Geometric Invariants Under the Möbius Action of the Group SL(2;R)

In this paper we have introduced new invariant geometric objects in the homogeneous spaces of complex, dual and double numbers for the principal group SL(2; ℝ), in the Klein’s Erlangen Program. We have considered the action as the Möbius action and have taken the spaces as the spaces of complex, dual and double numbers. Some new decompositions of SL(2; ℝ) have been used.

Geometric Midpoint Algorithm for Device-Free Localization in Low-Density Wireless Sensor Networks

Device-free localization (DFL) is a technique used to track a target transporting no electronic devices. Radiofrequency (RF) tomography based DFL technology in wireless sensor networks has been a popular research topic in recent years. Typically, high-tracking accuracy requires a high-density wireless network which limits its application in some resource-limited scenarios. To solve this problem, a geometric midpoint (GM) algorithm based on the computations of simple geometric objects is proposed to realize effective tracking of moving targets in low-density wireless networks. First, we proposed a signal processing method for raw RSS signals collected from wireless links that can detect the fluctuations caused by a moving target effectively. Second, a geometric midpoint algorithm is proposed to estimate the location of the target. Finally, simulations and experiments were performed to validate the proposed scheme. The experimental results show that the proposed GM algorithm outperforms the geometric filter algorithm, which is a state-of-the-art DFL method that yields tracking root-mean-square errors up to 0.86 m and improvements in tracking accuracy up to 67.66%.

THE EFFECT OF 3D TECHNOLOGIES IN STEREOMETRY TRAINING

The software proposed in the report can be used as a technological tool in the teaching and learning of the discipline of stereometry. The aim is to improve the learning process by supporting the development of students’ creativity and spatial imagination, qualities needed in the study of spatial geometric bodies. A new boundary method is used in the generation of geometric objects. This new method uses elements of the Cavalieri Indivisible method and Isaac Newton's boundary method, thus achieving higher accuracy with respect to generated objects compared to 3D systems that use the rules of trigonometry in the construction of geometric bodies. This article performs a comparative analysis between a traditional and a new method for generating 3D geometric objects according to certain parameters and criteria. The new method involved in the analysis was proposed by the author of the report. While the traditional one is based on trigonometry. Two parameters were studied, one for accuracy in generating objects, and the second determining speed. In order to generate cylindrical bodies, are used the quadratic Bézier curve and the 3D modeling technique known as an extrusion, which transforms two-dimensional objects into three-dimensional ones.. One way to generate a prism and pyramid is by extruding polygons. This report presents a new way of constructing edged bodies. The considered technique for 3D modeling participates in the analysis of the studied methods.

MODELISASI KOTAK TISU DENGAN PENGGABUNGAN KURVA BEZIER, KURVA HERMIT DAN HASIL DEFORMASI BENDA GEOMETRI

The tissue box is a place to store tissues to make them look neat and protect the tissues from dirt and dust. Tissue boxes are often used in households, restaurants and also as room decorations. Therefore, the shapes of tissue boxes that are being developed are increasingly varied according to consumer interests. The tissue box consists of three main parts, namely the cover, body and base of the box. This research was carried out by developing variations in the shape of the tissue box components using the Bezier curve, the Hermit curve and the results of the deformation of geometric objects. The deformation techniques used are rotation, dilation, and curve interpolation. Tissue box modeling processes are divided into four stages. The first, modeling the tissue box by dividing into three models, namely model A, model B and model C. The second, determining the size of the tissue box components based on the model. The third, modeling tissue box components. Finally, visualizing the results of the tissue box model by combining the components so that a variety of tissue box models are produced.

MODELISASI KURSI DENGAN PENGGABUNGAN HASIL DEFORMASI BENDA-BENDA RUANG MENGGUNAKAN KURVA BEZIER

Chairs are needed by humans to do some work, especially students and office workers. The parts contained in the chair are the chair legs, chair legs eats and chair backs. The purpose of this study is to obtain variations in the shape of office chairs using Bezier curves and incorporate the results of deformation of space geometric objects. In modeling this chair, it is divided into several stages, namely first, building the chair leg components. This chair leg component consists of chair wheels, connecting two wheels with tube deformation, modeling the chair leg branch components and modeling chair leg supports. Second, namely the model of the chair leg seat. Chair leg seat consists of regular hexagon prism deformation and regular quadrangle prism deformation. The third is the modelization of the back of the chair by using a rectangular prism model. The result of combining several components of the chair using one modeling axis produces 36 chair models, with special provisions, namely that the seat support parts can only be joined using a tube.

Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

A novel representation of spin 1/2 combines in a single geometric object the roles of the standard Pauli spin vector operator and spin state. Under the spin-position decoupling approximation it consists of three orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) is Hermitian; (2) shows handedness; (3) reproduces all standard expectation values, including the total one-particle spin modulus 𝑆tot = (ℏ/2)√3; (4) constrains basis opposite spins to have same handedness; (5) allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled pairs have precisely related gauge phases; (2) the dimensionality of the spin space doubles due to variation of handedness; (3) the four maximally entangled states are naturally defined by the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) and inversion (singlet). Cross-product terms in the expression for the squared total spin of two particles relate to experiment and they yield all standard expectation values after measurement. Here spin is directly defined and transformed in 3D orientation space, without use of eigen algebra and tensor product as done in the standard formulation. The formalism allows working with whole geometric objects instead of only components, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘on lab’ (uncontrolled gauge phase) spin transformations.

GEOMETRIC MODELING OF ADAPTIVE ALGEBRAIC CURVES PASSING THROUGH PREDETERMINED POINTS

In this paper, the geometric theory of multidimensional interpolation was further developed in terms of modeling and using adaptive curves passing through predetermined points. A feature of the proposed approach to modeling curved lines is the ability to adapt to any initial data for high-quality interpolation, which excludes unplanned oscillations, due to the uneven distribution of parameter values, the source of which are the initial data. This is the improvement of the previously proposed method for constructing and analytically describing arcs of algebraic curves passing through predetermined points, obtained on the basis of Bezier curves, which are compiled taking into account the expansion coefficients of the Newton binomial. The paper gives an example of using adaptive algebraic curves passing through predetermined points for geometric modeling of the stress-strain state of membrane coatings cylindrical shells using two-dimensional interpolation. The given example an illustrative showed the advantages of the proposed adaptation of algebraic curves passing through predetermined points and obtained on the basis of Bezier curves for geometric modeling of multifactor processes and phenomena. The use of such adaptation allows not only to avoid unplanned oscillations, but also self-intersection of geometric objects when generalized to a multidimensional space. Adaptive algebraic curves can also be effectively used as formative elements for constructing geometric objects of multidimensional space, both as guide lines and as generatrix’s.

Identifying Concepts Created for Geometric Objects: Mind Map

The purpose of this research is to determine the knowledge, perceptions and misconceptions of preservice primary school teachers about geometric objects through mind maps. For this purpose, the research was designed with a case study, one of the qualitative research methods. The study group of the research consists of 52 pre-service primary school teachers studying at a state university in the Aegean Region of Turkey. After giving information about mind maps to pre-service primary school teachers, they were asked to create mind maps about the concept of “Geometric Objects”. The created mind maps were subjected to content analysis and codes and categories were created. The mind maps of pre-service primary school teachers were analysed one by one, starting from the centre towards the outer branches and separated as related and unrelated concepts. As a result of the research, it was revealed that most of the pre-service primary school teachers who participated in the study used the concepts of geometric objects and geometric shapes interchangeably. In addition, as a result of the analysis of the first branches of mind maps, it was determined that unrelated concepts were more than related concepts. Very few of them have reached the fourth branch in the mind maps created by pre-service primary school teachers, and they had difficulty explaining the concept. In line with the results obtained from the research, the concepts related to geometric objects should be taught in the undergraduate education of pre-service primary school teachers to provide meaningful learning about geometric concepts in pre-service primary school teachers.

The Use of Non-Standard Problems with Matches When Teaching Younger Schoolchildren

The article deals with the actual problem of the intellectual development of primary schoolchildren by including in the work of non-standard tasks of a search nature, the use of sign-symbolic means of presenting information, which allow you to create models of the objects and processes under study. Among the non-standard are the problems with matches described by the authors in the form of visual calculating material, the search for a solution to which makes it possible to intensify the mental activity of students of this age category, to develop their fine motor skills of hands and mathematical observation. The article describes a methodology for the step-by-step familiarization of junior schoolchildren with simple and more complex transformations and manipulations with matches as geometric objects, the consistent formation of an independent creative approach in children to the choice of schemes for solving non-standard educational and practical problems.

The Fourth Fundamental Form IV of Dini-Type Helicoidal Hypersurface in the Four Dimensional Euclidean Space

We introduce the fourth fundamental form of a Dini-type helicoidal hypersurface in the four dimensional Euclidean space E4. We find the Gauss map of helicoidal hypersurface in E4. We obtain the characteristic polynomial of shape operator matrix. Then, we compute the fourth fundamental form matrix IV of the Dini-type helicoidal hypersurface. Moreover, we obtain the Dini-type rotational hypersurface, and reveal its differential geometric objects.