a collection of links and notes of varying quality.

see also the straightforward software implementation sph-ga.

• youtube: from zero to geo
• dorst, fontijne, mann - geometric algebra for computer science explanations and consideration for software
• versor.js, javascript port of libvsr
• wikipedia: conformal geometric algebra
• wikipedia: geometric algebra

additional links

• youtube

  • geometric algebra video series by mathoma
  • wild linear algebra youtube video series by n j wildberger about linear algebra, which includes some of the fundamental knowledge for understanding geometric algebra
• gaviewer for visualizations
• gaviewer tutorial
• bleyer.org list of resources
• clifford algebra: a visual introduction
• clifford/geometric algebra on euclideanspace.com introduction to elements and operations
• colapinto - articulating space dissertation. particularly for context
• ga-explorer list of resources
• lecture slides by dr chris doran website went down while posting this link
• libvsr c++ template based implementation, which means algebras for different metrics are pre-compiled and fast but perhaps can not easily be run-time created. some additional usage information may be found in the text "articulating space"
• pablo colapintos talk from c++ now 2014
• reference implementation from the book "geometric algebra for computer science"
• suter - geometric algebra primer introduction to the elements and operations. mentions some important details that are often assumed

• the geometric and inner product of the pseudoscalar with itself can quickly uncover flaws in null vector handling
• the sign can vary depending on the specific metric one uses. other libraries or references may use different conventions which flips all signs compared to your implementation
• in cga references, one can often see a scaling factor applied to the pseudoscalar "pseudoscalar = scaling_factor * (e1_2_3_0_∞)". sometimes this scaling factor is chosen to be "-1/2". the geometric product of the pseudoscalar with itself, mechanically calculated, is 1 using the conformal metric. only with the scaling factor is it sometimes modified

the conformal metric is a split-signature metric with two off-diagonal negative terms.

[1, 1, 1, 0, 0]

it aligns with the use of the no and ni null vector bases.

no * no = 0
ni * ni = 0
no * ni = -1
e_i * e_i = 1
e_i * e_j (where i != j) = 0

• scalars: real-number elements of the algebra
• vectors: elements of a vector space, forming the grade-1 components
• basis vectors: linearly independent vectors spanning the space
• null vectors: special basis vectors whose self-inner product vanishes, adding conformal or additional structure
• blades: simple multivectors obtained as the exterior product of linearly independent vectors
• bivectors: grade-2 elements representing oriented plane segments
• trivectors: grade-3 elements representing oriented volume segments
• multivectors: general elements of the algebra, linear combinations of scalars, vectors, bivectors, trivectors, and higher-grade elements
• pseudoscalar: the highest-grade element in the algebra, encoding oriented n-volumes (where n is the space dimension)
• spinors: multivectors representing transformations, such as rotations, reflections, and dilations
• null space: subspace spanned by null vectors
• dual: a multivector's complement with respect to the pseudoscalar
• grades: homogeneous components of a multivector corresponding to its dimensionality (e.g., scalars are grade-0, vectors grade-1)
• conformal points: vectors representing positions in conformal space via null vector embedding
• directions: null vectors that represent points at infinity
• rounds: elements describing spheres or circles in conformal geometric algebra
• flats: elements representing hyperplanes or lines in the conformal framework
• tangent vectors: null vectors encoding directions tangent to a conformal round

• exterior product (wedge product, outer product): an associative, antisymmetric product producing higher-grade blades
• inner product (dot product): reduces the grade of multivectors, encoding projections and magnitudes
• geometric product: the associative fundamental product combining the inner and exterior products, unifying algebraic operations
• reversion: reversing the order of vectors in a product, generalizing conjugation
• grade involution: negates all odd-grade components of a multivector
• dualization: mapping a multivector to its dual via multiplication with the pseudoscalar
• grade projection: extracting components of a multivector corresponding to specific grades
• norm: measuring the magnitude of a multivector using its geometric product
• sandwich product: encoding transformations (e.g., rotations) via spinor conjugation: A -> R A R⁻¹
• meet: the intersection of two geometric objects, derived via dualization
• join: the union of two geometric objects, derived via the exterior product
• reflection: mapping a vector across a hyperplane using a spinor: v -> -RvR⁻¹
• rotation: generalizing reflections using spinors to perform rotations
• inversion: mapping points or vectors relative to spheres in conformal space
• translation: embedding translations as specific spinor operations in conformal space
• dilation: scaling transformations represented by spinors
• exponential map: constructing rotations and dilations via exponentiation of bivectors
• logarithm map: extracting bivectors from spinors encoding rotations or dilations
• commutator product: measuring the "twist" between two elements, defined as [A, B] = AB - BA
• anticommutator product: symmetric product measuring alignment, defined as {A, B} = AB + BA
• orthogonal complement: constructing orthogonal spaces or duals relative to a flat or round
• null decomposition: separating null components of a vector or multivector for conformal analysis

in conformal geometric algebra (cga), a point in euclidean space is represented as a null vector in a higher-dimensional space. specifically, for an n-dimensional euclidean space, cga embeds this space into an (n + 2)-dimensional space with a signature (n + 1, 1). the point p with coordinates (x1, x2, ..., xn) is represented as:

p = x1 * e1 + x2 * e2 + ... + xn * en + e0 + 0.5 * (x1 ** 2 + x2 ** 2 + ... + xn ** 2) * e∞

where 'e1, e2, ..., en' are the basis vectors corresponding to the original euclidean space. e0 and e∞ are additional basis vectors with specific properties to facilitate the embedding, introduced to handle infinity and origin, and operations like translation, rotation, dilation, and reflection. e0 is often associated with the origin in the extended space. e∞ represents the "point at infinity," enabling the conformal model to handle infinite points, essential for representing directions and ideal points.

the null vector condition is given by:

dot_product(p, p) = 0

this condition ensures that the representation is a null vector in the conformal space, preserving the geometric properties of the original euclidean point.

rotor = e ** (-axis_orthogonal_plane * angle / 2)

rotor = cos(angle / 2) + bivector * sin(angle / 2)

rotor = 1 + 1 / 2 * translation_vector * e∞

rotor = e ** (-axis * angle / 2) * (1 + 1 / 2 * translation_vector * e∞)

e ** ((angle / 2) * n * e4) = cos(angle / 2) + sin(angle / 2) * n * e4

e ** ((1 / (2 * d)) * n * e0) = exterior_product(1 + (1 / (2 * d) * n, e0))

result = rotor * point * reverse(rotor)