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The Study Variety of Conformal Kinematics
Bahar Kalkan1, Zijia Li2, Hans-Peter Schröcker3
1Department of Mathematics, Van Yuzuncu Yil University, Van, Turkey.
Advances in Applied Clifford Algebras
|July 25, 2022
Summary
We introduce the Study variety, a new mathematical object generalizing rigid body kinematics. This variety allows for concrete calculations using conformal geometric algebra, simplifying complex motion analysis.
Area of Science:
- Mathematics
- Geometry
- Kinematics
Background:
- The Study quadric is a known model for rigid body kinematics.
- Existing methods for analyzing conformal motions can be complex due to high dimensions.
Purpose of the Study:
- Introduce and investigate the properties of the Study variety.
- Develop a new framework for analyzing conformal kinematics.
- Explore the relationship between straight lines on the Study variety and specific conformal motions.
Main Methods:
- Utilizing conformal geometric algebra (CGA) for calculations.
- Employing a four quaternion representation, extending dual quaternions.
- Analyzing the geometric properties of the Study variety in real projective space.
Main Results:
- The Study variety is a 10-dimensional, degree 12 projective variety.
- Straight lines on the Study variety correspond to a class of one-parametric conformal motions.
- The framework facilitates concrete calculations despite the variety's complexity.
Conclusions:
- The Study variety provides a powerful tool for understanding conformal kinematics.
- The four quaternion representation simplifies calculations within CGA.
- This work connects geometric structures to specific types of motion decomposition.
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