geometric algebra computing - gaalop · 2014-01-22 · chapter 3 „an interactive introduction to...
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![Page 1: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/1.jpg)
Dr. Dietmar Hildenbrand
Technische Universität Darmstadt
Geometric Algebra ComputingMathematical Introduction16.11.2012
![Page 2: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/2.jpg)
Montag, 19.11.2012 fällt die Übung aus
-> individuelle Besprechungstermine
16.11.2012 | Technische Universität Darmstadt | Dietmar Hildenbrand | 2
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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
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Nachtrag (CLUCalc)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 4
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Nachtrag
� Zwei identische Kugeln?
� Kugeln mit identischem Mittelpunkt, aber unterschiedlichen Radien?
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 5
![Page 6: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/6.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 6
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
![Page 7: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/7.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 7
Overview
� Calculations in 3D euclidean GA
� The sine rule
� Calculations in 5D conformal GA
� Reflection/projection in both spaces
3D 5D
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DAGM-Tutorial
� Start.clu …
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 8
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16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 9
Calculations in 3D Euclidean GA
![Page 10: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/10.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 10
The blades of 3D euclidean geometric algebra
![Page 11: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/11.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 11
The main products of geometric algebra
� Outer Product
vector bivector trivector
� Inner Product
� Geometric Product
![Page 12: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/12.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 12
Properties of the outer product
![Page 13: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/13.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 13
Properties of the outer product
� Note: the outer product can be used as a measure of parallelness
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16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 14
Example bivectorE3.clu
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16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 15
Computation example
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16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 16
The outer product of 2 vectors in 3D
321
321
211332
211221131331322332
333323231313
323222221212
313121211111
332211332211
)()()(
)()(
bbb
aaa
eeeeee
eebabaeebabaeebaba
eebaeebaeeba
eebaeebaeeba
eebaeebaeeba
ebebebeaeaeaba
∧∧∧
=
∧−+∧−−∧−=
∧+∧+∧+
∧+∧+∧+
∧+∧+∧=
++∧++=∧
![Page 17: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/17.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 17
trivectorE3.clu
![Page 18: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/18.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 18
trivectorE3.clu
![Page 19: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/19.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 19
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
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16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 20
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
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16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 21
Reverse, norm of subspaces
12...~
~||||
aaaA
with
AAA
k∧∧∧=
⋅=
Example:
In CLUCalc:
![Page 22: Geometric Algebra Computing - Gaalop · 2014-01-22 · Chapter 3 „An Interactive Introduction to Geometric Algebra Kapitel 1 „Introductions to Clifford Algebra“ und 2 „Geometries“](https://reader033.vdokument.com/reader033/viewer/2022042914/5f4e81c44769377874127916/html5/thumbnails/22.jpg)
16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 22
The inner product and perpendicularity
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16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 23
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
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16.11.2012 | Technische Universität Darmstadt | Computer Science Department | Dietmar Hildenbrand | 24
Thanks for your attention