This document provides an overview and review of geometric algebra, its applications, and the current state of the field. It discusses how geometric algebra provides a unified mathematical framework that simplifies diverse topics like transformations, projections, and modeling. The document reviews concepts like the geometric product and multivectors. It also summarizes several applications of geometric algebra like the homogeneous and conformal models, Voronoi diagrams, physical modeling, and benchmarks comparing it to other methods. Overall, the document demonstrates how geometric algebra is gaining recognition and being used in diverse areas as an efficient computational framework.

1. Geometric Algebra
Part 2: Applications and the state of the art
“... provides a single, simple mathematical framework which
eliminates the plethora of diverse mathematical descriptions and
techniques”
[McRobie and Lasenby, 1999]
Vitor Fernando Pamplona

2. Review: Geometric Product
• Outer product spans e2
2D
 1 e1 ∧  2 e2 = 1  2 e1 e2
• Inner product projects O e1
 12 e1 ∧ e2 ⋅ e2 =  12 e1 e1 ∧ e2
• Geometric Product
vu = v ⋅u  v ∧ u
Inner product Outer product

3. Review: Multivector
• Unique n­dimensional structure
vℜ 3 =  Scalar
  e1   e2   e3 Vector
  e1 e2   e1 e3   e2 e3 2­Blade
  e1 e2 e3 3­Blade
mv 3 = 3 e1 ∧ e3 − 2 e2 ∧ e3

4. Overview
• Homogeneous Model
• Conformal Model
• Meet / Join
• Framework Gaigen
• nD­Voronoi Diagram
• CliffoSor: GA processor
• Physical Modelling using GA
• Future Readings

5. Homogeneous Model
• Generalization to homogeneous coordinates of VA
• Affine and projective transformations
• Adds one dimension to working space
• There are no changes in basic operations
e0 e0
p p
p q P
P Q
R² R²
pΛq

6. Conformal Model
• Adds 2 new orthogonal dimensions e0 e∞
• It is a Minkowski space.
• Changes some operations e2 =1
∞ e2=−1
0 e∞⋅e0=0
• Euclidean to Conformal GA Model
a =0, ∞ ,  x ,  y , z =1, 1/2 a 2 ,a x ,a y ,a z
C= p11∧p22∧p333∧ e∞
E =p ∧ p2∧ p ∧ p4
N 1 ∧p ∧p

7. Meet and Join

8. Framework Gaigen
• Geometric Algebra C/C++ Code Generator
• Version 1.0 released 1,5 years ago.
• Generates the same code of GAP framework.
• 8­Dimensional Limit.
• Free Software :: http://gaigen.sourceforge.net
• From Daniel Fontijne at University of Amsterdan
• Version 2.0 will generate Java code too.

9. Why an nD­Voronoi Diagram?
• Raytracer benchmark
[Fontijne, D. & Dorst,2003]
Model Implem. Full Rend Time (s) Memory (MB)
3DLA Standard 1.00 6.2
3DGA Gaigen 2.56 6.7
4DLA Standard 1.05 6.4
4DGA Gaigen 2.97 7.7
5DGA Gaigen 5.71 9.9
• 3DLA: Linear Algebra
• 3DGA: Geometric Algebra
• 4DLA: Homogeneous coordinates
• 4DGA: Homogeneous model
• 5DGA: Conformal model

10. Edelsbrunner's idea
• Algorithm
VoronoiDiagram edelsbrunner(Points S)
O
1. Create a new orthogonal dimension
2. Build the paraboloid R
n− 1 http://www.ics.uci.edu/~eppstein/junkyard/nn.html
2
x n= ∑ x i
i= 0
3. For each point P
4. P' = projection of P on the paraboloid R
5. H = the hyperplane tangent to paraboloid R in P'
6. End for
7. Project the upper­envelope of H's on initial dimension.

