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CG UFRGS
Visgraf - Summer School in Computer Graphics - 2010
Introduction to Geometric Algebra Lecture IV
Leandro A. F. Fernandes laffernandes@inf.ufrgs.br
Manuel M. Oliveira oliveira@inf.ufrgs.br
Visgraf - Summer School in Computer Graphics - 2010
Checkpoint
Lecture IV
2
c
Visgraf - Summer School in Computer Graphics - 2010
Checkpoint, Lecture I
Multivector space
Non-metric products
The outer product
The regressive product
3
Checkpoint, Lecture II
Metric spaces
Bilinear form defines a metric on the
vector space, e.g., Euclidean metric
Metric matrix
Some inner products
Inner product of vectors
Scalar product
Left contraction
Right contraction
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The scalar product is a particular
case of the left and right contractions
These metric products are
backward compatible for 1-blades
Checkpoint, Lecture II
Dualization
Undualization
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Venn Diagrams
By taking the undual, the dual
representation of a blade can be correctly
mapped back to its direct representation
Checkpoint, Lecture III
Duality relationships between products
Dual of the outer product
Dual of the left contraction
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Checkpoint, Lecture III
Some non-linear products
Meet of blades
Join of blades
Delta product of blades
Visgraf - Summer School in Computer Graphics - 2010 Venn Diagrams 7
Today
Lecture IV – Mon, January 18
Geometric product
Versors
Rotors
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Visgraf - Summer School in Computer Graphics - 2010
Geometric Product
Lecture IV
9
Visgraf - Summer School in Computer Graphics - 2010
Geometric product of vectors
10
Inner Product Outer Product
Unique Feature
An invertible product for vectors!
Denoted by a white space, like
standard multiplication
Visgraf - Summer School in Computer Graphics - 2010
Geometric product of vectors
11
Unique Feature
An invertible product for vectors!
The Inner Product is not Invertible
γ
** Euclidean Metric
Visgraf - Summer School in Computer Graphics - 2010
Unique Feature
An invertible product for vectors!
Geometric product of vectors
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The Outer Product is not Invertible
Visgraf - Summer School in Computer Graphics - 2010
Geometric product of vectors
13
Inverse geometric product,
denoted by a slash,
like standard division
Unique Feature
An invertible product for vectors!
Visgraf - Summer School in Computer Graphics - 2010
Intuitive solutions for simple problems
14
t
r
p
q
?
** Euclidean Metric
Visgraf - Summer School in Computer Graphics - 2010
Geometric product and multivector space
The geometric product of two vectors is
an element of mixed dimensionality
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Scalars Vector Space Bivector Space Trivector Space
Visgraf - Summer School in Computer Graphics - 2010
Geometric product and multivector space
The geometric product of two vectors is
an element of mixed dimensionality
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Scalars Vector Space Bivector Space Trivector Space
Geometric Meaning
The interpretation of the resulting
element depends on the operands.
Visgraf - Summer School in Computer Graphics - 2010
Properties of the geometric product
Scalars commute
Distributivity
Associativity
Neither fully symmetric
nor fully antisymmetric
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Geometric product of basis blades
Lecture IV
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Geometric product of basis blades
Let’s assume an orthogonal metric, i.e.,
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Kronecker
delta function
With an orthogonal metric,
there are two cases to be handled
Metric factor
Geometric product of basis blades
Let’s assume an orthogonal metric, i.e.,
Case 1: blades consisting of different orthogonal factors
Case 2: blades with some common factors
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The geometric product
is equivalent to
the outer product
The dependent-basis
factors are replaced
by metric factors
Geometric product of basis blades
Let’s assume a non-orthogonal metric, e.g.,
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Apply the spectral theorem from
linear algebra and reduce the problem
to the orthogonal metric case
** The spectral theorem states that a
matrix is orthogonally diagonalizable
if and only if it is symmetric.
