62 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
The Inverse Iteration Method for Julia Sets in the 3-Dimensional Space
In this article, we introduce the adapted inverse iteration method to
generate bicomplex Julia sets associated to the polynomial map . The
result is based on a full characterization of bicomplex Julia sets as the
boundary of a particular bicomplex cartesian set and the study of the fixed
points of . The inverse iteration method is used in particular to
generate and display in the usual 3-dimensional space bicomplex Julia sets that
are dendrites.Comment: 16 pages, 4 figure
Basins of attraction for a quadratic coquaternionic map
In this paper we consider the extension, to the algebra of coquaternions, of a complex quadratic map with a real super-attractive 8-cycle. We establish that, in addition to the real cycle, this new map has sets of non-isolated periodic points of period 8, forming four attractive 8-cycles. Here , the cycles are to be interpreted as cycles of sets and an appropriate notion of attractivity is used. Some characteristics of the basins of attraction of the five attracting 8-cycles are discussed and plots revealing the intertwined nature of these basins are shown.Research at CMAT was financed by Portuguese Funds through FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, within the Project
UID/MAT/00013/2013. Research at NIPE has been carried out within the funding with COMPETE reference number POCI-01-
0145-FEDER-006683, with the FCT/MEC’s (Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P.) financial support through national
funding and by the ERDF through the Operational Programme on “Competitiveness and Internationalization – COMPETE 2020”
under the PT2020 Partnership Agreement.info:eu-repo/semantics/publishedVersio
Fractals with arbitrary segment lengths
Work in the area of fractal geometry has generally focused on a specific facet of the discipline at the expense of other interesting features. This approach often generates more questions than answers for the general audience due to the lack of unification across all views. It appears that a common thread to relate all aspects of fractal characteristics is missing. This paper addresses this question and presents some new and fascinating results. For example, in-depth mathematical analysis often defers to the intriguing and attractive graphical displays produced by mapping the complex plane to the pixel field on a CRT. Both area, mathematics and graphics, are generally developed or presented independently. the development of common attribute linkages is done separately or perhaps not at all. First, a completely modular survey of the state of the art concerning regular fractal geometry is given. In addition, a method for calculating the fractal dimension of asymmetric fractals is proposed, where a symmetric fractal is a special case of an asymmetric fractal --page iii
Visualising Volumetric Fractals
Fractal images have for many years been a richsource of exploration by those in computer science who also havean interest in graphics. They often served as a way of testing theperformance of new computing hardware and to explore thecapabilities of emerging display technologies. While there havebeen forays by some into 3D geometric fractals, the 3Dequivalents of the Mandelbrot set have been largely ignored. Thisis largely due to the lack of suitable tools for rendering these setsexcept perhaps as isosurfaces, a rather unsatisfactory and limitedrepresentation. The following will illustrate the application ofGPU based raycasting, a now relatively standard approach tovolume rendering, to the representation of volumetric fractals.Leveraging existing software that has been designed for generalvolume visualisation allows the interested 3D fractal explorer tofocus on the mathematical generation of the volume data ratherthan reinventing the entire volume rendering pipeline
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
- …