Spherical Trigonometry Research Articles

Reducing carbon footprint of ships in voyage planning: A case study of Atlantic passage

Determining the most efficient ship routes is crucial for ensuring both safety and fuel savings during voyage planning. Great circle Navigation (GC) and weather routing (WR) are the two most common methods used by navigators for ocean crossings. GC Navigation is renowned for its potential to save time and fuel by following shorter routes over long distances; however, it may expose vessels to adverse weather conditions and sea states due to navigating at higher latitudes. On the other hand, weather routing integrates pilot charts and meteorological data to identify safer routes, albeit potentially longer, minimizing risks associated with rough weather. This study focuses on route planning for a handy-sized tanker contracting a westbound Atlantic ocean voyage within a specific timeframe. The main objective of the study is to evaluate the efficacy of these two methods. The GC route components were computed using spherical trigonometry equations, while weather routing planning relied on pilot charts and meteorological data. Environmental conditions were simulated and tested in the Transas Full Mission Simulator (NTPRO 4000). The results reveal that the WR method provides 21.3% higher energy efficiency than GC. The insights derived from this research contribute significantly to enhancing the operational efficiency and safety standards of commercial vessels.

THREE DIMEI\SIONAL INTERSECTIOIN SBY AhIGULAR OBSERVATIONS

The research introduces a new mathematical model for computing the position of points in space(X, Y, Z) by the intersection of inclined angles derived from the measured horizontal and verticalangles using spherical trigonometry principles with the aid of the conventional surveying instrument(Theodolite), The most probable value (m.p.v) of the position of a point should be computed by themethod of ieast squares when the redundancy in observations exists and then it provides us with theprecision indices for the computed coordinates.

COMPARISON OF SPHERICAL TRIGONOMETRY METHOD, JEAN MEEUS ALGORITHM AND GOOGLE QIBLA FINDER IN DETERMINING OF THE QIBLA DIRECTION OF ISLAMIC HOSPITAL

Accuracy in facing the Qibla is an essential part of performing prayers. This vital value is evident when many mosques are built in public places. This article is qualitative with field data sources, namely coordinate points at the Jemursari Islamic Hospital mosque, Surabaya Islamic Hospital, and Al-Irsyad Hospital Surabaya. Once collected, the data was analyzed using three methods for calculating Qibla direction, namely Spherical Trigonometry, Jean Meeus, and Google Qibla Finder. This article found that the three methods obtained the same results at the Jemursari Islamic Hospital at 294°3'5", at the Surabaya Islamic Hospital at 294° 3'6", and at the Al-Irsyad Surabaya Hospital at 294°3'5 ". However, there is a difference between calculations and field measurements of 2°–7°, including within the Qibla deviation tolerance. It can be concluded that these three methods can accurately determine the Qibla direction in various locations. However, re-checking is required if the measurements exceed the tolerance limits.

Mythology, Cosmogonies, and Indian Science Fiction

This paper examines the cultural significance of the intersection between Indian mythology, cosmogonies, and science fiction. India has a rich history of diverse religious traditions and corresponding world views, many of which have influenced each other. The Hindu pantheon has included the Sun from early times. With the planets also gaining in importance after the rise of astrology, the need to predict their paths saw the development of astronomical observations and spherical trigonometry. These serve as examples of how scientific and religious ideas have interacted throughout Indian history. Despite a strong tradition of fiction in India, there is a lack of science fiction that combines elements of mythology and astronomy except in superficial ways. This paper explores the potential reasons for this gap and argues that an examination of this genre can offer insight into the ways in which science and religion are perceived and valued in contemporary Indian society. The paper also offers a commentary on the current state of Indian science fiction that blends mythology and astronomy

Bistable reconfigurable origami metamaterials with high load-bearing and low state-switching forces

Bistable shape-reconfigurable metamaterials can achieve large morphological changes under a relatively low state-switching force and can maintain the configuration autonomously after removing the force. However, the state-switching force and the load-bearing forces are usually coupled in the same direction, resulting in low loadability at the deployed state. Here we separate the load-bearing direction and the state-switching direction of bistable unit, and combine the directional folding feature of origami to design a shape-reconfigurable metamaterial with high bearing force and low switching force, named mechanical relay metamaterial. The feasibility of the design concept is first validated experimentally. The bearing force of the mechanical relay metamaterial is at least 4 times the switching force and 100 times the gravity of self-weight. Then the geometric design domain of the unit is derived by spherical trigonometry. Finally, the effect of geometric parameters on the load carrying capacity of the unit is investigated by finite element simulations, and the optimal geometric parameters are determined. This work provides an effective solution for bistable shape-reconfigurable metamaterials with high bearing force and low switching force.

Wave description of geometric phase.

