Introduction to Geometric Algebra Lecture IV

Leandro A. F. Fernandes Manuel M. Oliveira [email protected] [email protected]

Visgraf - Summer School in Computer Graphics - 2010

CG UFRGS Lecture IV Checkpoint

Visgraf - Summer School in Computer Graphics - 2010 2 Checkpoint, Lecture I

 Multivector space

 Non-metric products . The outer product . The regressive product

c

Visgraf - Summer School in Computer Graphics - 2010 3 Checkpoint, Lecture II

 Metric spaces . Bilinear form defines a metric on the vector space, e.g., Euclidean metric . Metric matrix

 Some inner products The scalar product is a particular . Inner product of vectors case of the left and right contractions . Scalar product . Left contraction These metric products are . Right contraction backward compatible for 1-blades

Visgraf - Summer School in Computer Graphics - 2010 4 Checkpoint, Lecture II

 Dualization

 Undualization

By taking the undual, the dual representation of a blade can be correctly mapped back to its direct representation Venn Diagrams

Visgraf - Summer School in Computer Graphics - 2010 5 Checkpoint, Lecture III

 Duality relationships between products . Dual of the outer product

. Dual of the left contraction

Visgraf - Summer School in Computer Graphics - 2010 6 Checkpoint, Lecture III

 Some non-linear products . Meet of blades

. Join of blades

. Delta product of blades

Visgraf - Summer School in ComputerVenn GraphicsDiagrams - 2010 7 Today

 Lecture IV – Mon, January 18 . Geometric product . Versors . Rotors

Visgraf - Summer School in Computer Graphics - 2010 8 Lecture IV Geometric Product

Visgraf - Summer School in Computer Graphics - 2010 9 Geometric product of vectors

Denoted by a white space, like Inner Product Outer Product standard multiplication

Unique Feature An invertible product for vectors!

Visgraf - Summer School in Computer Graphics - 2010 10 Geometric product of vectors

The Inner Product is not Invertible

Unique Feature An invertible product for vectors!

γ

** Euclidean Metric

Visgraf - Summer School in Computer Graphics - 2010 11 Geometric product of vectors

The Outer Product is not Invertible

Unique Feature An invertible product for vectors!

Visgraf - Summer School in Computer Graphics - 2010 12 Geometric product of vectors

Unique Feature An invertible product for vectors!

Inverse geometric product, denoted by a slash, like standard division

Visgraf - Summer School in Computer Graphics - 2010 13 Intuitive solutions for simple problems

q

r ?

p

t

** Euclidean Metric

Visgraf - Summer School in Computer Graphics - 2010 14 Geometric product and multivector space

 The geometric product of two vectors is an element of mixed dimensionality

Scalars Vector Space Bivector Space Trivector Space

Visgraf - Summer School in Computer Graphics - 2010 15 Geometric product and multivector space

 The geometric product of two vectors is an element of mixed dimensionality

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 16 Properties of the geometric product

Scalars commute

Distributivity

Associativity

Neither fully symmetric nor fully antisymmetric

Visgraf - Summer School in Computer Graphics - 2010 17 Lecture IV Geometric product of basis blades

Visgraf - Summer School in Computer Graphics - 2010 18 Geometric product of basis blades

 Let’s assume an orthogonal metric, i.e.,

Kronecker delta function With an orthogonal metric, there are two cases to be handled Metric factor

Visgraf - Summer School in Computer Graphics - 2010 19 Geometric product of basis blades

 Let’s assume an orthogonal metric, i.e.,

. Case 1: blades consisting of different orthogonal factors

The geometric product is equivalent to the outer product . Case 2: blades with some common factors

The dependent-basis factors are replaced Visgraf - Summer School in Computer Graphics - 2010 by metric factors 20 Geometric product of basis blades

 Let’s assume a non-orthogonal metric, e.g.,

Apply the spectral theorem from ** The spectral theorem states that a linear algebra and reduce the problem matrix is orthogonally diagonalizable to the orthogonal metric case if and only if it is symmetric.

