Geometric Algebra and Geometric Calculus 04/13/2015 ...files.meetup.com/14851982/Intro to GA and...
Transcript of Geometric Algebra and Geometric Calculus 04/13/2015 ...files.meetup.com/14851982/Intro to GA and...
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
● 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