11. Edelsbrunner with Geometric Algebra
1. D+2 Geometric Algebra Solution ... edelsbrunner(Points S)
• e
Edelsbrunner paraboloid ∞ 1. New dimension
Homogeneous model e 0
2. Build paraboloid R
•
3. For each point p
4. p'= proj. of p on R
5. H= tangent to R in p'
2. Step 2 is not needed
6. End for
7. Proj. the upper­envelope
4. Project p on R
p 
p '=  p 2 e∞

  2 p 2 e∞
p
H T= p '∧d u a l  ∧
p
∣∣∗ 1 4 p 2
5. Find hyperplane tangent p 

12. Edelsbrunner with Geometric Algebra
7. Project the upper­envelope of ... edelsbrunner(Points S)
hyperplanes on the d­2 1. New dimension
dimension. 2. Build paraboloid R
3. For each point p
backTracking(hypercube F) 4. p'= proj. of p on R
1. if (lowestGrade(F) == 2) 5. H= tangent to R in p'
2. F=meet(F,each out of stack HP) 6. End for
3. add lowestGrade(F) to Voronoi. 7. Proj. the upper­envelope
4. else
5. for each out of stack HP
Flag: Hierarchy of half­
6. F=meet(F, HP)
7. BackTracking(F)
spaces with the lowest
8. F = removeLowestGrade(F) level is a pair of points.
backTracking(hypercube bounding box)

13. The Benchmark
42 points in 2D 16 points in 2D
Qhull 0.004 + /­ 0.001 0.003 + /­ 0.000
Zaharia 32.127 + /­ 0.546 1.512 + /­ 0.005
Pam plona 0.592 + /­ 0.006 0.080 + /­ 0.004
●
Time in seconds
●
Each test was performed 5 times
●
Test Architecture:
●
Processor: Athlon 64, 2800+, 1800Mhz, 97% Free
●
Memory: 1 GB RAM / 600 MB Free
●
OS: Gentoo Linux, install from Stage 1.
●
Qhull version: 3.1­r1
●
Zaharia GAP version

14. Problems
●
Precision problems
●
Work with high precision numbers: 10­16 ... 10 8
●
Easy to get Double.NAN
●
Easy to get unreported edges
●
Infinity edges, no bounding box
●
Work with higher dimensions?
●
Need backtracking algorithm?
●
Search Hyperplane neighbors?
●
Kd­trees, Quadtrees, Octrees

15. CliffoSor
• Parallel Embedded Architecture for GA.
• Multiplications, additions in 4D and 3D rotations.
• Operations in Parallel
• Gaigen 1.0 vs CliffoSor
– Gaigen: 256 clock cycles
– CliffoSor: 64 clock cycles
– Test made with 500.000 geometric products.

16. Physical Modelling using GA
• Particle Dynamics without GA
• Rigid Body Dynamics with GA
• Collision Detection with GA
– Bounding Spheres
– Segment Triangle Intersection
• Collision Response with GA
• Conclusions
– Mathematics simplification in the collision detection
– Needs experience with GA

17. Physical Modelling: Future Works
• Hardware parallelism
– outer, inner and geometric products
• Advanced Collision Detection:
– Oct­trees
– Sphere­trees

18. Future Works
• CGA in Computer Vision
• Clifford Fourier Transform
• Derivatives and Integrals
• Generalization of Quaternions
• Inverse Kinematics with dual of Quaternions
• Interpolation with CGA
• Mesh deformation

19. References
• Dorst, L. & Mann, S. Geometric algebra: a computational framework for geometrical
applications (part II: aplications) IEEE Computer Graphics and Applications, 2002, 1
• Dorst, L. & Mann, S. Geometric algebra: a computational framework for geometrical
applications (part I: algebra) IEEE Computer Graphics and Applications, 2002, 1, 24­31
• Fontijne, D. & Dorst, L. Modeling 3D Euclidean Geometry IEEE Computer Graphics and
Applications, 2003
• Macdonald, A. A Survey of Geometric Algebra and Geometric Calculus, 2005
• Vaz, J.J. A álgebra geométrica do espaço euclidiano e a teoria de Pauli Revista
Brasileira de Ensino de Física, 1997, 19, 234­259
• Zaharia, M.D. & Dorst, L. The Interface Spec. and Implementation Internals of a
Program Module for Geometric Algebra University of Amsterdam, 2003