Geometric Algebra Computing in Engineering and Computer Science: A Comprehensive Exploration

Geometric algebra (GA) is a mathematical system that combines the power of vector calculus, matrix algebra, and Grassmann algebra. It provides a unified framework for representing and manipulating geometric objects and their relationships in a consistent and efficient manner. GA has found widespread applications in various fields of engineering and computer science, including robotics, computer graphics, machine learning, and more.

GA is based on the concept of a geometric product, which combines two vectors to produce a new vector, bivector, or pseudoscalar. The geometric product is defined by the following rules:

• Vector product:a ∧ b = a × b (vector cross product)
• Scalar product:a ⋅ b = a · b (scalar dot product)
• Bivector product:a ∧ b = a⊗b - b⊗a (exterior product)

These operations allow for the concise representation of geometric transformations and relationships, such as rotations, translations, and projections. GA also introduces the concept of multivectors, which are objects that can represent a combination of vectors, bivectors, and pseudoscalars.

Geometric Algebra Computing: in Engineering and Computer Science

by Judith Gautier

5 out of 5

Language	:	English
File size	:	23235 KB
Screen Reader	:	Supported
Print length	:	548 pages
Paperback	:	1152 pages
Item Weight	:	2.65 pounds
Dimensions	:	6.1 x 1.57 x 9.49 inches

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• Robotics: GA provides a powerful tool for representing and manipulating robot kinematics and dynamics. It enables efficient computations of joint angles, velocities, and accelerations, and simplifies the control of complex robotic systems.
• Computer graphics: GA is used in computer graphics for modeling, animation, and rendering. It allows for the creation of realistic and physically accurate simulations, as well as efficient algorithms for collision detection, intersection testing, and ray tracing.
• Mechanical engineering: GA is employed in mechanical engineering for analyzing the dynamics of rigid bodies, fluid mechanics, and heat transfer. It provides a unified framework for representing forces, moments, and velocities, and enables the efficient solution of complex mechanical systems.
• Electrical engineering: GA has applications in electrical engineering for modeling and simulating electromagnetic fields, antennas, and circuits. It allows for the concise representation of Maxwell's equations and simplifies the analysis of complex electromagnetic systems.

• Machine learning: GA is used in machine learning for feature extraction, dimensionality reduction, and classification. It provides a powerful framework for representing complex data structures and relationships, and enables the development of efficient algorithms for machine learning tasks.
• Deep learning: GA is employed in deep learning for modeling geometric relationships in data. It is used in convolutional neural networks for image processing, recurrent neural networks for time series analysis, and graph neural networks for representing and processing graph-structured data.
• Artificial intelligence: GA is used in artificial intelligence for representing and reasoning about spatial relationships. It provides a powerful framework for developing intelligent systems that can understand and manipulate their environment.

• Conciseness: GA provides concise representations for geometric objects and transformations, which can significantly reduce the complexity of computations and improve code readability.
• Efficiency: GA operations are typically more efficient than traditional vector and matrix operations, which can lead to faster execution times and improved performance.
• Unification: GA unifies the representation and manipulation of vectors, bivectors, and pseudoscalars in a single framework, making it easier to handle geometric relationships and transformations.
• Generalizability: GA is generalizable to higher dimensions and can be applied to a wide range of geometric problems, making it a powerful tool for solving complex engineering and computer science challenges.

Geometric algebra computing is a powerful and versatile tool that has significant applications in engineering and computer science. Its ability to simplify complex problems, provide efficient solutions, and unify geometric representations makes it an essential tool for researchers, engineers, and computer scientists. As the field continues to advance, we can expect to see even wider adoption of GA computing and the development of new and innovative applications in a variety of domains.

Geometric Algebra Computing: in Engineering and Computer Science

by Judith Gautier

5 out of 5

Language	:	English
File size	:	23235 KB
Screen Reader	:	Supported
Print length	:	548 pages
Paperback	:	1152 pages
Item Weight	:	2.65 pounds
Dimensions	:	6.1 x 1.57 x 9.49 inches

FREE