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Geometric Algebra Computing 23.10.2014 Dr.-Ing. Dietmar Hildenbrand Technische Universität Darmstadt

Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

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Page 1: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Geometric Algebra Computing

23.10.2014

Dr.-Ing. Dietmar HildenbrandTechnische Universität Darmstadt

Page 2: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Overview

� Foundations of Geometric Algebra (GA)

� CLUCalc tutorial (3D-GA)

� GA Applications

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Page 3: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

History of Geometric Algebra&Calculus

� [David Hestenes 2001]

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Page 4: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

� [David Hestenes 2001]

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

History of Geometric Algebra&Calculus

Page 5: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

� Outer Product

vector bivector trivector …

� Inner Product

Hermann G. Grassmann

Page 6: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

� Outer Product

vector bivector trivector …

� Inner Product

cross product and scalar product are

special cases of these general products

Hermann G. Grassmann

Page 7: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Hermann G. Grassmann

� Vorwort zur Ausdehnungslehre von 1862:

Page 8: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Hermann G. Grassmann

� Preface of „Ausdehnungslehre“ (Extensive Algebra) of 1862:

"I remain completely confident that the labor I have expended on the science presented here and which has demanded a signicant part of my life, as well as the most strenuous application of my powers, will not be lost. It is true that I am aware that the form which I have given the science is imperfect and must be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if this work should again remain unused for another seventeen years or even longer, without entering into the actual development ofscience, still that time will come when it will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit." and he went on to say:"there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into a living communication with contemporary developments."

Page 9: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Hermann G. Grassmann

� Preface of „Ausdehnungslehre“ (Extensive Algebra) of 1862:

"I remain completely confident that the labor I have expended on the science presented here and which has demanded a signicant part of my life, as well as the most strenuous application of my powers, will not be lost. It is true that I am aware that the form which I have given the science is imperfect and must be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if this work should again remain unused for another seventeen years or even longer, without entering into the actual development ofscience, still that time will come when it will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit." and he went on to say:"there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into a living communication with contemporary developments."

Page 10: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Hermann G. Grassmann

� Preface of „Ausdehnungslehre“ (Extensive Algebra) of 1862:

"I remain completely confident that the labor I have expended on the science presented here and which has demanded a signicant part of my life, as well as the most strenuous application of my powers, will not be lost. It is true that I am aware that the form which I have given the science is imperfect and must be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if this work should again remain unused for another seventeen years or even longer, without entering into the actual development ofscience, still that time will come when it will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit." and he went on to say:"there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into a living communication with contemporary developments."

Page 11: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

William K. Clifford

� Geometric Product

� For vectors

� Quaternions of Hamilton

� Normally called Clifford algebra in honor of Clifford

� He called it geometric algebra

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Page 12: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

David Hestenes

� realized geometric algebra as a general language for physics

(„New Foundations of Classical Mechanics“, …)

� developed calculus

("Clifford Algebra to Geometric Calculus:

A Unified Language for

Mathematics and Physics")

� developed the Conformal Geometric Algebra

� Geometric Algebra <-> Clifford Algebra?

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Page 13: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

3D CLUCalc Tutorial

� Chapter 6.1 – 6.3 of „Foundations of Geometric Algebra Computing“

16.10.2014 | Technische Universität Darmstadt | Dietmar Hildenbrand

Page 14: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Goal of geometric algebra

� Mathematical language

close to the geometric

intuition

Page 15: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Conformal Geometric Algebra (CGA)

�The 5 basis vectors:

Page 16: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Basis elements of CGA

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Page 17: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Plane and sphere as vector expression

� sphere

� plane

Page 18: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Intersection of geometric objects

� a,b are sphere

� a ^ b

� describes the intersection of the spheres

� represents a circle

Page 19: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Intersection of sphere and line

Page 20: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

23.10.2014 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 20

Circle and line

�Circle (a^b^c)

�Circle -> Line

( )

Page 21: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Geometric Operations

� Rotation about L (through theorigin)

� Rotor

= Quaternion

� Note.: the line can be arbitrary-> Rotation about an arbitraryaxis(dual quaternion)

Page 22: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

The inner product

� a,b 3D vectors

-> scalar product

Page 23: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Distances

One formula for

� distance of two points

� distance between point and plane

� is a point inside or outside of a sphere?

Page 24: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Angles

One formula for the angle between

� 2 lines

� 2 planes

� 2 circles

� …

Page 25: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Properties of Conformal Geometric Algebra

� Easy calculations with geometric objects and transformations

� Geometric intuitiveness

� Simplicity

� Compactness

� Unification of mathematical systems

� Complex numbers

� Vector algebra

� Quaternions

� Projective geometry

� Plücker coordinates

� ….

Page 26: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

First commercial product

� since 2007:

� Real time lighting engine Enlighten

� Geomerics, Cambridge UK

� Unreal Game Engine

Page 27: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Our Applications

� Robot kinematics

[Craig]

Page 28: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Application

� Computer animation/

Virtual reality

[Tolani et al. 2000]

Page 29: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Application

� Approximation of geometric objects to point clouds of laser

scans

Page 30: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Application

� Finite Element Solver

� compact expressions for

� velocities

� forces

� combining rotational and linear

parts

[Elmar Brendel et al]

Page 31: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Application

� Registration of multichannel images

Page 32: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Application

� Catadioptric cameras

[Perwass, 2006]

Page 33: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Application

� Pose estimation

[Perwass, 2006]

Page 34: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

Dietmar Hildenbrand | "Foundations of Geometric Algebra Computing" | TU Darmstadt

Application

� Molecular dynamics simulation

Page 35: Geometric Algebra Computing - Gaalop · 2014-10-23 · Geometric Algebra Computing 23.10.2014 Dr.-Ing. ... realized geometric algebra as a general language for physics ... ("Clifford

16.10.2014 | Technische Universität Darmstadt | Dietmar Hildenbrand

Thanks a lot …