2018-10-29

learning about geometric algebra 2017

a collection of notes

links

resources that helped me the most so far:

software

  • gaviewer for experimentation with calculations and visualisation
  • libvsr c++ template based implementation, which means algebras for different metrics are pre-compiled and fast but perhaps cant easily be run-time created. some additional usage information may be found in the text "articulating space"
  • reference implementation from the book "geometric algebra for computer science"
  • sph-cga experimental implementation in scheme based on this text
  • versor.js javascript port of libvsr

more

dependent concepts

with currently available material, understanding of the following concepts tends to be assumed in definitions and explanations: abstract algebra, algebra, bilinearity, codirectionality, coefficient, colinearity, commutativity, coordinate free, covector, dual, eigenbasis, eigenvalue, euclidean geometry, field, homogeneous space, ideal number, linear algebra, linear algebra minor, linear dependence, mathematical closure, matrix, matrix determinant, matrix multiplication, metric, metric signature, metric space, orthogonal, orthonormal, parametric equation, parity of a permutation, permutation, projection, real numbers, scalar, subspace, tensor, tensor product, vector algebra, vector basis, vector cross product, vector dot product, vector spaces, vectors

concepts

  • k-vector

    • blade
    • versor
    • pseudoscalar
    • multivector

      • bivector
      • trivector
  • geometric product

    • outer product
    • inner product
  • meet
  • join
  • dual
  • grade
  • grade inversion
  • grade reversal
  • rotor
  • motor
  • spinor

operations

geometric product

function signature

op :: k-vector k-vector -> scalar k-vector

perpendicularity and parallelness

perpendicularity and parallelness lead to zeros for either the defining inner product or outer product part is inner product if parallel. is outer product if perpendicular

gp(a, a): outer product part equals zero

result

  • gp(a, b) -> scalar + vector -> multivector
  • sum between scalar and bivector
  • the geometric product produces multivectors of mixed grade

fundamentality

  • the geometric product is more fundamental than the inner or outer product
  • inner and outer product can be expressed in terms of the geometric product
  • the geometric product can be expressed in terms of the inner and outer product
  • the gp is the fundamental product in geometric algebra, you will not need any other product, since it contains all geometric relationships between its arguments [1]

other

  • linear and associative
  • commutative only in the case of collinear (op(a, b) = 0) or orthogonal (ip(a, b) = 0) vectors
  • inverse gp: gp division
  • reflection: gp(mirror, igp(object, mirror))
  • the gp seems to be very much defined by properties of the traditional cross product and dot product. the one collapses at perpendicularity, the other at parallelity.

equations

gp(a, b) = ip(a, b) + op(a, b)
gp(a, b) = ip(b, a) - op(b, a)
ip(a, b) = gp(1/2, gp(a, b) + gp(b, a)) = ip(b, a)
op(a, b) = gp(1/2, gp(a, b) - gp(b, a)) = -op(b, a)
gp(a, b) = -gp(b, a) + gp(2, ip(a, b))
gp(b, a) = gp(a, b) - gp(2, op(a, b))
gp(a, a) = norm(a) ** 2
gp(scalar-x, scalar-y) = scalar-x * scalar-y
gp(a, a) = ip(a, a) = scalar
gp(a, gp(b, c)) = gp(gp(a, b), c)

orthonormal unit vectors: e1, e2

gp(e1, e1) = 0 + ip(e1, e1) = 1
gp(e1, e2) = op(e1, e2) + 0

outer product

function signature

op :: k-vector k-vector -> k-vector

properties

associativity, distributivity/linearity, antisymmetry, scalar operation, commutativity not required

equations

x, y: scalar

norm(op(a, b)) = norm(a) * norm(b) * sin(angle(a, b))
op(a, b) = -op(b, a)
op(a, a) = -op(a, a) = 0
op(x, y) = x * y
op(x, a) = op(a, x) = x * a
op(x + y, c) = op(x, c) + op(y, c)
op(a, op(b, c)) = op(op(a, b), c)
op(a, x * b + y * c) = x * op(a, b) + y * op(a, c)

colinear: op(a, b) = 0 coplanar: op(a, b, c) = 0

outer product for 3d vectors

op(a, b)
= op(a1 * e1 + a2 * e2 + a3 * e3,
      b1 * e1 + b2 * e2 + b3 * e3)
= (a1 * b2 - a2 * b1) * op(e1, e2) +
  (a2 * b3 - a3 * b2) * op(e2, e3) +
  (a3 * b1 - a1 * b3) * op(e3, e1)