Since Pancharatnam's 1956 discovery of optical geometric phase and Berry's 1984 discovery of geometric phase in quantum systems, researchers analyzing geometric phase have focused almost exclusively on algebraic approaches using the Jones calculus, or on spherical trigonometry approaches using the Poincaré sphere. The abstracted mathematics of the former and the abstracted geometry of the latter obscure the physical mechanism that generates geometric phase. We show that optical geometric phase derives entirely from the superposition of waves and the resulting shift in the location of the wave maximum. This wave-based model provides a way to visualize how geometric phase arises from relationships between waves, and from the transformations induced by optical elements. We also derive the relationship between the geometric phase of a wave by itself and the phase exhibited by an interferogram, and provide the conditions under which the two match one another.

Webgis Application For Searching Nu Mosque In East Java Using The Haversine Method

Indonesia is the country with the largest number of mosques in the world. In East Java, there are many Islamicorganizations, one of which is Nahdhotul Ulama (NU). Nahdhotul Ulama (NU) is one of the socio-religiousorganizations in Indonesia which was founded on 16 Rajab 1344 or to coincide with January 31, 1926 in Surabaya onthe initiative of two leading traditional scholars at that time, K.H Hasyim Asy'ari and K.H Abdul Wahab Hasbullah.Geographic Information System is a computer system that is used to collect, check, integrate, and analyze informationrelated to the earth's surface. Basically, the term geographic information system is a combination of three mainelements, namely system, information, and geography. Thus, an understanding of these three main elements will bevery helpful in understanding GIS. The haversine is an important equation in navigation, giving the great circle thedistance between any two points on the sphere of longitude and latitude. It is a special case of the more commonformula in spherical trigonometry, the law of haversines, relating the sides and angles of a spherical triangle (SphericalTrigonometry).From the calculation results of the haversine algorithm the first trial formula is at the user's location at 1. Jl. RayaSurabaya - Malang, Sawahan, Buduran, Buduran District, Sidoarjo Regency, East Java 61252 found the closest mosquebased on the heversine formula is Darussalam Mosque, jl. Kedemang Singomenggolo no 98 Buduran with a distanceof 1,396 Km. From the calculation results of the haversine algorithm, the first trial formula at the user's location on Jl.Raya Suko No.55, RT.01/RW.01, Ngemplak, Cemeng Kalang, Sidoarjo District, Sidoarjo Regency, East Java 61224found the closest mosque based on the heversine formula is Miftahul Jannah Mosque, jl. Banjar poh village BanjarBendo with a distance of 1,878 Km

Differences of Salat Time Recording: Analytical Contemporary Method of Kitab al-Khulashah al-Wafiyyah

The discussion on the reckoning of prayer times contained in the Khulashah book has its own characteristics. The initial calculation process for prayer times has referred to the triangular ball algorithm system. Besides, there are also daqoiqut tamkin corrections presented through tables, and calculation methods using calculators. This interesting calculation can be studied; how is the transformation system of prayer time coordinates and how is the accuracy level of the results of the initial prayer time calculation. This study uses a library research paradigm with a qualitative theory verification approach using an arithmetic approach and the results are presented descriptively, then compared with similar methods. The results show that the coordinate transformation has referred to the modern astronomical system and the calculation system has used spherical trigonometry (spherical triangle). The calculation results have differences ranging from 0 to 3 minutes with contemporary methods. It is caused by differences in data and methods. If the calculation using the contemporary method there is a difference of 0 seconds to 12 seconds, this indicates that when the calculation is made modifications to the contemporary calculation system will produce a more accurate calculation.

Diferensiasi Hisab Waktu Salat (Studi Kitab al-Khulashah al-Wafiyyah dan Kontemporer)

The discussion on the reckoning of prayer times contained in the Khulashah book has its own characteristics. The initial calculation process for prayer times has referred to the triangular ball algorithm system. Besides, there are also daqoiqut tamkin corrections presented through tables, and calculation methods using calculators. This study uses a library research paradigm with a qualitative theory verification approach using an arithmetic approach and the results are presented descriptively, then compared with similar methods. The results show that the coordinate transformation has referred to the modern astronomical system and the calculation system has used spherical trigonometry (spherical triangle). The calculation results have differences ranging from 0 to 3 minutes with contemporary methods. It is caused by differences in data and methods. If the calculation using the contemporary method there is a difference of 0 seconds to 12 seconds, this indicates that when the calculation is made modifications to the contemporary calculation system will produce a more accurate calculation

A Method for Calculating the Equation of Noon (an English translation of Methodus Computandi Aequationem Meridiei)

In this paper Euler presents a method for determining solar noon, the time at which the Sun crosses the meridian. The method requires the times of two observations of the Sun, one in the morning and one in the afternoon, at equal altitudes above the horizon. Solar noon is approximately the midpoint between two such observations, but, since the declination of the Sun will have changed during the day, a correction term, called the equation of noon, is required. Euler explains that this term is too large to ignore and discusses the table of values constructed by de la Hire; this table applies only at the latitude of Paris and relies on laborious calculations. For his own method, Euler describes the apparent motion of the Sun using spherical trigonometry and then uses differentials to complete the calculation with sufficient accuracy for his purposes. He provides examples and claims that his method makes it practical to construct a table at whatever latitude is required.

