Hugo Hadfield
Cambridge University PhD student, Signal Processing and Communications Laboratory
Intro to GA for CS
Geometric/Clifford Algebra
- GA is about studying the properties of the real Clifford algebras
- We typically describe which algebra we are working in with the following syntax:
Cl(p,q,r)
Cl(p, q, r)
- This syntax defines a set of basis vectors
- For these basis vectors we define the geometric product:
eiei=1
e_ie_i = 1
eiei=−1
e_ie_i = -1
eiei=0
e_ie_i = 0
p vectors
q vectors
r vectors
- For vectors this product is anticommutative:
eiej=−ejei
e_ie_j = -e_je_i
Example 3D GA
Cl(3,0,0)
Cl(3, 0, 0)
- Consider the algebra:
- This creates 3 basis vectors that square to 1:
e1e1=1,e2e2=1,e3e3=1
e_1e_1 = 1, \, e_2e_2 = 1, \, e_3e_3 = 1
e1e2=−e2e1=e12
e_1e_2 = -e_2e_1 = e_{12}
- We will write the product of two vectors with the shorthand:
(e1+e2)(e1+e3)=e1e1+e2e1+e2e3+e1e3=1−e12+e23+e13
(e_1 + e_2)(e_1 + e_3) = e_1e_1 + e_2e_1 + e_2e_3 + e_1e_3 \\ = 1 - e_{12} + e_{23} + e_{13}
- The geometric product is distributive and associative:
e123e1=e1e2e3e1=−e2e1e3e1=e2e3e1e1=e2e3
e_{123}e_1 = e_1e_2e_3e_1 = -e_2e_1e_3e_1 = e_2e_3e_1e_1 = e_2e_3
Grade Selection
(e1+e2)(e1+e3)=1−e12+e23+e13
(e_1 + e_2)(e_1 + e_3) = 1 - e_{12} + e_{23} + e_{13}
- Define another operation, the grade selection operation:
⟨1−e12+e23+e13⟩0=1
\left< 1 - e_{12} + e_{23} + e_{13}\right>_0 = 1
⟨1−e12+e23+e13⟩2=−e12+e23+e13
\left< 1 - e_{12} + e_{23} + e_{13}\right>_2 = - e_{12} + e_{23} + e_{13}
- If there is nothing of the grade in there, then it gives 0:
⟨1−e12+e23+e13⟩1=0
\left< 1 - e_{12} + e_{23} + e_{13}\right>_1 = 0
- Consider the result:
(e1+e2)(e1+e3)=⟨(e1+e2)(e1+e3)⟩0+⟨(e1+e2)(e1+e3)⟩2=1−e12+e23+e13
(e_1 + e_2)(e_1 + e_3) = \left< (e_1 + e_2)(e_1 + e_3) \right>_0 + \left< (e_1 + e_2)(e_1 + e_3) \right>_2\\
= 1 - e_{12} + e_{23} + e_{13}
- The result of the geometric product can be written with grade selection:
Inner and Outer Products
- Consider the result:
(e1+e2)(e1+e3)=⟨(e1+e2)(e1+e3)⟩0+⟨(e1+e2)(e1+e3)⟩2=1−e12+e23+e13
(e_1 + e_2)(e_1 + e_3) = \left< (e_1 + e_2)(e_1 + e_3) \right>_0 + \left< (e_1 + e_2)(e_1 + e_3) \right>_2\\
= 1 - e_{12} + e_{23} + e_{13}
Inner (dot) product
- This motivates us to define two new products:
Outer (wedge) product
Xa∧Yb=⟨XaYb⟩a+b
X_a\wedge Y_b = \left< X_a Y_b \right>_{a+b}
Xa⋅Yb=⟨XaYb⟩∣a−b∣
X_a\cdot Y_b = \left< X_a Y_b \right>_{|a-b|}
- For vectors in GA this leads us to the result:
ab=a⋅b+a∧b
ab = a \cdot b + a \wedge b
What is it good for?
- Each algebra can be viewed as a domain specific language for some specific geometry
- All the algebras share the same basic operations, learn them once, use them everywhere
- Cl(3,0,0) - 3D vectors, quaternions, projections
- Cl(4,0,0) - Homogeneous points, lines, planes, rotations, projections
- Cl(3,0,1) - Homogeneous points, lines, planes, projections, intersections, dual quaternions, screw theory, rigid body dynamics
- Cl(4,1,0) - Everything in Cl(3,0,1) + line-segments, circles, spheres, all conformal transformations, distance geometry
- Cl(4,2,0) - 3D Lie Sphere Geometry
- Cl(4,4,0) - Homogeneous points, lines, quadrics, rotations, shear transformations
- Cl(8,2,0) - Everything in Cl(4,1,0) + Quadrics, Dupin Cyclides + probably a lot more
Algebras for 3D Euclidean space (there are lots more):
Some algebras for non-Euclidean space (there are lots more):
- Cl(1,3,0) - Space time algebra, Lorentz transformations, electromagentism
- Cl(4,0,0) - Spherical and hyperbolic space, quantum mechanics
Serial Robot Forward Kinematics CGA
Can we construct a rotor from origin to endpoint?
