Quaternion-based curves and surfaces for enhanced spatial motion generation using geometric algebra

Abstract This paper presents a novel framework for generating quaternion-based curves and surfaces, leveraging the power of geometric (Clifford) algebra to enhance spatial motion generation. The proposed approach enables the representation and manipulation of complex spatial motions in a mathematically rigorous and computationally efficient manner. Through the integration of quaternion algebra and geometric concepts, our method significantly expands the capabilities of traditional spatial modeling techniques, facilitating the creation of smooth and accurate motion representations. By extending traditional techniques to encompass both curves and surfaces, this framework overcomes the limitations of conventional spline (point)-based methods, particularly when dealing with non-polynomial and transcendental functions. The proposed approach simplifies the handling of rotational and translational motions, making it particularly well-suited for real-time applications in computer graphics, robotics, and virtual reality. The paper also comprehensively analyzes dual quaternion conjugation, exploring its different types and their applications in representing complex spatial transformations. We demonstrate the effectiveness of our approach by applying it to various simulation scenarios, showcasing its potential to contribute to the advancement of computer graphics, virtual reality, and related fields. The availability of open-source code and datasets further promotes the reproducibility and accessibility of our research, fostering a deeper understanding and exploration of quaternion-based spatial motion generation. Fig. 1: https://www.geogebra.org/m/rf8wsvxb DOI: 10.13140/RG.2.2.26124.53120. Fig. 2: https://maple.cloud/app/6278312446525440/lox+and+circle DOI: 10.13140/RG.2.2.12702.75840. Fig. 3: https://www.geogebra.org/3d/cs4ehrx9 DOI:10.13140/RG.2.2.14380.48008. Fig. 4: https://www.geogebra.org/m/hqttazwf DOI: 10.13140/RG.2.2.10605.60648. Fig. 5: https://www.geogebra.org/calculator/aqh7wvzg DOI: 10.13140/RG.2.2.21091.36644. Fig. 6: https://maple.cloud/app/6235802336624640/deformative+motion+dual+quat DOI:10.13140/RG.2.2.24446.80963. Fig. 7-8-9: https://github.com/Ferhajj/Matlab DOI: 10.13140/RG.2.2.16058.20167, DOI: 10.13140/RG.2.2.29479.97441, DOI: 10.13140/RG.2.2.22769.08804.