Rotation of subspaces How to rotate vector a in the plane by radians. 55
Rotation of subspaces How to rotate vector a in the plane by radians. 56
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. 57
Versor product for general multivectors Grade Involution The sign change under the grade involution exhibits a + - + - + - … pattern over the value of t . 58
Checkpoint The geometric product is the most fundamental product of geometric algebra It is an invertible product The other linear products can be derived from it Versors encode linear transformations Reflections Rotations (rotors) Rotors generalize quaternions to n -D spaces 59
Coffee Break 60
Checkpoint Module I Subspaces, the outer product, and the multivector space The resulting subspace is a primitive for computation! 61
Checkpoint Module I Subspaces, the outer product, and the multivector space Scalars Vector Space Bivector Space Trivector Space 62
Checkpoint Module I Subspaces, the outer product, and the multivector space Module II Metric and some inner products The inner product of vectors The scalar product The left contraction 63
Checkpoint Module I Subspaces, the outer product, and the multivector space Module II Metric and some inner products Module III Geometric product 64
Checkpoint Module I Subspaces, the outer product, and the multivector space Module II Metric and some inner products Module III Geometric product Module IV Orthogonal transformations as versors 65
Module V Models of Geometry 66
Geometric Algebra models Assumes a metric to the space Gives a geometrical interpretation to subspaces Directions, points, straight lines, circles, etc. Makes versors behave like some transformation type Scaling, rotation, translation, etc. 67
Euclidean vector space model Euclidean metric matrix 68
Euclidean vector space model Euclidean metric matrix Subspaces k -D directions Versors Reflections Rotations 69
2-D Working Space Homogeneous model One extra basis vector interpreted as point at origin Euclidean metric matrix 70
2-D Working Space Homogeneous model One extra basis vector interpreted as point at origin Euclidean metric matrix How Stuff Works 71
2-D Working Space Homogeneous model One extra basis vector interpreted as point at origin Euclidean metric matrix Subspaces Directions How Stuff Works 72
2-D Working Space Homogeneous model One extra basis vector interpreted as point at origin Euclidean metric matrix Subspaces Directions Flats (point, line, plane) How Stuff Works 73
2-D Working Space Homogeneous model One extra basis vector interpreted as point at origin Euclidean metric matrix Subspaces Directions Flats (point, line, plane) How Stuff Works 74
Homogeneous model One extra basis vector interpreted as point at origin Euclidean metric matrix Subspaces Directions Flats (point, line, plane) Versors Rotations around the origin 75
2-D Working Space Conformal model Two extra basis vectors Point at origin Point at infinity Non-Euclidean metric 76
2-D Working Space Conformal model Two extra basis vectors Point at origin Point at infinity Non-Euclidean metric How Stuff Works 77
2-D Working Space Conformal model Subspaces Points How Stuff Works 78
2-D Working Space Conformal model Subspaces Points Rounds (point pair, circle, sphere) How Stuff Works 79
2-D Working Space Conformal model Subspaces Points Rounds (point pair, circle, sphere) Flats (flat point, line, plane) How Stuff Works 80
2-D Working Space Conformal model Subspaces Tangent (point, tangent k -D direction) Rounds (point pair, circle, sphere) Flats (flat point, line, plane) How Stuff Works 81
2-D Working Space Conformal model Subspaces Tangent (point, tangent k -D direction) Rounds (point pair, circle, sphere) Flats (flat point, line, plane) Free ( k -D direction) How Stuff Works 82
Conformal model Subspaces Tangent (point, tangent k -D direction) Rounds (point pair, circle, sphere) Flats (flat point, line, plane) Free ( k -D direction) Versors Reflections 83
Conformal model Subspaces Tangent (point, tangent k -D direction) Rounds (point pair, circle, sphere) Flats (flat point, line, plane) Free ( k -D direction) Versors Reflections Rotations 84
Conformal model Subspaces Tangent (point, tangent k -D direction) Rounds (point pair, circle, sphere) Flats (flat point, line, plane) Free ( k -D direction) Versors Reflections Rotations Translations 85
Conformal model Subspaces Tangent (point, tangent k -D direction) Rounds (point pair, circle, sphere) Flats (flat point, line, plane) Free ( k -D direction) Versors Reflections Rotations Translations Isotropic scaling 86
Module VI History Application in Visual Computing Concluding Remarks 87
Why I never heard about GA before? Geometric algebra is a “new” formalism Paul Dirac (1902-1984) William R. Hamilton Hermann G. Grassmann William K. Clifford David O. Hestenes (1805-1865) (1809-1877) (1845-1879) (1933-) Wolfgang E. Pauli (1900-1958) 1980s, 2001 1920s 1840s 1877 1878 D. Hestenes (2001) Old wine in new bottles..., in Geometric algebra with 88 applications in science and engineering, chapter 1, pp. 3-17, Birkhäuser, Boston.
