# Introduction to Geometric Algebra

Professors - Leandro A. F. Fernandes, UFRGS (Home Page), and Manuel M. Oliveira, UFRGS (Home Page)

Abstract - Geometric problems in visual computing (computer graphics, computer vision, and image processing) are typically modeled and solved using linear algebra (LA). Thus, vectors are used to represent directions and points in space, while matrices are used to model transformations. LA, however, presents some wellknown limitations for performing geometric computations. As a result, one often needs to aggregate different formalisms (e.g., quaternions and Plücker coordinates) to obtain complete solutions. Unfortunately, such extensions are not fully compatible among themselves, and one has to get used to jumping back and forth between formalisms, filling in the gaps between them. Geometric algebra (GA), on the other hand, is a mathematical framework that naturally generalizes and integrates useful formalisms such as complex numbers, quaternions and Plücker coordinates into a high-level specification language for geometric operations. Due to its consistent structure, GA equations are often universal and generally applicable. They extend the same solution to higher dimensions and to all kinds of geometric elements, without having to handle special cases, as it happens in conventional techniques. This lecture series aims at introducing the fundamental concepts of GA as a powerful mathematical tool to describe and solve geometric problems in visual computing.

Lecture I (Slides)

• Subspaces
• Multivector space
• Some non-metric products

Lecture II (Slides)

• Metric spaces
• Some inner products
• Dualization and undualization

Lecture III (Slides)

• Duality relationships between products
• Some non-linear products

Lecture IV (Slides)

• Geometric product
• Versors
• Rotors

Lecture V (Slides)

• Models of geometry
• Euclidean vector space model
• Homogeneous model

Lecture VI (Slides)

• Conformal model
• Concluding remarks

Extra I (Slides)

• Tensor representation of products
• Matrix notation of products

Extra II (Slides)

• Implementation approaches
• Libraries and toolkits

Reference Material

L. Dorst, D. Fontijine, and S. Mann, Geometric algebra for computer science: an object oriented approach to geometry. Morgan Kaufmann Publishers, 2007. (Home Page)

C. Perwass, Geometric algebra with applications in engineering. Springer Publishing Company, 2009. (Home Page)

G. Sommer, Geometric computing with Clifford algebras. Springer Publishing Company, 2001. (Home Page)

L. A. F. Fernandes, M. M. Oliveira, "Geometric algebra: a powerful tool for solving geometric problems in visual computing," Tutorials of Sibgrapi 2009 (XXII Brazilian Symposium on Computer Graphics and Image Processing), Rio de Janeiro, Brazil, 2009. (Manuscript, Supplementary Material A, Supplementary Material B, Slides)

L. Dorst, S. Mann, "Geometric algebra: a computational framework for geometrical applications, Part 1," IEEE Computer Graphics and Applications, vol. 22, no. 3, pp. 24-31, 2002. (DOI Bookmark)

S. Mann, L. Dorst, "Geometric algebra: a computational framework for geometrical applications, Part 2," IEEE Computer Graphics and Applications, vol. 22, no. 4, pp. 58-67, 2002. (DOI Bookmark)

D. Hildenbrand, D. Fontijne, C. Perwass, L. Dorst, "Geometric algebra and its application to computer graphics ," Tutorial 3 of Eurographics 2004, Grenoble, France, 2004. (Lecture Notes)