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Keywords = equilateral hyperbola

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30 pages, 24914 KB  
Article
Algorithm to Find and Analyze All Configurations of Four-Bar Linkages with Different Geometric Loci Degenerate Forms
by Giorgio Figliolini, Chiara Lanni and Luciano Tomassi
Symmetry 2025, 17(8), 1171; https://doi.org/10.3390/sym17081171 - 22 Jul 2025
Viewed by 992
Abstract
A general algorithm to determine the coupler link geometric loci, such as centrodes, inflection and return circles, as well as circling-point and centering-point curves, is formulated to analyze any type of four-bar linkages with the main target to find all mechanism configurations, in [...] Read more.
A general algorithm to determine the coupler link geometric loci, such as centrodes, inflection and return circles, as well as circling-point and centering-point curves, is formulated to analyze any type of four-bar linkages with the main target to find all mechanism configurations, in which at least one of the above-mentioned loci degenerates. Thus, different types of four-bar linkages, such as crank-rocker, double-crank, double-rocker and triple-rocker, are classified according to Grashof’s law, in order to distinguish and analyze their corresponding geometric loci. In particular, the proposed algorithm is based on four diagrams of the angular velocity ratios versus the mechanism driving angle, which consider the links pairs of input/output, input/coupler, and output/coupler, along with those of coupler/input and coupler/output for their relative motion. These diagrams allow the determination of all mechanism configurations according to Freudenstein’s theorems, where the aforementioned geometric loci degenerate into straight lines, including the line at infinity, ϕ-curves, and/or equilateral hyperbolas. This algorithm has been implemented in Matlab in order to run several examples regarding different four-bar linkages, according to Grashof’s law, and analyzing the degenerate forms of their inflection and return circles, as well as the circling-point and centering-point curves, that are also validated by using the collineation axis. Full article
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9 pages, 609 KB  
Article
On Yiu’s Equilateral Triangles Associated with a Kiepert Hyperbola
by Cherng-tiao Perng
Geometry 2025, 2(3), 10; https://doi.org/10.3390/geometry2030010 - 1 Jul 2025
Viewed by 483
Abstract
In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are [...] Read more.
In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we provide proof of his assertions. Furthermore, as an analogue of Lemoine’s problem, we formulated and answered the question of how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle. Full article
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16 pages, 322 KB  
Article
Quaternionic Product of Equilateral Hyperbolas and Some Extensions
by Mircea Crasmareanu and Marcela Popescu
Mathematics 2020, 8(10), 1686; https://doi.org/10.3390/math8101686 - 1 Oct 2020
Cited by 2 | Viewed by 8993
Abstract
This note concerns a product of equilateral hyperbolas induced by the quaternionic product considered in a projective manner. Several properties of this composition law are derived and, in this way, we arrive at some special numbers as roots or powers of unit. Using [...] Read more.
This note concerns a product of equilateral hyperbolas induced by the quaternionic product considered in a projective manner. Several properties of this composition law are derived and, in this way, we arrive at some special numbers as roots or powers of unit. Using the algebra of octonions, we extend this product to oriented equilateral hyperbolas and to pairs of equilateral hyperbolas. Using an inversion we extend this product to Bernoulli lemniscates and q-lemniscates. Finally, we extend this product to a set of conics. Three applications of the given products are proposed. Full article
(This article belongs to the Special Issue Geometric Methods and their Applications)
18 pages, 6974 KB  
Article
A New Ellipse or Math Porcelain Service
by Valery Ochkov, Massimiliano Nori, Ekaterina Borovinskaya and Wladimir Reschetilowski
Symmetry 2019, 11(2), 184; https://doi.org/10.3390/sym11020184 - 4 Feb 2019
Cited by 5 | Viewed by 6740
Abstract
Egglipse was first explored by Maxwell, but Descartes discovered a way to modify the pins-and-string construction for ellipses to produce more egg-shaped curves. There are no examples of serious scientific and practical applications of Three-foci ellipses until now. This situation can be changed [...] Read more.
Egglipse was first explored by Maxwell, but Descartes discovered a way to modify the pins-and-string construction for ellipses to produce more egg-shaped curves. There are no examples of serious scientific and practical applications of Three-foci ellipses until now. This situation can be changed if porcelain and ellipses are combined. In the introduced concept of the egg-ellipse, unexplored points are observed. The new Three-foci ellipse with an equilateral triangle, a square, and a circle as “foci” are presented for this application and can be transformed by animation. The new elliptic-hyperbolic oval is presented. The other two similar curves, hyperbola and parabola, can be also used to create new porcelain designs. Curves of the order of 3, 4, 5, etc. are interesting for porcelain decoration. An idea of combining of 3D printer and 2D colour printer in the form of 2.5D Printer for porcelain production and painting is introduced and listings functions in Mathcad are provided. Full article
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