Visgraf - Summer School in Computer Graphics - 2010 21 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

Visgraf - Summer School in Computer Graphics - 2010 22 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 See the Supplementary Material A of the • The eigenvalues specifyTutorial the atnew Sibgrapi metric 2009 for a purest 4. Get back to the originaltreatment basis of the geometric product • Apply a change of basis using the original eigenvector matrix

Visgraf - Summer School in Computer Graphics - 2010 23 Lecture IV Subspace Products from Geometric Product

Visgraf - Summer School in Computer Graphics - 2010 24 The most fundamental product of GA

 The subspace products can be derived from the geometric product The “grade extraction” operation extracts grade parts from multivector

A general multivector variable in

Visgraf - Summer School in Computer Graphics - 2010 25 The most fundamental product of GA

 The subspace products can be derived from the geometric product

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 26 Lecture IV Orthogonal Transformations as Versors

Visgraf - Summer School in Computer Graphics - 2010 27 Reflection of vectors

Mirror

Input vector

Vector a was reflected in vector v, resulting in vector a´

Visgraf - Summer School in Computer Graphics - 2010 28 k-Versor

V is a k-versor. It is computed as the geometric product of k invertible vectors.

Visgraf - Summer School in Computer Graphics - 2010 29 Rotation of subspaces

How to rotate vector a in the plane by radians.

Visgraf - Summer School in Computer Graphics - 2010 30 Rotation of subspaces

How to rotate vector a in the plane by radians.

Visgraf - Summer School in Computer Graphics - 2010 31 Rotation of subspaces

Rotors How to rotate vector a in Unit versors encoding rotations. the plane by radians. They are build as the geometric product of an even number of unit invertible vectors.

Visgraf - Summer School in Computer Graphics - 2010 32 Versor product for general multivectors

Grade Involution The sign change under the grade involution exhibits a + - + - + - … pattern over the value of t.

Visgraf - Summer School in Computer Graphics - 2010 33 The structure preservation property

The structure preservation of versors holds for the geometric product, and hence to all other products in geometric algebra. The ○ symbol represents any product of geometricof geometric algebra, algebra and, as a consequence, any operation defined from the products

Visgraf - Summer School in Computer Graphics - 2010 34 Inverse of versors The inverse of versors is computed as for the inverse of invertible blades

Squared (reverse) norm Inverse

Reverse (+ + – – + + – – … pattern over k)

The norm of rotors is equal to one, so the inverse of a rotor is its reverse

Visgraf - Summer School in Computer Graphics - 2010 35 Multivector classification

 It can be used for blades and versors . Use Euclidean metric for blades . Use the actual metric for versors

 Test if 푀 푀 푀 is truly the inverse of the multivector

−1 −1 −1 grade 푀 푀 = 0 푀 푀 = 푀 푀

 Test the grade preservation property

grade 푀 퐞푖 푀 = 1

 If the multivector is of a single grade then it is a blade; otherwise it is a versor

Visgraf - Summer School in Computer Graphics - 2010 36 Credits

William R. Hamilton Hermann G. Grassmann William K. Clifford (1805-1865) (1809-1877) (1845-1879)

W.Grassmann, R.Clifford, Hamilton W. H. K. (1844) G. (1878) (1877) On Applications aVerwendung new species of derGrassmann's of imaginaryAusdehnungslehre extensivequantities fur algebraconnected die allgemeine. Am. with J. theMath. Theorie theory, der Polarenof quaternionsund denWalter Zusammenhang. Inde Proc. Gruyter of theUnd algebraischer Royal Co., vol. Irish 1, Gebilde Acad.n. 4, 350, .vol. J.-358 Reine 2, 424 Angew.-434 Math. (Crelle's J.), Walter de Gruyter Und Co., 84, 273-283

Visgraf - Summer School in Computer Graphics - 2010 37 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

Visgraf - Summer School in Computer Graphics - 2010 38