Where geometric algebra has been applied Perwass (2004) detect corners, line segments, lines, crossings, y-junctions and t-junctions in images C. B. U. Perwass ( 2004) Analysis of local image structure using…, Instituts für Informatik 89 und Praktische Mathematik der Universität Kiel, Germany, Tech. Rep. Nr. 0403.
Where geometric algebra has been applied Bayro-Corrochano et al . (1996) analyze the projective structure of n uncalibrated cameras E. Bayro-Corrochano, et al. ( 1996 ) Geometric algebra: a framework for computing point 90 and line correspondences and projective…, in Proc. of the 13th ICPR, pp. 334– 338.
Where geometric algebra has been applied Rosenhahn and Sommer (2005) define 2-D/3-D pose estimation of different corresponding entities B. Rosenhahn, G. Sommer (2005) Pose estimation in conformal geometric algebra part II: 91 real- time pose estimation using…, J. Math. Imaging Vis., 22:1, pp. 49– 70.
Where geometric algebra has been applied Hildenbrand et al. (2005) apply the conformal model on the inverse kinematics of a human-arm-like robot D. Hildenbrand et al. (2005), Advanced geometric approach for graphics and visual 92 guided robot object manipulation, in Proc. of the Int. Conf. Robot. Autom., pp. 4727 – 4732.
Where geometric algebra has been applied Jourdan et al. (2004) perform automatic tessellation of quadric surfaces F. Jourdan et al. (2004), Automatic tessellation of quadric surfaces using Grassmann- 93 Cayley algebra, in Proc. Int. Conf. Comput. Vis. Graph., pp. 674 – 682.
Where geometric algebra has been applied Lasenby et al. (2006) model higher dimensional fractals J. Lasenby et al. (2006), Higher dimensional fractals in geometric algebra, Cambridge 94 University Engineering Department, Tech. Rep. CUED/F-INFENG/TR.556.
Drawbacks There are some limitations yet Versors do not encode all projective transformations Projective Affine Similitude Linear Rigid / Euclidean Scaling Identity Isotropic Scaling Reflection Translation Rotation Shear Perspective 95
Drawbacks There are some limitations yet Versors do not encode all projective transformations Efficient implementation of GA is not trivial Multivectors may be big (2 n coefficients) Storage problems Numerical instability Custom hardware is optimized for linear algebra There is an US patent on the conformal model A. Rockwood, H. Li, D. Hestenes (2005) System for encoding and 96 manipulating models of objects, U.S. Patent 6,853,964.
Concluding remarks Consistent framework for geometric operations Geometric elements as primitives for computation Geometrically meaningful products Extends the same solution to Higher dimensions All kinds of geometric elements An alternative to conventional geometric approach It should contribute to improve software development productivity and to reduce program errors 97
How to start Geometric Algebra for Geometric Algebra with Computer Science Applications in Engineering L. Dorst – D. Fontijne – S. Mann C. Perwass Morgan Kaufmann Publishers (2007) Springer Publishing Company (2009) 98
Questions? θ 99
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