Introduction toGeometric Algebra and

Geometric CalculusFrank Palazzolo

Detroit Physics Meetup04/13/2015

Outline- What is it?- History: Clifford, Grassmann, Hestenes- Geometric Algebra in 2D and 3D

- Intro & Applications- Geometric Calculus

- Intro & Applications- Implementations & References

● Mathematical framework, which unifies and extends many concepts in mathematics and physics

● Examples: Complex Numbers, Quaternions, Rotations, Reflections, Linear Algebra, Vector Calculus

What is it?

Motivation- Unifies and extends many areas of

mathematics/physics in one framework- Can operate at the conceptual level,

without resorting to coordinates*- Provide new insights into physics- Easier to learn

● Hermann Grassmann (1809-1877)○ Wedge (Outer Product)

● William Clifford (1845-1879)○ Geometric Product, Clifford Algebra

● David Hestenes (1933-)○ Rediscoverer and promoter of GA

● Others - Hamilton, Gibbs

History

History (Cont’d)

● Scalar (Dot, Inner) Product● Cross Product● Wedge (Outer) Product● Complex Numbers● Geometric Product

Five ways to multiply Vectors

● Result is a scalar● Result is Symmetric

○ |a||b|cos(theta)● Clearly not invertible

Scalar (Dot, Inner) Product

● Result is a Vector● Result is Antisymmetric

○ magnitude is |a||b|sin(theta)● Only exists in 3 dimensions● Related to problem of

○ “Axial” vectors vs “Polar” vectors

Cross Product

● Result is a Bivector (Directed Area)● Result is Antisymmetric● Exists in any number of dimensions● Solves problem of

○ “Axial” vectors vs “Polar” vectors

Wedge Product (^) (Grassmann)

● Can treat complex numbers as vectors● Can interpret multiplication of complex

numbers as vector multiplication● For example a x b● Another interesting example... a x b*

○ Real part is scalar product○ Imaginary part is wedge product

Complex Numbers

- Sum of Scalar and Wedge Products- Symmetric and Antisymmetric parts- Invertible -> Inv(a) = a/|a|- Can define all other products with this

one

Geometric Product

How it works in 2D● Lets define our own algebra

○ e1 and e2, orthogonal unit basis vectors○ define e1e1 = e2e2 = 1○ e1e2 = e1^e2 = -e1e2 = I

● Turns out that’s it for 2D○ e1e2e1 = -e1e1e2 = -e1○ e1e2e2 = e1○ (e1e2)(e1e2) = -e1e1e2e2 = -1

How it works - 2D Vectors● Multivector in 2D● v = a + b*e1 + c*e2 + d*I

○ 1 scalar○ 2 vectors e1 and e2○ 1 bivector (pseudoscalar, e1e2 = I)

● Geometric Product to Multivectors○ This space is G2, vector space, superset of

R2

How it works - 2D Vectors● e1e2 = I● Left multiply by I

○ Ia = rotation 90 degree counter clockwise● Right multiple by I

○ aI = rotate by 90 degrees clockwise● II = -1● Complex numbers form a subalgebra of

G2

● 3D Multivector○ 1 scalar part○ 3 vectors (e1, e2, e3)○ 3 bivectors (e1e2, e2e3, e3e1)○ 1 trivector (pseudoscalar, e1e2e3 = i)

● bivectors square to -1● Quaternions form even subalgebra of G3● i really acts like i now

3D Vectors

● Dual of a Multivector A*○ cross product is really A/I

● Projections, Rejections○ para(a) = (a.B)/B, perp(a) = (a^B)/B

● Rotation - Rotors○ theta is scalar, i is unit bivector…○ Rotor = R = exp(-i*theta/2)○ R(a) = -RaR

3D Vectors

● Applications to Linear Algebra○ Determinants, Eigenvectors

● Applications in Classical Mechanics○ Angular Momentum, E and B fields

● Applications in QM○ Pauli Spin Matrices

3D Vectors

● sigma matrices are just○ bivector parts of G3○ no mysterious vector of matrices○ same as quaternion space

● Separates Unique Math from Physics

Application: Pauli Spin Matrices

● 4D and Special Relativity○ Very similar to what we have seen, but can

start with some vector defined with aa = -1● 5D Conformal Model

○ Application to Computer Science○ Basic elements of the algebra are points,

lines, planes, circles, spheres

Other spaces

● Operations on vector fields● Gradient

○ Apply to scalar field, result is vector field● Divergence (like dot product)

○ Apply to vector field, result is scalar field● Curl (like cross product)

○ Apply to vector field, result is vector field

Vector Calculus (Gibbs)

● Operations on vector fields● Gradient

○ Apply to multivector field, result is multivector field

● Gradient of Vector field is Divergence + Curl

Geometric Calculus (Hestenes)

● Single equation captures all Physics○ Gradient of the 4-current density J is the

Field F. ○ E field is the vector component○ B field is the bivector component

Application: Maxwells Equation

● Conceptually Useful○ Better illustrates math connections○ Separates unique physics from math○ Can do things by hand without coordinates

● Computationally Useful?○ Takes more horsepower to compute with

Multivectors, but optimizations are possible○ Chris Doran - Geomerics

Future…?

● Cambridge - Intro papers, software○ http://geometry.mrao.cam.ac.uk/

● Alex MacDonald○ http://faculty.luther.edu/~macdonal/

● David Hestenes site○ http://geocalc.clas.asu.edu/

References - Websites

● Linear and Geometric Algebra - MacDonald

● Vector and Geometric Calculus - MacDonald

● Geometric Algebra for Physicists - Doran● Geometric Algebra for Computer Science

- Dorst

References - Books

● Geometric Algebra with Applications to Engineering - Perwass

● Foundations of Geometric Algebra Computing - Hildenbrand

References - More Books

● https://staff.science.uva.nl/l.dorst/clifford/index.html

● Siggraph 2001● GAIGEN● The Power of Geometric Algebra

Computing for Mathematica○ GAALOP - GA -> OpenCL -> GPU, FPGA, etc.○ http://shar.es/1gQTHw

References - Software