• Journal for History of Mathematics
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• v.36 no.2
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• pp.21-34
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• 2023

Spherical trigonometry refers to the geometry related to spherical triangles. It has been an important tool for studying astronomy since ancient times. In trigonometry, concepts such as trigonometric functions naturally emerge from the relationship between arcs and chords of a circle. In this paper, we briefly examine the origin of spherical trigonometry. To introduce the basics of spherical trigonometry, we present fundamental and important theorems such as Menelaus's theorem, the law of sines and the law of cosines on a sphere, along with their proofs. Furthermore, we discuss the educational value and potential applications of spherical trigonometry.

• Journal for History of Mathematics
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• v.37 no.5
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• pp.107-121
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• 2024

This paper presents the findings of a research project on Sanhag Seubyu, a work by mid-19^th-century Joseon mathematician Jo Hui-sun. Joseon's efforts to adopt China's new calendrical system began in the mid-17^th century and continued for nearly two centuries, becoming fully documented by the 1860s. While mathematical records from this period are largely the collaborative work of Nam Byeong-gil and Lee Sang-hyeog, Jo Hui-sun's Sanhag Seubyu indicates that astronomical mathematics was of interest among scholars, suggesting that other scholars may have reached similar levels of expertise. Sanhag Seubyu comprises seven chapters, with all but the first and last focusing on spherical trigonometry. This paper presents our investigation into the content of the first two of the five chapters.

• Journal for History of Mathematics
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• v.35 no.3
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• pp.73-113
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• 2022

In this paper we survey on the geometric development in the history of Joseon mathematics. We have relatively many research papers on the history of equations in Joseon but the history of geometry is limited to that of trigonometry (gugosul). We survey on the results on the whole geometry including the introduction of western geometry in Joseon. Joseon mathematics developed differently during several different periods. We investigate how geometric theories developed during those periods and the meaning behind them. We do not claim that our survey is anywhere close to a complete one. This is rather a preliminary attempt to collect research results to plan our research following those of our predecessors.

• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1099-1133
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• 2009

We study the hyperbolic cosine and sine laws in the extended hyperbolic space which contains hyperbolic space as a subset and is an analytic continuation of the hyperbolic space. And we also study the spherical cosine and sine laws in the extended de Sitter space which contains de Sitter space S$^n_1$ as a subset and is also an analytic continuation of de Sitter space. In fact, the extended hyperbolic space and extended de Sitter space are the same space only differ by -1 multiple in the metric. Hence these two extended spaces clearly show and apparently explain that why many corresponding formulas in hyperbolic and spherical space are very similar each other. From these extended trigonometry laws, we can give a coherent and geometrically simple explanation for the various relations between the lengths and angles of hyperbolic polygons, and relations on de Sitter polygons which lie on S$^2_1$, and tangent laws for various polyhedra.

• Journal of Advanced Navigation Technology
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• v.3 no.2
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• pp.139-146
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• 1999

A flight simulator is composed of engine, navigation systems and instrument modules. However, two problems exist here. First, the coordinate of each independent module is not same. To solve this problem, we design a method that mutual coordinates are capable of transformation each other. Second, the distance and bearing between two points on the earth are computed in a sphere shape using the spherical trigonometry. However, the computing time is very severe. In this paper, we project the sphere into the planar to reduce the computing time. An experimental result shows that the performance of the proposed method is excellent to both distance and bearing calculations in close region. Also, the computing time is reduced from $4.95{\times}10^{-4}$ seconds to $1.648{\times}10^{-4}$ seconds.

• The Journal of the Acoustical Society of Korea
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• v.35 no.6
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• pp.411-418
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• 2016

In this paper, we developed a new method estimating the location of ballistic missile launching using infrasound signals. Infrasound signal generated from the North Korea's ballistic missile launch was used as source data and its signal was recorded at KMA (Korea Meteorological Administration) infrasound stations located at Cheorwon and Yanggu. Time-frequency analysis, TDOA (Time Delay of Arrival) method and spherical trigonometry were applied for data processing of signal recorded and occurrence location detection. We could confirm the outstanding performance of the algorithm estimating source location which was only 3 km apart from the actual launching site.

• Journal of Astronomy and Space Sciences
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• v.23 no.1
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• pp.71-90
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• 2006

This study is about , which constitutes Honcheonui(an armillary sphere) put of $\ulcorner$Uigijipseol$\lrcorner$ (Volume I) together with and -dealt with earlier than this subject. The study's construction on the text of is organized into 15 categories, including installation, observations, instructions for use, astronomical theories and formulas, etc. This study provides easy-to-understand illustrations of the figures shown in the original and contains detailed descriptions of the related calculation procedures. In the 'Instructions for Use of Honcheonui' discussed here, the readers are introduced to astronomical computation systems, mainly based on spherical trigonometry and proportional methods. The section also provides systematic and detailed reviews of astronomical theories and calculation procedures, allowing you to assess the level of astronomy knowledge at that time.

• Publications of The Korean Astronomical Society
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• v.39 no.1
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• pp.13-26
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• 2024

The armillary sphere, an astronomical observation device embodying the Orbital Heaven Theory of the Later Han Dynasty in China, holds both historical and scientific significance. It has been produced in various forms by many individuals since its inception in the era of King Sejong in the Joseon Dynasty. A prominent figure in this field was Nam Byeong-cheol (南秉哲, 1817-1863), known for his work 'Uigijipseol' (儀器輯說), published in 1859, which detailed the history, production methods, and usage of the armillary sphere. This text particularly highlights 21 applications of the armillary sphere, divided into 33 measurements, covering aspects like installation, time, and positional measurements, supplemented with explanations of spherical trigonometry. Despite numerous records of the armillary sphere's design during the Joseon Dynasty, detailed usage information remains scarce. In this study, the 33 measurements described in 'Uigijipseol' (儀器輯說) were systematically classified into six for installation, nineteen for position measurement, seven for time measurement, and one for other purposes. Additionally, the measurement methods were analyzed and organized by dividing them into the ecliptic ring, moving equatorial ring, and fixed equatorial ring of the armillary sphere. In other words, from a modern astronomical perspective, the results of schematization for each step were presented by analyzing it from the viewpoint of longitude, right ascension, and solar time. Through the analysis of Nam's armillary sphere, this study not only aims to validate the restoration model of the armillary sphere but also suggests the potential for its use in basic astronomical education based on the understanding of the 19th-century Joseon armillary sphere.

• The Journal of The Korea Institute of Intelligent Transport Systems
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• v.15 no.1
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• pp.66-76
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• 2016

Determination of type of a vehicle is being used in various areas such as collecting tolls, collecting statistical traffic data and traffic prognosis. Because most of the vehicle type classification systems depend on vehicle length indirectly or directly, highly reliable automatic vehicle length measurement system is crucial for them. This study makes use of a pencil beam laser rangemeter and devises a mechanical device which rotates the laser rangemeter. The implemented system measures the range between a point and the laser rangemeter then indicates it as a spherical coordinate. We obtain several silhouettes of cross section of the vehicle, the rate of change of the silhouettes, signs of the rates then squares the rates to apply cell averaging constant false alarm rate (CA-CFAR) technique to find out where the border is between the vehicle and the background. Using the border and trigonometry, we calculated the length of the vehicle and confirmed that the calculated vehicle length is about 94% of actual length.