other

  • an orthogonal vector with a norm that corresponds to the area of a parallelogram between two vectors
  • the outer product is not defined to be replaceable by a number except in the zero or scalar case. in many cases it is not further reducible. any possible vector element multiplication or subtraction follows from its properties
  • the outer product of two 2d vectors is an e1e2 (or e12) bivector that can be represented as an object with only one value, a scale factor. there would be six scale factors for 4d bivectors, three for 3d bivectors. the number of scale factors is determined by the number of irreducible outer products when equation elements are distributed. this corresponds to the grade of blades
  • grade increasing. k-vector -> (k + 1)-vector
  • generates a new type op(e1, e2) which cant be further simplified. if we then multiply by another vector, say e3, then this will produce yet a further new type op(e1, e2, e3) and so on
  • allows the formal specification of subspaces (blades)
  • null result if and only if the factors are linearly dependent
  • associated geometric properties: direction, orientation, magnitude
  • inner product: distance and orthogonality/perpendicularity. outer product: area and parallelism
  • op(e1, e2) represents the right-angled turn from the direction e1 to e2
  • similar to the cross product
  • alternative names: exterior product, wedge product. the name outer product is often used because of the symmetry with inner product and because interior product has another perhaps more commonly used meaning compared to outer product, which also has different meaning in other contexts

inner product

function signature

ip :: k-vector k-vector -> (k - 1)-vector/scalar

equations

ip(a, b) = norm(a) * norm(b) * cos(angle)
ip(a, b) = sum(a1 * b1, ..., an * bn)
sqrt(ip(a, a)) = norm(a)
ip((0, 1), (1, 0)) = 0
ip((0, 20), (3, 0)) = 0
projection(a, b) = ip(a, b) / norm(b)
angle = arccos(ip(a, b) / ip(norm(a), norm(b)))
norm(a) * norm(b) * (cos(pi/2) = 0) = 0
ip(a, b) = multiply(projection(a, b), norm(b))

properties

  • positivity: ip(a, a) >= 0
  • symmetry: ip(a, b) = ip(b, a)
  • linearity, x and y being scalars: ip(a, x * b + y * c) = x * ip(a, b) + y * ip(a, c)

descriptions

  • equal to scaling the vector "b" by a factor equal to the length of the orthogonal projection of "a" on "b" (where the right angle is on the "b" side)
  • two vectors are orthogonal if and only if their inner product is zero
  • the inner product is the dot product generalised to higher dimensional spaces
  • in geometric algebra the inner product is not only defined for vectors
  • grade decreasing. k-vector -> (k - 1)-vector
  • can be used to calculate lengths/distances and angles between vectors

links

dual, pseudoscalar

pseudoscalar: i

equations

i = op(e1, ..., edim)
gp(i, i) = -1
dual(a) = gp(a, i)

in r3

i = e1e2e3 = op(e1, e2, e3)
gp(e1, e1e2e3) = e2e3
gp(e1, e2) = gp(i, e3)
gp(e2, e3) = gp(i, e1)
gp(e3, e1) = gp(i, e2)
gp(i, i) = e1e2e3e1e2e3 = e1e2e1e2 = -1
gp(i, op(e1, e2)) = gp(i, op(e1, e2, e3, e3)) = gp(i, i, e3) = -e3

in odd dimensional space: gp(i, a) = gp(a, i)

  • undual: dedualisation, dual(dual(x)) (perhaps sign issues)
  • multiplying (gp) by the pseudoscalar collapses all vectors that are non-zero
  • multiplication (gp) by the pseudoscalar represents a 90 degree turn in all three dimensions. therefore two such 90 degree turns in each of the three dimensions results in a total reversal of direction, which is equivalent to a negation, and thus gp(i, i) = -1
  • in odd dimension, the pseudoscalar commutes. in even dimension, it anticommutes

meet, join

  • calculates intersections of lines, planes, spheres, cylinders, etc
  • similarities: (meet: intersection) (join: union)
  • meet(a, b) = ip(dual(a), b)

structures

bivectors

  • magnitude and direction
  • are not specific in shape
  • magnitude can be thought of as the area
  • are equal as long as the area is equal. example: op(a, b) = op(2 * a, b / 2)

multivector

  • are combinations of possibly different k-vector types: scalar, bivector, trivector, etc
  • defined to be the sum of different-grade k-blades, such as the summation of a scalar, a vector, and a 2-vector. a sum of only k-grade components is called a k-vector, or a homogeneous multivector
  • if a given element is homogeneous of a grade k, then it is a k-vector, but not necessarily a k-blade. such an element is a k-blade when it can be expressed as the wedge product of k vectors
  • sum of elements of different dimensions

