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Geometric algebra for quadrics-transformations and their efficient implementation
Aleš Návrat1, Ludvík Procházka2, Petr Vašík1
1Brno University of Technology Faculty of Mechanical Engineering , Brno, Czech Republic.
Summary
This study introduces geometric algebra for quadrics (GAQ) to represent Euclidean transformations like translations and rotations. A C++ implementation is provided for practical application of these geometric algebra methods.
Area of Science:
- Mathematics
- Computer Science
- Robotics
Background:
- Geometric algebra offers a unified framework for geometric transformations.
- Representing Euclidean transformations efficiently is crucial in various scientific domains.
Purpose of the Study:
- To describe geometric algebra for quadrics (GAQ) for representing Euclidean transformations.
- To provide general methods for deriving transformation generators and computing versors.
- To present a C++ implementation for practical use.
Main Methods:
- Derivation of generators for translations and rotations within GAQ.
- Numerical computation of versors in the high-dimensional GAQ space.
- Development of a C++ library for GAQ-based transformations.
Main Results:
- General methods for deriving Euclidean transformation generators in GAQ.
- Successful verification of derived generators through examples.
- A practical C++ implementation for numerical computation of versors.
Conclusions:
- GAQ provides a powerful framework for Euclidean transformations.
- The C++ implementation simplifies the application of GAQ for transformations.
- This work contributes to modern applications of geometric algebra.
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