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.
Transcript: New from BookNet Canada for 2024: BNC SalesData and LibraryData -...
Geometric Algebra Applications and State of the Art
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
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
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
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