k-vector

  • multivector parts can be recovered, not like normal addition
  • the exact parallelogram of bivectors can often not be recovered
  • multivector: sum of different-grade k-blades
  • k-vector: a sum of only k-grade components. homogeneous multivector

blade

  • are k-vectors that can be build using the outer product
  • blades represent unit lines, unit areas and unit volumes
  • often defined to be orthonormal, which means all are orthogonal, which means they are perpendicular to each other with no overlap in dimensions
  • pseudoscalar: the blade of maximum grade
  • bi-/trivector: k-vector
  • only one combination of lower-grade basis vectors is relevant per grade even though multiple combinations/orderings are possible in higher dimensions
  • all "permutations" of dimensions
  • basis blades of grade represent grade-dimensional vector subspaces
  • grade(op(a, b)) = grade(a) + grade(b)

number of basis blades

2 ** dimensions

number of blades per grade

d: dimensions, g: grade

d! / (d - g)! / g!

example

  • 0-blade: scalar
  • 1-blade: vector
  • 2-blade: op(vector, vector)
  • 3-blade: op(vector, vector, vector)

conformal geometric algebra

  • uses more dimensions for calculation then projects to lower dimensions. operations are particularly generic
  • no: null vector at origin
  • ni: null vector at infinity (a point that is infinitely far away from any other point), closing point at infinity
  • null vector: vector with norm zero
  • points are represented by null vectors
  • inner product is proportional to the squared point distance

equations

ip(no, no) = 0
ip(ni, ni) = 0
ip(no, ni) = -1
ip(ni, no) = -1
ip(p, p) = 0
ip(p / ip(-inf, p), q / ip(-inf, q)) = -1/2 square(distance(P, Q))
pt(p) = no, p, 1/2 * p * p * ni)

vectors for shapes

multiple vectors can describe shapes. the vectors can be imagined as being chained, with each going into a different dimension. because of associativity, different combinations of sides can describe the same shape

vector space

any set of objects satisfying the following: scalar multiplication, vector addition and null object defined. a vector space seems to define the size/dimensionality of vectors, the type for values and the possible values

orthonormal

the direction of the vectors is perpendicular to each other so they point in separate directions like axis

software implementations

  • basis blade representation and operations. bitmaps for identifiers, operations like grade, reverse, compare
  • multivector representation and operations
  • product calculations: geometric/outer/inner product
  • use a metric signature

vector projection/rejection

description 1

let c and d be vectors with tails at an origin. let e be a vector with its tail at the head of c meeting d at a right angle. e is the rejection. a vector from the tail of d to the head of of e is the projection

description 2

start with two vectors, introduce a third vector to complete a right triangle. if the first two vectors are not perpendicular, the triangle has two catheti, with the one lying on the second vector being the projection, and the third vector being the rejection

equations

projection(a, b) = ip(a, b) / norm(b)
rejection(a, b) = a - projection(a, b)

vector base notation

elements are often written as multiples of a unit/basis vector, which can be for example a vector that goes 1 into a distinct direction. base vectors represent directions

a * e1 + b * e2 + c * e3 == [a * e1, b * e2, c * e3] == [a, b, c]

example

  • e1: (1, 0, 0)
  • e2: (0, 1, 0)
  • e3: (0, 0, 1)

if orthonormal / linearly independent

  • inner product with themselves is 1
  • e1 ip e1 = e2 ip e2 = e3 ip e3 = 1
  • inner product with each other is 0
  • e1 ip e2 = e2 ip e3 = e3 ip e1 = 0

metric

  • a function that associates a scalar to each possible pair of vectors
  • distance, length, area, angle, orthogonality, orthogonal maps, projections, rotations and other concepts are based on it
  • norm = metric(vector-a, vector-a)
  • oblique coordinate system: the bases are not necessarily unit vectors and the inner product of different base vectors may not be 0

signature

  • p q r notation: positive negative zero
  • matrix notation: matrix from all inner product combinations of basis vectors
  • vector notation: numbers from the diagonal of the inner product matrix. length of the signature usually corresponds to dimensionality
  • only sign and not the magnitude might matter

nullspace

  • set of all vector pairs that evaluate to zero for some function
  • inner product nullspace: infinite orthogonal plane. where op(a, b) = 0
  • outer product nullspace: infinite colinear line. where op(a, b) = 0