A Novel Analytical Solution Method for Celestial Positioning

Traditional celestial navigation adopts the intercept method or Sumner line method in order to solve the fix problems of celestial sight reduction. While an estimated position is often needed in order to complete celestial positioning, these methods are not rigorous and they have many shortcomings. Thus, this study has analyzed the geometry of the circle of equal altitude and the two-body problem. Furthermore, vector methods are proposed without using spherical trigonometry, in order to meet navigation needs. Finally, web mapping was used to display the results. The results show that the two methods that are proposed in this study had better accuracy, practical convenience, and faster computing speed than those which have been proposed by other studies. The two methods’ capability to avoid complicated manual calculations and chart work processes makes them highly suitable as an alternative to the use of global navigation satellite systems. These methods enable maritime school students to quickly understand the principles of celestial navigation through computing and graphical interfaces.

AUTOMATIC EXTRACTION OF SOLAR AND SENSOR IMAGING GEOMETRY FROM UAV-BORNE PUSH-BROOM HYPERSPECTRAL CAMERA

Abstract. Calculating solar-sensor zenith and azimuth angles for hyperspectral images collected by UAVs are important in terms of conducting bi-directional reflectance function (BRDF) correction or radiative transfer modeling-based applications in remote sensing. These applications are even more necessary to perform high-throughput phenotyping and precision agriculture tasks. This study demonstrates an automated Python framework that can calculate the solar-sensor zenith and azimuth angles for a push-broom hyperspectral camera equipped in a UAV. First, the hyperspectral images were radiometrically and geometrically corrected. Second, the high-precision Global Navigation Satellite System (GNSS) and Inertial Measurement Unit (IMU) data for the flight path was extracted and corresponding UAV points for each pixel were identified. Finally, the angles were calculated using spherical trigonometry and linear algebra. The results show that the solar zenith angle (SZA) and solar azimuth angle (SAA) calculated by our method provided higher precision angular values compared to other available tools. The viewing zenith angle (VZA) was lower near the flight path and higher near the edge of the images. The viewing azimuth angle (VAA) pattern showed higher values to the left and lower values to the right side of the flight line. The methods described in this study is easily reproducible to other study areas and applications.

Gravitational lens on de Sitter background

Gravitational lenses are examined in a de Sitter (dS) background, for which the existence of the dS horizon is taken into account and hyperbolic trigonometry is used together with the hyperbolic angular diameter distance. Spherical trigonometry is used to discuss a gravitational lens in anti--de Sitter (AdS) background. The difference in the form among the dS/AdS lens equations and the exact lens equation in Minkowski background begins at the third order, when a small angle approximation is used in terms of lens and source planes. The angular separation of lensed images is decreased by the third-order deviation in the dS lens equation, while it is increased in AdS. In the present framework on the dS/AdS backgrounds, we discuss also the deflection angle of light, which does not include any term of purely the cosmological constant. Despite the different geometry, the deflection angle of light rays in hyperbolic and spherical geometry can take the same form. Through a coupling of the cosmological constant with lens mass, the separation angle of multiple images is larger (smaller) in dS (AdS) than in the flat case, for a given mass, source direction, and angular diameter distances among the lens, receiver and source.

Efficiency Comparison of Quaternions and Spherical Trigonometry Formulas for 3D Rotations

Efficiency Comparison of Quaternions and Spherical Trigonometry Formulas for 3D Rotations

Implementation of Ibn al-Haytham’s Method for Determining Qibla Direction Using Raspberry Pi

Ibn al-Haytham or better known as Alhazen (965-1040 AD) was a great Muslim scientist who mastered many disciplines. One of his works entitled "Qawl fi Samt al-Qibla bi al-Hisab" describes a mathematical method for determining the direction of Qibla. In this paper, Ibn al-Haytham's algorithm is modified, thus the algorithm can determine the Qibla direction on the entire surface of the globe. Then, Ibn al-Haytham's mathematical method is compared with modern spherical trigonometry methods and our previous research using al-Biruni’s method. The computational results show that all methods have the same accuracy, and show that ibn al-Haytham's method is still relevant now. Therefore, the modified ibn al-Haytham method algorithm was implemented to develop a Qibla direction device and interface based on Raspberry Pi 4, GPS module, digital compass, and Processing 3 software. The implementation results show that the device can display the Qibla direction interface according to numerical calculations with high accuracy, real-time, and dynamic interface.

