• The Mathematical Education
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• v.44 no.2 s.109
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• pp.297-306
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• 2005

In this paper, we propose a computer environment class for student's mental representations about non-Euclidean geometry. Through the software Cinderella, students construct knowledge about non-Euclidean geometry and recognize differentness between Euclidean and non-Euclidean geometry. Also they recognize an existence of non-Euclidean geometry newly and its mental representations with images represented in Cinderella. In geometry class, we make students can use many representations systematically and can figure a visual internal image by emphasizing a transform process. And then students can reason about non-Euclidean geometry.

• IEMEK Journal of Embedded Systems and Applications
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• v.16 no.5
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• pp.215-223
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• 2021

3D data can be categorized into two parts : Euclidean data and non-Euclidean data. In general, 3D data exists in the form of non-Euclidean data. Due to irregularities in non-Euclidean data such as mesh and point cloud, early 3D deep learning studies transformed these data into regular forms of Euclidean data to utilize them. This approach, however, cannot use memory efficiently and causes loses of essential information on objects. Thus, various approaches that can directly apply deep learning architecture to non-Euclidean 3D data have emerged. In this survey, we introduce various deep learning methods for mesh and point cloud data. After analyzing the operating principles of these methods designed for irregular data, we compare the performance of existing methods for shape classification and segmentation tasks.

• Korean Journal of Mathematics
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• v.31 no.4
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• pp.495-504
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• 2023

This article deals with non-degenerate linear transformations on Euclidean Jordan algebras. First, we study non-degenerate for cone invariant, copositive, Lyapunov-like, and relaxation transformations. Further, we study that the non-degenerate is invariant under principal pivotal transformations and algebraic automorphisms.

• International Journal of CAD/CAM
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• v.5 no.1
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• pp.59-68
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• 2005

Euclidean Voronoi diagram of spheres in 3-dimensional space has not been explored as much as it deserves even though it has significant potential impacts on diverse applications in both science and engineering. In addition, studies on the data structure for its topology have not been reported yet. Presented in this, paper is the topological representation for Euclidean Voronoi diagram of spheres which is a typical non-manifold model. The proposed representation is a variation of radial edge data structure capable of dealing with the topological characteristics of Euclidean Voronoi diagram of spheres distinguished from those of a general non-manifold model and Euclidean Voronoi diagram of points. Various topological queries for the spatial reasoning on the representation are also presented as a sequence of adjacency relationships among topological entities. The time and storage complexities of the proposed representation are analyzed.

• Honam Mathematical Journal
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• v.34 no.1
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• pp.103-111
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• 2012

Wetzel[5] proved if ${\Gamma}$ is a closed curve of length L in $E^n$, then ${\Gamma}$ lies in some ball of radius [L/4]. In this paper, we generalize Wetzel's result to the non-Euclidean plane with much stronger version. That is to develop a lower bound of length of a triangle inscribed in a circle in non-Euclidean plane in terms of a chord of the circle.

• Journal for History of Mathematics
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• v.24 no.4
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• pp.21-31
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• 2011

In this paper I claim that Lobachevsky's philosophy of mathematics is a kind of reservoir of contemporary philosophy of mathematics. I discuss how his philosophy contributed to the rise of non-Euclidean geometry.

• Journal of the Korean Society of Costume
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• v.60 no.5
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• pp.117-127
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• 2010

In this study, hyperspace is a result of imagination created by means of facts and fiction, represents a transfer to determination and indetermination, and means an extension to an open form. In other words, hyperspace is a high dimensional space expanded to imagination through the combination of the viewpoint on facts in this dimension and fiction. When the 2D plane surface or 3D symmetry is destroyed, or when the frame is twisted or entangled, the non-Euclidean geometry is created eventually. And when the twisting leads to transmutation and the destruction of the form reaches the extreme; this in turn became the twisting like Mbius band. Likewise, the non-Euclidean geometry is co-related to the asymmetry of the Higgs mechanism. When the 'destruction of symmetry' is considered, symmetric theory and asymmetric world can be connected. The asymmetry in turn can maintain balance by arranging the uneven weights at different distances from the shaft. Moreover, at this the concept of the upper, lower, left and right, which was included in the original form, may be crumbled down. The destruction of the symmetry is essential in order to present forecast that coincides with the phenomenon of the real world. Non-Euclidean geometry characteristic is expressed by asymmetry, twists, and deconstruction and its representative characteristic is ambiguity. The boundary between the front, back, upper, lower, inner and outer is unclear, and it is difficult and vague to pinpoint specific location. The design that does not clearly define or determine the direction of wearing costume is indeed the non-oriented design that can be worn without getting restricted by specific direction such as front and back. Non-Euclidean geometry characteristic of hyperspace have been applied to create new shapes through the modification of the substance from traditional clothing of the eastern world to modern fashion. The way of thinking in the 'hyperspace' that used to be expressed in the costumes of the east and the west in the past became the forum for unlimited creation.

• Phonetics and Speech Sciences
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• v.5 no.4
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• pp.201-207
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• 2013

This study is a follow-up study of Han and Kang (2013) and Kang and Han (2013) which examined cross-generational changes in the Korean vowels /o/ and /u/ using acoustic analyses of the vowel formants of these two vowels, their Euclidean distances and the overlap fraction values generated in SOAM 2D (Wassink, 2006). Their results showed an on-going approximation of /o/ and /u/, more evident in female speakers and non-initial vowels. However, these studies employed non-words in a frame sentence. To see the extent to which these two vowels are merged in real words in spontaneous speech, we conducted an acoustic analysis of the formants of /o/ and /u/ produced by two age groups of female speakers while reading a letter sample. The results demonstrate that 1) the younger speakers employed mostly F2 but not F1 differences in the production of /o/ and /u/; 2) the Euclidean distance of these two vowels was shorter in non-initial than initial position, but there was no difference in Euclidean distance between the two age groups (20's vs. 40-50's); 3) overall, /o/ and /u/ were more overlapped in non-initial than initial position, but in non-initial position, younger speakers showed more congested distribution of the vowels than in older speakers.

• Journal of Science Education
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• v.47 no.3
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• pp.263-272
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• 2023

We consider how a pre-service teacher can understand and utilize various concepts of Euclidean geometry by learning conic sections using mathematical definitions in non-Euclidean geometry. In a third-grade class of D University, we used mathematical definitions to demonstrate that learning conic sections in non-Euclidean space, such as taxicab geometry and Minkowski distance space, can aid pre-service teachers by enhancing their ability to acquire and accept new geometric concepts. As a result, learning conic sections using mathematical definitions in taxicab geometry and Minkowski distance space is expected to contribute to enhancing the education of pre-service teachers for Euclidean geometry expertise by fostering creative and flexible thinking.

• Journal of applied mathematics & informatics
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• v.9 no.1
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• pp.441-447
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• 2002

We show that the conharmonical1y flat K-contact find cosymplectic manifolds are local1y Euclidean. Evidently non locally Euclidean conharmonically flat Sasakian manifold does not exist. Moreover we see that conharmonically flat Kenmotsu manifold does not exist and conharmonically flat fibred quasi quasi Sasakian space is locally Euclidean if and only if the scalar curvature of each fibre vanishes identically.