R0=cos(θ0/2)+sin(θ0/2)e1∧e3
R_0 = \cos(\theta_0/2) + \sin(\theta_0/2)e_1\wedge e_3
R1=cos(θ1/2)+sin(θ1/2)e1∧e2
R_1 = \cos(\theta_1/2) + \sin(\theta_1/2)e_1\wedge e_2
Rρ=1−21ρe1∧n∞
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\nn}{\nonumber}
\newcommand{\ninf}{n_{\infty}}
\newcommand{\einf}{e_{\infty}}
\newcommand{\no}{n_{0}}
\newcommand{\eo}{e_{0}}
\newcommand{\wdg}{\wedge}
\newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2} }
R_\rho = 1 - \frac{1}{2}\rho e_1\wedge \ninf
R2=cos(θ2/2)+sin(θ2/2)e1∧e2
R_2 = \cos(\theta_2/2) + \sin(\theta_2/2)e_1\wedge e_2
Rl=1+21le1∧n∞
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\nn}{\nonumber}
\newcommand{\ninf}{n_{\infty}}
\newcommand{\einf}{e_{\infty}}
\newcommand{\no}{n_{0}}
\newcommand{\eo}{e_{0}}
\newcommand{\wdg}{\wedge}
\newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2} }
R_l = 1 + \frac{1}{2}l e_1\wedge \ninf
R=R0R1RρR2Rl
R = R_0R_1R_\rho R_2R_l
Y=Rn0R~
\newcommand{\no}{n_{0}}
Y = R\no\tilde{R}
Construct the base rotors
Construct the first link's translation rotor
Construct the elbow and link 2 translation rotor
The combined rotor is the product of all of them
ρ
\rho
l
l
y
y
θ0
\theta_0
θ1
\theta_1
θ2
\theta_2
x
x
Serial Robot Inverse Kinematics
ρ
\rho
l
l
y
y
θ0
\theta_0
θ1
\theta_1
θ2
\theta_2
SY=I5(Y−21l2n∞)
\newcommand{\ninf}{n_{\infty}}
S_Y = I_5(Y - \frac{1}{2}l^2\ninf)
Sbase=I5(n0−21ρ2n∞)
\newcommand{\ninf}{n_{\infty}}
S_{\text{base}} = I_5(n_0 - \frac{1}{2}\rho^2\ninf)
C=SY∨Sbase
C = S_Y\vee S_{\text{base}}
Π=n0∧up(e3)∧Y∧n∞
\newcommand{\ninf}{n_{\infty}}
\Pi = n_0\wedge up(e_3)\wedge Y\wedge \ninf
P=Π∨C
\newcommand{\ninf}{n_{\infty}}
P = \Pi\vee C
X=[1+PP~P](P⋅n∞)
\newcommand{\ninf}{n_{\infty}}
X = \left[1 + \frac{P}{\sqrt{P\tilde{P}}}\right](P\cdot\ninf)
x
x
(If P2<0 then y is out of reach)
Construct a sphere at the base, n0
Construct a sphere at the endpoint, y
Intersect the spheres to give a circle
Define a vertical plane through the endpoint and base
Intersect the circle and the plane to give a point pair
Choose one of the elbow position solutions
x=−X⋅n∞−∑j=1j=3(X⋅ej)ej
x = -\frac{-\sum_{j=1}^{j=3}(X\cdot e_j)e_j}{X\cdot n_\infty}
A∨B is just I5(I5A∧I5B)
I5=e1e2e3e4e5
Model of a Delta Robot
Forward Kinematics
ai=(rb−re+lcos(θi))si+lsin(θi)e3
a_i = (r_b - r_e + l\cos(\theta_i))s_i + l\sin(\theta_i)e_3
Get the pseudo-elbow point Ai
Y=−P~(T⋅n∞)Py=Y⋅n∞−∑j=1j=3(Y⋅ej)ej
Y = -\tilde{P}(T\cdot n_\infty)P\\
y = \frac{-\sum_{j=1}^{j=3}(Y\cdot e_j)e_j}{Y\cdot n_\infty}
Ai=21ai2n∞+ai−n0
A_i = \frac{1}{2}a_i^2n_\infty + a_i - n_0
Σi=Ai−21ρ2n∞
\Sigma_i = A_i - \frac{1}{2}\rho^2n_\infty
Construct a sphere about the pseudo-elbow
T=I5⋀i=0i=2Σi
T = I_5\bigwedge_{i=0}^{i=2}\Sigma_i
P=1+T2T
P = 1 + \frac{T}{\sqrt{T^2}}
The intersection of the three spheres (one from each limb) correspond to the two possible possitions of the centre of the end plate
re
r_e
Y
Y
Ai
A_i
rb
r_b
l
l
ρ
\rho
θ
\theta
Implementations
- Writing a fast GA implementation can be hard as multivectors are typically very sparse and some algebras can be very high dimensional
- People get around this in a few ways:
- Really good data structures: GARAMON
- C++ templating - GAL, versor, GATL
- JIT compilers - clifford, ganja.js
- Symbolic optimisation - GAALOP
- Only support one algebra and hand optimise it with SIMD etc. to be very fast - Klein
- Sparse representation and codegen - Gaigen2
- Load it all on the GPU with tensorflow - tfga
GA and proof systems
- There seems to have been some work on using proof systems for GA:
- There are a few symbolic GA packages:
- Maple (various)
- Python - galgebra
- There has also been work done on using geometric/clifford algebras for proof systems:
Additional reading
- Clifford Algebra to Geometric Calculus (Hestenes and Sobczyk)
- Geometric Algebra for Physicists (Chapter 4 contains axiomatic construction)
- Geometric Algebra For Computer Science, An Object Oriented Approach to Geometry
Intro to GA for CS Hugo Hadfield June 2020 hh409.user.srcf.net
Intro to GA for CS
By Hugo Hadfield
Intro to GA for CS
An introduction to Geometric Algebra from a more traditional computer science perspective. It presents different subalgebras as domain specific programming languages and discusses the various implementation approaches common in the GA community.
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