Coda

These interesting situations in which generics play a key role in interactional pragmatics sparked my memory of solid geometry and spherical trigonometry class at Stuyvesant High School in the early 1960s. Each morning our instructor, the somewhat irascible Mr. Burns, would start off by asking a question on the day's material, calling for a response by ‘[student surname]’. Stuyvesant, in those days an all-male institution, functioned, like prep schools, on a surname basis for both reference and address; the teachers’ names were prefaced by Mr. or Mrs. or Miss, while student names had no prefixed title.

Antoni Wyrwicz – professor of higher mathematics at the Vilnius university (1817–1831)

Antoni Wyrwicz (1791–1865) was a professor of higher mathematics at the Vilnius university in the period from 1817 until 1831. Wyrwicz taught algebra by tractate of Sniadecki and Burdon, spherical trigonometry by Sniadecki, Cagnoli, Delambre Neper, astronomy by Delambre and Biot. Wyrwicz in 1819 published translation of Biot Analytical geometry. From 1824 until 1831 Wyrwicz taught differential and integral calculus by Lacroix and Lagrange. He translated 6 school books for algebra from Lacroix and 6 school books for geometry from Legendre.

An Axiom of True Courses Calculation in Great Circle Navigation

Based on traditional expressions and spherical trigonometry, at present, great circle navigation is undertaken using various navigational software packages. Recent research has mainly focused on vector algebra. These problems are calculated numerically and are thus suited to computer-aided great circle navigation. However, essential knowledge requires the navigator to be able to calculate navigation parameters without the use of aids. This requirement is met using spherical trigonometry functions and the Napier wheel. In addition, to facilitate calculation, certain axioms have been developed to determine a vessel’s true course. These axioms can lead to misleading results due to the limitations of the trigonometric functions, mathematical errors, and the type of great circle navigation. The aim of this paper is to determine a reliable trigonometric function for calculating a vessel’s course in regular and composite great circle navigation, which can be used with the proposed axioms. This was achieved using analysis of the trigonometric functions, and assessment of their impact on the vessel’s calculated course and established axioms.

Metode Perhitungan Awal Bulan Kamariah Uzal Syahruna dalam Kitab Tasḥīl al-Mithāl wa al-Aqwāl Li ‘Amal al-Hilāl

Abstract: This study aims to determine the method of ḥisāb early in the lunar month in the book Tasḥīl al-Mithāl by Uzal Syahruna and assess its advantages and disadvantages compared to the same process in other books. It is undeniable that the dichotomization between the existence of classical ḥisāb books and contemporary ḥisāb books with the beginning to fade in people's belief in the results occurred in Falak Science. Through literature study and interviews, this paper finds that the method of determining the beginning of the lunar month in the book falls into the ḥisāb ḥaqīqī bi taḥqīq category. The theory and calculation system are based on modern astronomical formulas (spherical trigonometry), using calculators and computational programs. The advantage of this book is that it establishes its procedures on current astronomical formulas, the output of ḥisāb is not much different from other ḥaqīqī bi taḥqīq books. Simultaneously, the weakness is that it still uses unlimited astronomical data as a basis. Some have not been corrected, especially for determining irtifā 'al-hilāl and hilāl mar'i corrections.

Estudo das quantidades de soluções para os casos V e VI dos triângulos esféricos oblíquos

Os entes humanos ao se tornarem sedentários começaram a perceber a necessidade de se conhecer a própria localização. Ao estudarem as estrelas, perceberam um referencial seguro para tal localização e passaram a utilizá-las para o cálculo da posição, surgindo assim a Astronomia de Posição. Para o cálculo da posição, por sua vez, foi necessário um modelo matemático, sendo natural pensar-se na Esfera. Natural ainda foi estender a Trigonometria Plana à Esfera, surgindo assim a Trigonometria Esférica. O estudo da Trigonometria Esférica se faz necessário para a formação dos engenheiros cartógrafos e agrimensores, pois o conhecimento das ferramentas para a resolução do Triângulo Esférico vai ser fundamental para a compreensão dos mecanismos de posicionamento quer seja pela Astronomia de Posição quer seja pela Geodésia. De modo a contribuir com o ensino da Trigonometria Esférica, a revisão conceitual apresentada neste trabalho se propõe a estudar a quantidade de soluções para os triângulos esféricos oblíquos do Caso V e do Caso VI, respectivamente, quando são conhecidos dois lados e o ângulo oposto a um dos lados e quando são conhecidos dois ângulos e o lado oposto a um dos ângulos. Os triângulos pertencentes a estes casos podem apresentar nenhuma, uma solução, duas ou infinitas soluções. O conhecimento sobre determinado assunto somente pode ser de caráter científico quando há o conhecimento sobre as etapas dedutivas relacionadas àquele assunto, por isso apresenta-se o presente trabalho para que o estudante possa conhecer cientificamente partes da Trigonometria Esférica.