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Geometric algebra and computer graphics

Charles G. Gunn, Steven De Keninck
2019 ACM SIGGRAPH 2019 Courses on - SIGGRAPH '19  
These notes from a SIGGRAPH 2019 short course entitled "Geometric algebra for computer graphics" introduce projective geometric algebra (PGA) as a modern framework for this task.  ...  PGA includes vector, quaternion, dual quaternion, and exterior algebras as sub-algebras, simplifying the learning curve and transition path for experienced practitioners.  ...  Acknowledgements Thanks to Steven De Keninck for ganja.js, stimulating conversations and helpful feedback during the preparation of these notes.  ... 
doi:10.1145/3305366.3328099 dblp:conf/siggraph/GunnK19 fatcat:snfapnwxirgihlyhzxkluv4nbi

On the algebraic and geometric foundations of computer graphics

Ron Goldman
2002 ACM Transactions on Graphics  
Projective spaces fail to support computer graphics on two levels: theoretical and computational, geometric and algebraic. Theoretically, projective spaces are compact, nonorientable manifolds.  ...  We shall see that both of these basic problems-theoretical and computational, geometric and algebraic-can be overcome by replacing projective space with Grassmann space.  ...  Computer graphics has nothing to lose geometrically, and a great deal to gain algebraically, from adopting the mathematical framework of Grassmann space.  ... 
doi:10.1145/504789.504792 fatcat:qlb4niq4wzf47koaq5upkzmmdy

Applications of Conformal Geometric Algebra in Computer Vision and Graphics [chapter]

Rich Wareham, Jonathan Cameron, Joan Lasenby
2005 Lecture Notes in Computer Science  
This paper introduces the mathematical framework of conformal geometric algebra (CGA) as a language for computer graphics and computer vision.  ...  Specifically it discusses a new method for pose and position interpolation based on CGA which firstly allows for existing interpolation methods to be cleanly extended to pose and position interpolation  ...  These algorithms could form a useful basis for many applications in computer graphics and computer vision.  ... 
doi:10.1007/11499251_24 fatcat:bcf5mw6fd5b5dex5qe5eov64wy

Inverse Kinematics Computation in Computer Graphics and Robotics Using Conformal Geometric Algebra

Dietmar Hildenbrand, Julio Zamora, Eduardo Bayro-Corrochano
2008 Advances in Applied Clifford Algebras  
We focus on inverse kinematics applications in computer graphics and robotics based on Conformal Geometric Algebra.  ...  The main benefits of using geometric algebra in the virtual reality software Avalon are the easy, compact and geometrically intuitive formulation of the algorithms and the immediate computation of quaternions  ...  We transferred our 3D problem with the help of geometric intuition into the Conformal Geometric Algebra, performed the geometric computing in 5D and got our results back into the real 3D world.  ... 
doi:10.1007/s00006-008-0096-5 fatcat:nucahkjdjnaxjlnlu44tqq53ri

Geometric algebra: a computational framework for geometrical applications. 2

S. Mann, L. Dorst
2002 IEEE Computer Graphics and Applications  
Geometric algebra is a consistent computational framework in which to define geometric primitives and their relationships.  ...  This algebraic approach contains all geometric operators and permits specification of constructions in a totally coordinate-free manner.  ...  Introduction In the usual way of defining geometrical objects in fields like computer graphics, robotics and computer vision, one uses vectors to characterize the constructions.  ... 
doi:10.1109/mcg.2002.1016699 fatcat:obvxbqbicjhjjfkcidkvn2bhie

Geometric Algebra: A computational framework for geometrical applications Part 1

L. Dorst, S. Mann
2002 IEEE Computer Graphics and Applications  
T he traditional method of defining geometrical objects in fields like computer graphics, robotics, and computer vision routinely uses vectors to characterize constructions.  ...  If we could see subspaces as basic computational elements and do direct algebra with them, then algorithms and their implementation would not need to split their cases on dimensionality.  ...  His research interests are in splines and the mathematical foundations of computer graphics.  ... 
doi:10.1109/mcg.2002.999785 fatcat:n3hfg3wmt5deneq7p53wd53h2a

Modeling and visualization of 3D polygonal mesh surfaces using geometric algebra

M.D. Zaharia, L. Dorst
2004 Computers & graphics  
The language of geometric algebra can be used in the development of computer graphics applications.  ...  The paper emphasizes the elegance and generality of the geometric algebra approach referring also to the necessary computational resources. r  ...  Acknowledgements This work was developed as part of the research program ''Geometric algebra, a new foundation for geometric programming'', funded by The Netherlands Organization for Scientific Research  ... 
doi:10.1016/j.cag.2004.04.007 fatcat:rghnrbo64ve4xcam6hwuwauko4

ANALYSIS OF POINT CLOUDS - Using Conformal Geometric Algebra

2008 Proceedings of the Third International Conference on Computer Graphics Theory and Applications   unpublished
This paper presents some basics for the analysis of point clouds using the geometrically intuitive mathematical framework of conformal geometric algebra.  ...  In a nutshell, this paper provides a starting point for shape analysis based on this new, geometrically intuitive and promising technology.  ...  CONFORMAL GEOMETRIC ALGEBRA FOUNDATIONS Geometric Algebra of 3D Space Geometric algebra (GA) uses a dimension independent way of vector multiplication, adding inner and outer products. e i e j = e i  ... 
doi:10.5220/0001094100990106 fatcat:botel7emvjehlbfe5ujzn7jp7q