drawing

  • k-vectors can still have scalars for basis vectors (as far as the outer product it is built with are not further reducible), and the scalars/weights are the coordinates
  • the norm of the coordinate vector appears to be the diameter of circles and spheres or diagonal of squares

vector norm

calculated using multiplication from elementary arithmetic

(x ** 2 + y ** 2 + z ** 2) // 2

other

  • much of geometric algebra can be and is described in a coordinate free way, that means without information about how it would be mapped to coordinates
  • like with complex numbers, a plus sign might be used but the entities are meant to be kept separated, with the plus sign not having the meaning it has in elementary arithmetic
  • rules like associativity/distributivity/commutativity define how equations can be reformulated, solved or simplified
  • algebra concerns relations between entities and what the entities are supposed to represent and much less so how the entities or operations themselves can be represented using for example numbers and arithmetic
  • an algebra can be "generated"
  • multiplication from elementary arithmetic might be replaced with the geometric product. then letter combinations and exponents use the geometric product instead of the multiplication of elementary arithmetic. for example x ** 2 then means gp(x, x)
  • knowledge relation: linear-algebra -> geometric-algebra -> conformal-geometric-algebra
  • anticommutativity: each pairing has an anti-pair
  • addition of bivectors can be visualised as creating a plane like a hypotenuse between planes (similar to vector addition)
  • how is it possible to work with infinity (for example outer product with infinity) . answer: the outer product does not resolve it, it is used as a constant

questions

  • how to draw higher dimensional shapes in lower dimensions. just leave out parts of point vectors?
  • difference between k-vector, blade and versor
  • how to calculate angle between vectors
  • why is no/ni used instead of em/ep
  • what does the pqr notations, the inner product matrices and metrices really mean. how does the metric influence the products
  • how to map points from r3 to cga and back
  • how to use a degenerate/oblique metric
  • how to create complex shapes like paths and polygons
  • transformations. rotation, translation, scaling

unsorted

equations

gp(e1, e1) = 1
gp(op(e1, e2), op(e1, e2)) = -1
gp(a, a) = gp(norm(a), norm(a))
gp(a, a) / gp(norm(a), norm(a)) = 1
gp(a / gp(norm(a), norm(a)), a) = 1
inv(a) = a / gp(norm(a), norm(a))
inv(a) = a / gp(norm(a), norm(a)) = a / ip(a, a)
inverse is either shorter or longer
if norm(a) > 1 then inverse is shorter
if norm(a) < 1 then inverse is longer
if norm(a) = 1 then inverse the same
par(a, b) -> gp(a, b) = gp(b, a) -> commutes
perp(a, b) -> gp(a, b) = -gp(b, a) -> anticommutes
gp(a, b) = ip(a, b) + op(a, b)
gp(b, a) = ip(b, a) + op(b, a)
gp(b, a) = ip(a, b) - op(a, b)
gp(a, b) + gp(b, a) = gp(2, ip(a, b))
ip(a, b) = gp(1/2, gp(a, b) + gp(b, a))
gp(a, b) - gp(b, a) = gp(2, op(a, b))
op(a, b) = gp(1/2, gp(a, b) - gp(b, a))
a = par(a) + perp(a)
perp(a) = a - par(a)
gp(a, b) = gp(par(a), b) + gp(perp(a), b)
gp(a, b) = gp(par(a), b) - gp(b, perp(a))
gp(a, b) = gp(par(a), b) - gp(b, a - par(a))
gp(a, b) = gp(par(a), b) - gp(b, a) + gp(b, par(a))
gp(a, b) = gp(par(a), b) - gp(b, a) + gp(par(a), b)
gp(a, b) + gp(b, a) = gp(2, par(a), b)
gp(1/2, gp(a, b) + gp(b, a)) = gp(par(a), b)
ip(a, b) = gp(par(a), b)
gp(ip(a, b), inv(b)) = gp(par(a), b, inv(b))
par(a) = gp(ip(a, b), inv(b))
par(a) = ip(a, b) / ip(b, b) * b
if norm(a) = 1 then inv(a) = a
par(a) = gp(ip(a, b), inv(b))
perp(a) = gp(op(a, b), inv(b))

reflection

a = par(a) + perp(a)
ref(a) = par(a) - perp(a)
b ref(a) = b par(a) - b perp(a)
b ref(a) = par(a) b + perp(a) b
b ref(a) = (par(a) + perp(a)) b
b ref(a) = a b
inv(b) b ref(a) = inv(b) a b
ref(a) = inv(b) a b

sources

a: dorst, fontijne, mann - geometric algebra for computer science (2007)

1: [a] section 6.3 p151