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Keywords = Mollweide

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30 pages, 2813 KiB  
Article
Transformation of Trigonometric Functions into Hyperbolic Functions Based on Cable Statics
by Julian Garcia-Guarin
Appl. Sci. 2025, 15(5), 2647; https://doi.org/10.3390/app15052647 - 1 Mar 2025
Viewed by 930
Abstract
Trigonometric functions are widely used to express relationships between the sides and angles of triangles, being fundamental in a wide variety of fields in science and engineering. Previous research indicates that the Gudermann function connects hyperbolic and trigonometric functions without requiring complex numbers, [...] Read more.
Trigonometric functions are widely used to express relationships between the sides and angles of triangles, being fundamental in a wide variety of fields in science and engineering. Previous research indicates that the Gudermann function connects hyperbolic and trigonometric functions without requiring complex numbers, while the Mercator projection maps the Earth’s spherical surface onto a cylindrical plane. This article presents four key contributions derived from hyperbolic functions, with the main proof applying Newton’s first law in the static case of cables. First, a new method relates right triangle formulas to the sides of a right triangle, facilitating vector decomposition along the X and Y axes. Second, a right triangle function with a hyperbolic angle is proposed, relating the three sides of a right triangle and the hyperbolic angle, offering an alternative to the Pythagorean theorem. Third, the law of hyperbolic cosines and the law of hyperbolic tangents is applied to trigonometric problems. Fourth, the hyperbolic Mollweide’s formula is used to solve oblique triangles. These results demonstrate the potential of hyperbolic transformations in engineering and mathematical contexts, for both education and research. Future investigations should include experimental and analytical tests to further extend the applications to all branches based on mathematics. Full article
(This article belongs to the Section Applied Physics General)
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13 pages, 41711 KiB  
Article
Comparison of FOSS4G Supported Equal-Area Projections Using Discrete Distortion Indicatrices
by Luís Moreira de Sousa, Laura Poggio and Bas Kempen
ISPRS Int. J. Geo-Inf. 2019, 8(8), 351; https://doi.org/10.3390/ijgi8080351 - 9 Aug 2019
Cited by 11 | Viewed by 6111
Abstract
This study compares the performance of five popular equal-area projections supported by Free and Open Source Software for Geo-spatial (FOSS4G)—Sinusoidal, Mollweide, Hammer, Eckert IV and Homolosine. A set of 21,872 discrete distortion vindicatrices were positioned on the ellipsoid surface, centred on the cells [...] Read more.
This study compares the performance of five popular equal-area projections supported by Free and Open Source Software for Geo-spatial (FOSS4G)—Sinusoidal, Mollweide, Hammer, Eckert IV and Homolosine. A set of 21,872 discrete distortion vindicatrices were positioned on the ellipsoid surface, centred on the cells of a Snyder icosahedral equal-area grid. These indicatrices were projected on the plane and the resulting angular and distance distortions computed, all using FOSS4G. The Homolosine is the only projection that manages to minimise angular and distance distortions simultaneously. It yields the lowest distortions among this set of projections and clearly outclasses when only land masses are considered. These results also indicate the Sinusoidal and Hammer projections to be largely outdated, imposing too large distortions to be useful. In contrast, the Mollweide and Eckert IV projections present trade-offs between visual expression and accuracy that are worth considering. However, for the purposes of storing and analysing big spatial data with FOSS4G the superior performance of the Homolosine projection makes its choice difficult to avoid. Full article
(This article belongs to the Special Issue Open Science in the Geospatial Domain)
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