Preface for Special Issue on Geometric Algebra in Computer Science and Engineering

Dietmar Hildenbrand, Eckhard Hitzer, George Papagiannakis
2017 Advances in Applied Clifford Algebras  
The applications in computer graphics and computer vision apply Conformal Geometric Algebra in the areas of binocular computer vision, object detection in point clouds, construction of Voronoi diagrams  ...  The first workshop on Geometric Algebra in Computer Science and Engineering (GACSE 2016) was held on 28th of June 2016 as part of the 33rd Computer Graphics International conference (CGI 2016) in Heraklion  ... 
doi:10.1007/s00006-017-0799-6 fatcat:lt4vxa4fn5aabdxsw2c3uh7ocq

Engineering Graphics in Geometric Algebra [chapter]

Alyn Rockwood, Dietmar Hildenbrand
2010 Geometric Algebra Computing  
Geometric algebra unites many underpinning mathematical concepts in computer graphics such as vector algebra and vector fields, quaternions, kinematics and projective geometry, and it easily deals with  ...  We illustrate the suitability of geometric algebra for representing structures and developing algorithms in computer graphics, especially for engineering applications.  ...  Some applications Computer graphics, and the related areas of robotics and computer vision are active areas of research in geometric algebra.  ... 
doi:10.1007/978-1-84996-108-0_3 fatcat:wqxobrk44rcihdubqaw2hswhjm

Clipping and Intersection Algorithms: Short Survey and References [article]

Vaclav Skala
2022 arXiv   pre-print
Some algorithms use the projective extension of the Euclidean space and vector-vector operations, which supports GPU and SSE use.  ...  This contribution presents a brief survey of clipping and intersection algorithms in E2 and E3 with a nearly complete list of relevant references.  ...  Acknowledgments The author would like to thank colleagues and students at the University of West Bohemia in Plzen, VSB-Technical University and Ostrava University in Ostrava for their comments and recommendations  ... 
arXiv:2206.13216v1 fatcat:sncnvcatmfawpiazkekz66we4e

Using geometric algebra for 3D linear transformations

H. Zhang, C. Zhu, Q. Peng, J.X. Chen
2006 Computing in science & engineering (Print)  
derivations extend the geometric algebra research and applications to the computer graphics field.  ...  method to derive various geometrical transformations, which are different from the current ray-tracing and transformation methods implemented in graphics algorithms and hardware. 3 We hope that these  ...  However, applications of geometric algebra in 3D computer graphics are still undergoing study. 4 Basic Elements and Operations Geometric algebra has four basic computing elements in 3D physical space  ... 
doi:10.1109/mcse.2006.54 fatcat:2k2tabfsvjfv7kmvonibknp2pm

Geometric Algebra Computers [chapter]

Dietmar Hildenbrand
2012 Geometry and Computing  
On one hand, there is a recent development of geometric algebra to an easy handling of engineering applications, especially in computer graphics, computer vision and robotics.  ...  Furthermore there is a lot of potential for optimization and parallelization. In this paper, we investigate computers suitable for geometric algebra algorithms.  ...  Many other researchers followed and showed that applying geometric algebra in their field of research can be advantageous, e.g. in engineering areas like computer graphics, computer vision and robotics  ... 
doi:10.1007/978-3-642-31794-1_14 fatcat:itwitou6yvgznjr566fm3jnvte

Higher-order interpolation and least-squares approximation using implicit algebraic surfaces

Chandrajit Bajaj, Insung Ihm, Joe Warren
1993 ACM Transactions on Graphics  
We have implemented our algebraic surface-fitting algorithms, and included them in the distributed and collaborative geometric environment SHASTRA.  ...  MIX = O, where MI and MA are matrices for interpolation and least-squares approximation, respectively, and x is a vector containing coefficients of an algebraic  ...  ACKNOWLEDGMENT We wish to thank Professor Robert Lynch for his valuable comments on the matrix computations and Dr. Guoliang Xu for several discussions on geometric continuity.  ... 
doi:10.1145/159730.159734 fatcat:5sxkdrahdjfplnmwol45p4zxnq

An embedded, FPGA-based computer graphics coprocessor with native geometric algebra support

Silvia Franchini, Antonio Gentile, Filippo Sorbello, Giorgio Vassallo, Salvatore Vitabile
2009 Integration  
The representation of geometric objects and their transformation are the two key aspects in computer graphics applications.  ...  Geometric algebra (aka Clifford algebra) is attracting attention as a natural way to model geometric facts and as a powerful analytical tool for symbolic calculations.  ...  Salvatore Segreto and Dr. Ing. Vincenzo Vullo for their support of the FPGA coprocessor implementation and testing.  ... 
doi:10.1016/j.vlsi.2008.09.010 fatcat:jhslfijqafhf5dl4kbkajkrkny
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