129 research outputs found
Dual-Quaternion Julia Fractals
Fractals offer the ability to generate fascinating geometric shapes with all
sorts of unique characteristics (for instance, fractal geometry provides a
basis for modelling infinite detail found in nature). While fractals are
non-euclidean mathematical objects which possess an assortment of properties
(e.g., attractivity and symmetry), they are also able to be scaled down,
rotated, skewed and replicated in embedded contexts. Hence, many different
types of fractals have come into limelight since their origin discovery. One
particularly popular method for generating fractal geometry is using Julia
sets. Julia sets provide a straightforward and innovative method for generating
fractal geometry using an iterative computational modelling algorithm. In this
paper, we present a method that combines Julia sets with dual-quaternion
algebra. Dual-quaternions are an alluring principal with a whole range
interesting mathematical possibilities. Extending fractal Julia sets to
encompass dual-quaternions algebra provides us with a novel visualize solution.
We explain the method of fractals using the dual-quaternions in combination
with Julia sets. Our prototype implementation demonstrate an efficient methods
for rendering fractal geometry using dual-quaternion Julia sets based upon an
uncomplicated ray tracing algorithm. We show a number of different experimental
isosurface examples to demonstrate the viability of our approach
A Survey on Dual-Quaternions
Over the past few years, the applications of dual-quaternions have not only
developed in many different directions but has also evolved in exciting ways in
several areas. As dual-quaternions offer an efficient and compact symbolic form
with unique mathematical properties. While dual-quaternions are now common
place in many aspects of research and implementation, such as, robotics and
engineering through to computer graphics and animation, there are still a large
number of avenues for exploration with huge potential benefits. This article is
the first to provide a comprehensive review of the dual-quaternion landscape.
In this survey, we present a review of dual-quaternion techniques and
applications developed over the years while providing insights into current and
future directions. The article starts with the definition of dual-quaternions,
their mathematical formulation, while explaining key aspects of importance
(e.g., compression and ambiguities). The literature review in this article is
divided into categories to help manage and visualize the application of
dual-quaternions for solving specific problems. A timeline illustrating key
methods is presented, explaining how dual-quaternion approaches have progressed
over the years. The most popular dual-quaternion methods are discussed with
regard to their impact in the literature, performance, computational cost and
their real-world results (compared to associated models). Finally, we indicate
the limitations of dual-quaternion methodologies and propose future research
directions.Comment: arXiv admin note: text overlap with arXiv:2303.1339
Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics
Before we dive into the accessibility stream of nowadays indicatory
applications of octonions to computer and other sciences and to quantum physics
let us focus for a while on the crucially relevant events for todays revival on
interest to nonassociativity. Our reflections keep wandering back to the
two square identity and then via the four
square identity up to the eight square identity.
These glimpses of history incline and invite us to retell the story on how
about one month after quaternions have been carved on the bridge
octonions were discovered by , jurist and
mathematician, a friend of . As for today we just
mention en passant quaternionic and octonionic quantum mechanics,
generalization of equations for octonions and triality
principle and group in spinor language in a descriptive way in order not
to daunt non specialists. Relation to finite geometries is recalled and the
links to the 7stones of seven sphere, seven imaginary octonions units in out of
the cave reality applications are appointed . This way we are welcomed
back to primary ideas of , and other distinguished
fathers of quantum mechanics and quantum gravity foundations.Comment: 26 pages, 7 figure
Instantaneously trained neural networks with complex and quaternion inputs
Neural network architectures such as backpropagation networks, perceptrons or generalized Hopfield networks can handle complex inputs but they require a large amount of time and resources for the training process. This thesis investigates instantaneously trained feedforward neural networks that can handle complex and quaternion inputs. The performance of the basic algorithm has been analyzed and shown how it provides a plausible model of human perception and understanding of images. The motivation for studying quaternion inputs is their use in representing spatial rotations that find applications in computer graphics, robotics, global navigation, computer vision and the spatial orientation of instruments. The problem of efficient mapping of data in quaternion neural networks is examined. Some problems that need to be addressed before quaternion neural networks find applications are identified
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