Introduction to Geometric Algebra
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1 Introduction to Geometric Algebra Lecture 6 Intersection and Union of Subspaces Professor Leandro Augusto Frata Fernandes laffernandes@ic.uff.br Lecture notes available in The Blade Factorization Problem Find, for a given blade such that, a set of k vectors Why one may want to factorize a blade To use the factors as input to libraries that cannot handle blades To implement another low-level algorithm (e.g., meet and join products) Good News! We are only concerned with the outer product and consequently are allowed to choose any convenient metric e.g., Euclidean metric Introduction to Geometric Algebra (2013.1) 2 1
2 How to find the factors? One may project candidate vectors onto Introduction to Geometric Algebra (2013.1) 3 How to find the factors? One may project candidate vectors onto All nonzero blades are invertible under Euclidean metric By find k linearly independent vectors a factorization of is found (up to a scale) Introduction to Geometric Algebra (2013.1) 4 2
3 Blade Factorization Input nonzero blade and k > 0 By assuming Euclidean metric The algorithm also works for null blade in the actual metric The output is a scalar value and a set of orthonormal factors in Euclidean metric Introduction to Geometric Algebra (2013.1) 5 The Meet and Join of Blades Introduction to Geometric Algebra (2013.1) 6 3
4 The Meet and Join of Blades Meet of Blades Join of Blades Geometric Meaning The geometric version of intersection and union from set theory. Introduction to Geometric Algebra (2013.1) 7 The Meet and Join of Blades Common Subspace m Introduction to Geometric Algebra (2013.1) 8 4
5 The Meet and Join of Blades Common Subspace γ Introduction to Geometric Algebra (2013.1) 9 Relationships Between Meet and Join Don t worry about the inverse, because meet and join are independent of the particular metric Introduction to Geometric Algebra (2013.1) 10 5
6 Relationships Between Meet and Join This is not the dual relative to the pseudoscalar of the total space, but of the pseudoscalar within which the problem resides. Introduction to Geometric Algebra (2013.1) 11 The Delta Product of Blades Introduction to Geometric Algebra (2013.1) 12 6
7 The Delta Product of Blades Geometric Meaning The symmetric difference of the factors in and. Introduction to Geometric Algebra (2013.1) 13 Computing the Grade of the Meet and Join? Meet Join? Delta Introduction to Geometric Algebra (2013.1) 14 7
8 Tests for Containment This test returns true if and only if the vector. This test returns true if and only if. Introduction to Geometric Algebra (2013.1) 15 Tests for Containment OK Introduction to Geometric Algebra (2013.1) 16 8
9 Computing the Meet and Join of Blades Introduction to Geometric Algebra (2013.1) 17 Some Observations How the algorithm works It starts with a scalar, and build the common subspace by the outer product of potential factors until it arrives at the true meet. Potential factors of the meet They are factors of both input blades They are not factors of the delta product Meet Delta Introduction to Geometric Algebra (2013.1) 18 9
10 Some Observations How the algorithm works It starts with a pseudoscalar, and remove factors from it until the true join is obtained. Factors that should not be in the join They are not factors of the input blades They are factors of the dual of the delta product Dual of Join Dual of Delta Introduction to Geometric Algebra (2013.1) 19 The Algorithm Swap input blades when it is necessary. This may engender an extra sign: 1. Input: blades and, where 2. Compute the dual of the delta product and factorize it in factors 3. Set and 4. For each of the factors : The rejection is a vector that is perpendicular to. a. Compute the projection and the rejection b. If,. If the grade of is the required grade of the meet, then compute the join and break the loop. Otherwise continue with c. If,. If the grade of is the required grade of the join, then computer the meet from the join and break the loop. Otherwise continue with 5. Output: blades and Dual of Delta Introduction to Geometric Algebra (2013.1) 20 10
11 Efficient Factorization and Join of Blades Fontijne, D. (2008) Efficient algorithms for factorization and join of blades. In 3rd International Conference on Applied Geometric Algebras in Computer Science and Engineering, Grimma, Germany 5 to 10 times faster than earlier algorithms The factors are linearly independents, but they are not orthogonal in general Remeber: the meet can be computed from the join Introduction to Geometric Algebra (2013.1) 21 11
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