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)
- This syntax defines a set of basis vectors
- For these basis vectors we define the geometric product:
e_ie_i = 1
e_ie_i = -1
e_ie_i = 0
p vectors
q vectors
r vectors
- For vectors this product is anticommutative:
e_ie_j = -e_je_i
Example 3D GA
Cl(3, 0, 0)
- Consider the algebra:
- This creates 3 basis vectors that square to 1:
e_1e_1 = 1, \, e_2e_2 = 1, \, e_3e_3 = 1
e_1e_2 = -e_2e_1 = e_{12}
- We will write the product of two vectors with the shorthand:
(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:
e_{123}e_1 = e_1e_2e_3e_1 = -e_2e_1e_3e_1 = e_2e_3e_1e_1 = e_2e_3
Grade Selection
(e_1 + e_2)(e_1 + e_3) = 1 - e_{12} + e_{23} + e_{13}
- Define another operation, the grade selection operation:
\left< 1 - e_{12} + e_{23} + e_{13}\right>_0 = 1
\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:
\left< 1 - e_{12} + e_{23} + e_{13}\right>_1 = 0
- Consider the result:
(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:
(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
X_a\wedge Y_b = \left< X_a Y_b \right>_{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 \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?
R_0 = \cos(\theta_0/2) + \sin(\theta_0/2)e_1\wedge e_3
R_1 = \cos(\theta_1/2) + \sin(\theta_1/2)e_1\wedge e_2
\newcommand{\la}{\langle}
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\newcommand{\no}{n_{0}}
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\newcommand{\wdg}{\wedge}
\newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2} }
R_\rho = 1 - \frac{1}{2}\rho e_1\wedge \ninf
R_2 = \cos(\theta_2/2) + \sin(\theta_2/2)e_1\wedge e_2
\newcommand{\la}{\langle}
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\newcommand{\ninf}{n_{\infty}}
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\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 = R_0R_1R_\rho R_2R_l
\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
\rho
l
y
\theta_0
\theta_1
\theta_2
x
Serial Robot Inverse Kinematics
\rho
l
y
\theta_0
\theta_1
\theta_2
\newcommand{\ninf}{n_{\infty}}
S_Y = I_5(Y - \frac{1}{2}l^2\ninf)
\newcommand{\ninf}{n_{\infty}}
S_{\text{base}} = I_5(n_0 - \frac{1}{2}\rho^2\ninf)
C = S_Y\vee S_{\text{base}}
\newcommand{\ninf}{n_{\infty}}
\Pi = n_0\wedge up(e_3)\wedge Y\wedge \ninf
\newcommand{\ninf}{n_{\infty}}
P = \Pi\vee C
\newcommand{\ninf}{n_{\infty}}
X = \left[1 + \frac{P}{\sqrt{P\tilde{P}}}\right](P\cdot\ninf)
x
(If \(P^2 < 0 \) then y is out of reach)
Construct a sphere at the base, \(n_0\)
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 = -\frac{-\sum_{j=1}^{j=3}(X\cdot e_j)e_j}{X\cdot n_\infty}
\( A \vee B \) is just \( I_5(I_5A \wedge I_5B) \)
\( I_5 = e_1e_2e_3e_4e_5 \)
Model of a Delta Robot
Forward Kinematics
a_i = (r_b - r_e + l\cos(\theta_i))s_i + l\sin(\theta_i)e_3
Get the pseudo-elbow point \(A_i\)
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}
A_i = \frac{1}{2}a_i^2n_\infty + a_i - n_0
\Sigma_i = A_i - \frac{1}{2}\rho^2n_\infty
Construct a sphere about the pseudo-elbow
T = I_5\bigwedge_{i=0}^{i=2}\Sigma_i
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
r_e
Y
A_i
r_b
l
\rho
\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