Geometric product of basis blades
For non-orthogonal metrics
1. Compute the eigenvectors and eigenvalues of
the metric matrix
2. Represent the input with respect to the eigenbasis
• Apply a change of basis using the inverse of
the eigenvector matrix
3. Compute the geometric product on this new
orthogonal basis
• The eigenvalues specify the new orthogonal metric
4. Get back to the original basis
• Apply a change of basis using the original eigenvector matrix
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Geometric product of basis blades
For non-orthogonal metrics
1. Compute the eigenvectors and eigenvalues of
the metric matrix
2. Represent the input with respect to the eigenbasis
• Apply a change of basis using the inverse of
the eigenvector matrix
3. Compute the geometric product on this new
orthogonal basis
• The eigenvalues specify the new metric
4. Get back to the original basis
• Apply a change of basis using the original eigenvector matrix
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See the Supplementary Material A of the
Tutorial at Sibgrapi 2009 for a purest
treatment of the geometric product
Subspace Products from
Geometric Product
Lecture IV
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Visgraf - Summer School in Computer Graphics - 2010
The most fundamental product of GA
The subspace products can be derived
from the geometric product
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The “grade extraction” operation
extracts grade parts from multivector
A general multivector
variable in
Visgraf - Summer School in Computer Graphics - 2010
The most fundamental product of GA
The subspace products can be derived
from the geometric product
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Outer product Scalar product
Left contraction Right contraction
Delta product The largest grade such
that the result is not zero
Visgraf - Summer School in Computer Graphics - 2010
Orthogonal Transformations
as Versors
Lecture IV
27
Visgraf - Summer School in Computer Graphics - 2010
Reflection of vectors
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Vector a was reflected in
vector v, resulting in vector a´
Input vector
Mirror
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k-Versor
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V is a k-versor. It is computed
as the geometric product of
k invertible vectors.
Visgraf - Summer School in Computer Graphics - 2010
Rotation of subspaces
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How to rotate vector a in
the plane by radians.
Visgraf - Summer School in Computer Graphics - 2010
Rotation of subspaces
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How to rotate vector a in
the plane by radians.
Visgraf - Summer School in Computer Graphics - 2010
Rotation of subspaces
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Rotors
Unit versors encoding rotations.
They are build as the geometric product
of an even number of unit invertible vectors.
How to rotate vector a in
the plane by radians.
Visgraf - Summer School in Computer Graphics - 2010
Versor product for general multivectors
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Grade Involution
The sign change under the grade involution
exhibits a + - + - + - … pattern over the value of t.
The ○ symbol represents any product
of geometric algebra
The structure preservation property
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The structure preservation of
versors holds for the geometric product,
and hence to all other products
in geometric algebra.
as a consequence, any operation
defined from the products
The ○ symbol represents any product
of geometric algebra, and,
Inverse of versors
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Inverse
Reverse (+ + – – + + – – … pattern over k)
The inverse of versors is
computed as for the inverse of invertible blades
Squared (reverse) norm
The norm of rotors is equal to one,
so the inverse of a rotor is its reverse
Multivector classification
It can be used for blades and versors
Use Euclidean metric for blades
Use the actual metric for versors
Test if is truly the inverse of the multivector
Test the grade preservation property
If the multivector is of a single grade then it is a blade; otherwise it is a versor
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𝑀 𝑀 𝑀
grade 𝑀 𝑀−1 = 0 𝑀 𝑀−1 = 𝑀−1𝑀
grade 𝑀 𝐞𝑖 𝑀 = 1
Credits
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Hermann G. Grassmann
(1809-1877)
Grassmann, H. G. (1877) Verwendung der Ausdehnungslehre fur die allgemeine Theorie
der Polaren und den Zusammenhang algebraischer Gebilde. J. Reine Angew. Math.
(Crelle's J.), Walter de Gruyter Und Co., 84, 273-283
W. R. Hamilton (1844) On a new species of imaginary quantities connected with the theory
of quaternions. In Proc. of the Royal Irish Acad., vol. 2, 424-434
William R. Hamilton
(1805-1865)
Clifford, W. K. (1878) Applications of Grassmann's extensive algebra. Am. J. Math.,
Walter de Gruyter Und Co., vol. 1, n. 4, 350-358
William K. Clifford
(1845-1879)
Differences between algebras
Clifford algebra
Developed in nongeometric directions
Permits us to construct elements by a universal addition
Arbitrary multivectors may be important
Geometric algebra
The geometrically significant part of Clifford algebra
Only permits exclusively multiplicative constructions • The only elements that can be added are
scalars, vectors, pseudovectors, and pseudoscalars
Only blades and versors are important
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