Search Results (6)

Automated guided vehicles are widely used in warehousing environments for automated pallet handling, which is one of the fundamental parts to construct intelligent logistics systems. Pallet detection is a critical technology for automated guided vehicles, which directly affects production efficiency. A novel pallet detection method for automated guided vehicles based on point cloud data is proposed, which consists of five modules including point cloud preprocessing, key point extraction, feature description, surface matching and point cloud registration. The proposed method combines the color with the geometric features of the pallet point cloud and constructs a new Adaptive Color Fast Point Feature Histogram (ACFPFH) feature descriptor by selecting the optimal neighborhood adaptively. In addition, a new surface matching method called the Bidirectional Nearest Neighbor Distance Ratio-Approximate Congruent Triangle Neighborhood (BNNDR-ACTN) is proposed. The proposed method overcomes the problems of current methods such as low efficiency, poor robustness, random parameter selection, and being time-consuming. To verify the performance, the proposed method is compared with the traditional and modified Iterative Closest Point (ICP) methods in two real-world cases. The results show that the Root Mean Square Error (RMSE) is reduced to 0.009 and the running time is reduced to 0.989 s, which demonstrates that the proposed method has faster registration speed while maintaining higher registration accuracy. Full article

Through signs and symbols, maps represent geographic space in a generalized and abstracted way. Cartographic research is, therefore, concerned with establishing a mutually shared set of signs and semiotic rules to communicate geospatial information successfully. While cartographers generally strive for cognitively congruent maps, empirical research has only started to explore the different facets and levels of correspondences between external cartographic representations and processes of human cognition. This research, therefore, draws attention to the principle of contextual congruence to study the correspondences between shape symbols and different geospatial content. An empirical study was carried out to explore the (in)congruence of cartographic point symbols with respect to positive, neutral, and negative geospatial topics in monothematic maps. In an online survey, 72 thematic maps (i.e., 12 map topics × 6 symbols) were evaluated by 116 participants in a between-groups design. The point symbols comprised five symmetric shapes (i.e., Circle, Triangle, Square, Rhomb, Star) and one Asymmetric Star shape. The study revealed detailed symbol-content congruences for each map topic as well as on an aggregated level, i.e., by positive, neutral, and negative topic clusters. Asymmetric Star symbols generally showed to be highly incongruent with positive and neutral topics, while highly congruent with negative map topics. Symmetric shapes, on the other hand, emerged to be of high congruence with positive and neutral map topics, whilst incongruent with negative topics. As the meaning of point symbols showed to be susceptible to context, the findings lead to the conclusion that cognitively congruent maps require profound context-specific considerations when designing and employing map symbols. Full article

Ethnomathematics makes school mathematics more relevant and meaningful for students. The current research aims to study the effect of using ethnomathematics in the context of Islamic ornamentation on learning the topic of congruent triangles. To achieve this aim, 30 10th-grade students engaged in ethnomathematics by learning about congruent triangles using Islamic ornamentation. Data was gathered via (a) videotaping and transcribing students’ learning and (b) students answering two parallel questionnaires that included proof questions on the three congruence theorems. The students were required to answer one questionnaire before the learning process and one after it. The main results indicated that the students succeeded in constructing the concepts of congruence and congruent triangles via the ethnomathematics learning process. In addition, the students succeeded in arriving at and formulating the three congruence theorems. Moreover, findings obtained from the questionnaires indicated that the students improved their proving processes as a result of ethnomathematics-based learning. Furthermore, paired sample t-tests indicated significant differences between the students’ mean scores before and after the learning process. Full article

The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer $k\ge 2$ . Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the $p^e$ -Pascal’s triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form ${\displaystyle \frac{p^{em}-1}{p^e-1}}$ with some integer $m\ge 1$ . This is a generalization of a Lucas’ result asserting that the n-th row in the (2-)Pascal’s triangle consists of odd integers if and only if n is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence ${(x+1)}^{p^e}\equiv {(x^p+1)}^{p^{e-1}}(\mathrm{mod}{p}^e)$ of binomial expansions which we could prove for any prime number p and any positive integer e. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions. Full article

In astronomy, physics, climate modeling, geoscience, planetary science, and many other disciplines, the mass of data often comes from spherical sampling. Therefore, establishing an efficient and distortion-free representation of spherical data is essential. This paper introduces a novel spherical (global) coordinate system that is free of singularity. Contrary to classical coordinates, such as Cartesian or spherical polar systems, the proposed coordinate system is naturally defined on the spherical surface. The basic idea of this coordinate system originated from the classical planar barycentric coordinates that describe the positions of points on a plane concerning the vertices of a given planar triangle; analogously, spherical area coordinates (SACs) describe the positions of points on a sphere concerning the vertices of a given spherical triangle. In particular, the global coordinate system is obtained by decomposing the globe into several identical triangular regions, constructing local coordinates for each region, and then combining them. Once the SACs have been established, the coordinate isolines form a new class of global grid systems. This kind of grid system has some useful properties: the grid cells exhaustively cover the globe without overlapping and have the same shape, and the grid system has a congruent hierarchical structure and simple relationship with traditional coordinates. These beneficial characteristics are suitable for organizing, representing, and analyzing spatial data. Full article

The correspondence between right triangles with rational sides, triplets of rational squares in arithmetic succession and integral solutions of certain quadratic forms is well-known. We show how this correspondence can be extended to the generalized notions of rational θ-triangles, rational squares occurring in arithmetic progressions and concordant forms. In our approach we establish one-to-one mappings to rational points on certain elliptic curves and examine in detail the role of solutions of the θ-congruent number problem and the concordant form problem associated with nontrivial torsion points on the corresponding elliptic curves. This approach allows us to combine and extend some disjoint results obtained by a number of authors, to clarify some statements in the literature and to answer some hitherto open questions. Full article