A wide variety of media can be used in learning, including distance learning, such as print, lectures, conference sections, tutors, pictures, video, sound, and computers. Any one instance of distance learning will make choices among these media, perhaps using several.

Multimedia in Learning

Computational geometry

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

Second International Handbook of Mathematics Education

This article presents a novel data encryption technique suitable for Internet of Things (IoT) applications. The cryptosystem is based on the application of a Catalan object (as a cryptographic key) that provides encryption based on combinatorial structures with noncrossing or nonnested matching. The experimental part of this article includes a comparative analysis of the proposed encryption method with the Catalan numbers and data encryption standard (DES) algorithm, which is performed with machine learning-based identification of the encryption method using ciphertext only. These tests showed that it is much more difficult to recognize ciphertext generated with the Catalan method than one made with the DES algorithm. System reliability depends on the quality of the key, therefore, statistical testing proposed by National Institute of Standards and Technology was also performed. Twelve standard tests, the approximate entropy measurement, and random digression complexity analysis are applied in order to evaluate the quality of the generated Catalan key. A proposal for applying this method in e-Health IoT is also given. Possibilities of applying this method in the IoT applications for smart cities data storage and processing are provided.

Data Encryption for Internet of Things Applications Based on Catalan Objects and Two Combinatorial Structures

In this paper is presented a procedure for the application of one computational geometry algorithm in the process of generating hidden cryptographic keys from one segment of the 3D image. The presented procedure consists of three phases. In the first phase, is done the separation of one segment from the 3D image and determination of triangulation of the separated polygon. In the second phase, is done a conversion from the obtained triangulation of the polygon in the record which represent the Catalan key. In the third phase, the Catalan-key is applied in encryption of text based on the balanced parentheses combinatorial problem.

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Generation of Cryptographic Keys with Algorithm of Polygon Triangulation and Catalan Numbers

This paper presents a new technique of generation of convex polygon triangulation based on planted trivalent binary tree and ballot notation. The properties of the Catalan numbers were examined and their decomposition and application in developing the hierarchy and triangulation trees were analyzed. The method of storage and processing of triangulation was constructed on the basis of movements through the polygon. This method was derived from vertices and leaves of the planted trivalent binary tree. The research subject of the paper is analysis and comparison of a constructed method for solving of convex polygon triangulation problem with other methods and generating graphical representation. The application code of the algorithms was done in the Java programming language.

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Convex polygon triangulation based on planted trivalent binary tree\\ and ballot problem

This paper presents the authentication method using the Delaunay triangulation incremental algorithm and the Catalan objects. The proposed method is a combination of computational geometry and cryptography. This method presents a new step towards encoding the triangle coordinates using the Catalan-key. We provided specific suggestions for the application of this method in the authentication for bank clients by the image encryption. Client authentication verification is performed by asking the client to enter the (x,y) coordinate values of randomly selected indices of an array. If the entered coordinates match the index values in the banking system array, then the transaction or other operation is approved. If the matching fails, it means that we have an unidentified person who has followed the whole process and wants to break into the banking system. There are many advantages arising from a scenario for the user authentication by the assigned Catalan object and the stack permutation method. Also, we provided concrete examples for the Delaunay encryption of image with an authentication scenario and experimental results for the proposed method.

/pdf/authentication-based-on-the-image-encryption-using-delaunay-19xq4pshy2.pdf

Authentication Based on the Image Encryption using Delaunay Triangulation and Catalan Objects

Computational geometry is an integral part of mathematics and computer science deals with the algorithmic solution of geometry problems. From the beginning to today, computer geometry links different areas of science and techniques, such as the theory of algorithms, combinatorial and Euclidean geometry, but including data structures and optimization. Today, computational geometry has a great deal of application in computer graphics, geometric modeling, computer vision, and geodesic path, motion planning and parallel computing. The complex calculations and theories in the field of geometry are long time studied and developed, but from the aspect of application in modern information technologies they still are in the beginning. In this research is given the applications of computational geometry in polygon triangulation, manufacturing of objects with molds, point location, and robot motion planning.

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Computational Geometry Applications

This paper aims to examine the impact of digital tools in mathematics and the readiness of teachers and students applying these interactive tools in teaching. The data used in the research are obtained from the test results of 526 students, in five secondary schools in North Macedonia. The students in this research, are divided into two groups: mainly as a control group and an experimental group. The control group is the group of students who do not have access to the interactive tools at home and who use interactive tools only once a week in the school while the group of students in the experimental group have access to them and have the opportunity to use these interactive applications every day. The students in the control group and the experimental group were selected from the same year and the gender equality of the groups was taken into account. To further understand the relationship between teaching with digital tools and learning after testing was surveyed the participants. The results in our research suggest that interactive teaching tools have a positive impact on the teaching process and increase students' knowledge.

/pdf/impact-of-using-digital-tools-in-high-school-mathematics-a-zdhcv7nxao.pdf

Aybeyan Selimi

Papers

Generation of Cryptographic Keys with Algorithm of Polygon Triangulation and Catalan Numbers

Convex polygon triangulation based on planted trivalent binary tree\\ and ballot problem

Authentication Based on the Image Encryption using Delaunay Triangulation and Catalan Objects

Computational Geometry Applications

Impact of Using Digital Tools in High School Mathematics: A Case Study in North Macedonia