Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“

Dr. Dietmar Hildenbrand

Technische Universität Darmstadt

Geometric Algebra ComputingMathematical Introduction16.11.2012

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16.11.2012 | Technische Universität Darmstadt | Dietmar Hildenbrand | 2

Nachtrag

� Schnitt von Kugel und Kreis

� Was passiert, wenn der Kreis auf der Kugel liegt?

(KugelMalKreis.clu)

16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 3

Nachtrag (CLUCalc)


Nachtrag

� Zwei identische Kugeln?

� Kugeln mit identischem Mittelpunkt, aber unterschiedlichen Radien?



Literature

� [1] Christian Perwass and Dietmar Hildenbrand� “Aspects of Geometric Algebra in Euclidean, Projective

and Conformal Space”, Tutorial auf der DAGM 2003, Stand 14. Jan. 2004

� Chapter 3 „An Interactive Introduction to GeometricAlgebra

� Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“

� [2] Christian Perwass,� “Geometric Algebra with Applications in Engineering”,

� Springer 2009

� [3] John Vince� „Geometric Algebra: An Algebraic System for

Computer Games and Animation“, Springer, 2009

� Chapter 9


Overview

� Calculations in 3D euclidean GA

� The sine rule

� Calculations in 5D conformal GA

� Reflection/projection in both spaces

3D 5D

DAGM-Tutorial

� Start.clu …



Calculations in 3D Euclidean GA


The blades of 3D euclidean geometric algebra


The main products of geometric algebra

� Outer Product

vector bivector trivector

� Inner Product

� Geometric Product


Properties of the outer product


Properties of the outer product

� Note: the outer product can be used as a measure of parallelness


Example bivectorE3.clu


Computation example


The outer product of 2 vectors in 3D

321

321

211332

211221131331322332

333323231313

323222221212

313121211111

332211332211

)()()(

)()(

bbb

aaa

eeeeee

eebabaeebabaeebaba

eebaeebaeeba

eebaeebaeeba

eebaeebaeeba

ebebebeaeaeaba

∧∧∧

=

∧−+∧−−∧−=

∧+∧+∧+

∧+∧+∧+

∧+∧+∧=

++∧++=∧


trivectorE3.clu


trivectorE3.clu


The outer product of 3 vectors in 3D

321

321

321

321

321123213

321132312

321231321

332211332211332211

)(

)(

)(

)()()(

eee

ccc

bbb

aaa

eeecbacba

eeecbacba

eeecbacba

ecececebebebeaeaeacba

∧∧=

∧∧−+

∧∧−+

∧∧−=

++∧++∧++=∧∧

Note : linearly dependent vectors -> the outer product is 0


The inner product of two vectors

� Inner product = Scalar product

is true only for vectors!

� For vector and bivector:

� General rule in [1] page 6


Reverse, norm of subspaces

12...~

~||||

aaaA

with

AAA

k∧∧∧=

⋅=

Example:

In CLUCalc:


The inner product and perpendicularity


The general inner product

� Inner product not only defined for vectors!

� Example:

� Note: - the resulting vector is perpendicular to x InnerProductE3.clu

� - the inner product is grade decreasing


Thanks for your attention

Documents

Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“