Numbers working together in a unified mathematical form ...
Dual-Quaternions (Synergy of Numbers)
Combining numbers into a unified mathematical framework to showcase new systems and models, enable conversion and encoding, while addressing limitations and challenges inherent existing approaches.
Dual-quaternions combine the elegance of quaternions with the versatility of dual numbers, offering a compact mathematic form for describing the system. Hamilton developed the mathematics for quaternions in 1843 (scratching them on a bridge in Ireland), while Clifford introduced dual-number theory a bit later in 1873. Of course, the concept of using dual-number theory with quaternions was proposed by Clifford - who put forward dual-quaternions and proposing using them as a solution for representing rotations and translations in a unified form.
The reason dual-quaternions gained a lot of attention in recent years, is due to their applications in spatial transforms, as they offer an unparalleled solution with precision for describing rotations and translations in 3D space. Their compact yet robust formulation, they streamline complex transformations, propelling everything from robotics and computer graphics to aerospace engineering into a realm of efficiency and accuracy previously unimaginable. Say hello to a seamless fusion of numbers and precision with dual-quaternions - the ultimate tool for navigating the multidimensional problems.
Mathematical numbers representating real-world data does not usually function in isolation (typically connected or grouped together with synergistic characteristics). For example, pixels in an image, the surounding pixels play a part, by itself it has little meaning but together with the other pixels it has value; following this on, if you perform operations on the image, you need to consider and take into account the neighbouring pixels when calculating the transform (this is what dual-quaternions do).
No limit to the different types of uses and configurations - dual-quaternions aren't confined to spatial transforms - as Acrobatic Gymnastics aren't limited to only triangle pyramids (but they are good at them).
Resources and Articles
• A Survey on Dual-Quaternions [LINK]
• Dual-Quaternion Julia Fractals [LINK]
• A Beginners Guide to Dual-Quaternions [LINK]
• Dual-Quaternion Interpolation [LINK]
• Interactive Web-Page Dual-Quaternion Survey [LINK]
• Dual-quaternion App/interactive page (interpolation) [LINK]
• Dual-Quaternion Surfaces and Curves [LINK]
• Sound and Dual-Quaternions [LINK]
• Signal Analysis and Dual-Quaternion Fourier Transform [LINK]
• Dual-Quaternions and Computer Graphics [LINK]
Code Snippets
• Encoding RGB/Location within Dual-Quaternion [LINK]
Dual-quaternions is a broad topic and governs many areas - not just transforms and animation but graphics which includes a range of potential innovations.
Dual-Quaternions and Computer Graphics is an indispensable resource for mathematicians, engineers, and computer scientists seeking a deeper understanding of this revolutionary mathematical tool. Whether you are a student exploring the frontiers of algebra or a seasoned professional looking to enhance your skills, this book provides a comprehensive and accessible guide to the world of dual-quaternions. Unlock the potential of 4D mathematics and revolutionize your approach to geometric transformations. Embrace the dual-quaternion rainbow ;)
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