace.com/maths/algebra/clifford/index.htm)
* [colapinto  -  articulating space](http://versor.mat.ucsb.edu/ArticulatingSpace.pdf)
* [lecture slides by chris doran](http://geometry.mrao.cam.ac.uk/2016/10/geometric-algebra-2016) (may be offline)
* [reference implementation](http://www.geometricalgebra.net/reference_impl.html) from "geometric algebra for computer science"
* [suter  -  geometric algebra primer](http://www.jaapsuter.com/geometric-algebra)
* [pablo colapinto talk from c++ now 2014](https://youtu.be/W4p-e-g37tg)

# terminology
* scalars: grade-0 elements (real numbers)
* vectors: grade-1 elements (basis: e₁, e₂, ...)
* basis vectors: linearly independent set spanning the space
* null vectors: vectors with self-inner product zero
* blades: simple multivectors from exterior product of independent vectors
* bivectors: grade-2 blades (oriented planes)
* trivectors: grade-3 blades (oriented volumes)
* multivectors: general linear combination of grades
* pseudoscalar: highest-grade blade
* spinors: multivectors representing transformations (rotations, reflections, dilations)
* dual: multivector's complement w.r.t. pseudoscalar
* grades: homogeneous components by dimension
* conformal points: null vectors embedding Euclidean points
* directions: null vectors for points at infinity
* rounds: spheres/circles in CGA
* flats: lines/hyperplanes in CGA
* tangent vectors: null vectors tangent to rounds

# products
* geometric product: associative combination of inner and exterior products
* exterior product (wedge): antisymmetric, raises grade
* inner product (dot): reduces grade, encodes projections
* commutator product: `[A, B] = AB - BA`
* anticommutator product: `{A, B} = AB + BA`

# main operations
* reversion: reverse vector order in product
* grade involution: negate odd-grade parts
* dualization: multiply by pseudoscalar
* grade projection: extract specified grade(s)
* norm: magnitude from geometric product
* sandwich product: `A -> R A R⁻¹`
* meet: intersection via dualization
* join: union via exterior product
* reflection: `v -> -R v R⁻¹`
* rotation: spinor from bivector exponentials
* inversion: inversion in sphere/plane
* translation: spinor with e∞
* dilation: spinor scaling transformation
* exponential map: bivector -> rotation/dilation
* logarithm map: spinor -> bivector
* orthogonal complement: dual subspace
* null decomposition: split null and non-null components

# conformal geometric algebra (cga)

## metric
split-signature metric with two off-diagonal negatives:
```
[1, 1, 1, 0, 0]
```
basis products:
```
no * no = 0
ni * ni = 0
no * ni = -1
eᵢ * eᵢ = 1
eᵢ * eⱼ = 0 (i ≠ j)
```

## point embedding
for euclidean n-space:
```
p = x₁ e₁ + x₂ e₂ + ... + xₙ eₙ + e₀ + 0.5(x₁² + ... + xₙ²) e∞
```
null condition:
```
dot(p, p) = 0
```

## common rotors
rotation:
```
R = cos(angle / 2) + B sin(angle / 2)     # B is unit bivector
```
translation:
```
T = 1 + 0.5 * t * e∞
```
rotation + translation:
```
R * T
```

## rotor application
```
p' = R * p * reverse(R)
```

## perspective projection
see:
[computing perspective projections in 3-dimensions using rotors in the homogeneous and conformal models of clifford algebra](https://www.researchgate.net/publication/266979004_Computing_Perspective_Projections_in_3-Dimensions_Using_Rotors_in_the_Homogeneous_and_Conformal_Models_of_Clifford_Algebra)

# implementation notes
* pseudoscalar scaling factors vary; some use -1/2.
* geometric product of pseudoscalar with itself = 1 in conformal metric (without scaling).
* sign conventions differ between libraries; may flip all results.
* product of pseudoscalar with itself can reveal null vector handling bugs.