In one of his astronomical works the prominent arabic medieval scientists Thabit ibn Qurra （836-901） studied the visible motion of the Sun and found the points, where its velocity is maximum or minimum. He also lbun...In one of his astronomical works the prominent arabic medieval scientists Thabit ibn Qurra （836-901） studied the visible motion of the Sun and found the points, where its velocity is maximum or minimum. He also lbund the points on the ecliptic, where this velocity is equal to the average velocity of the Sun over all the ecliptic. For this purpose he used the idea of infinitely small arcs and their ratios in different points of the circle. The great scientist Leonard Euler （1707-1783） introduced in his works on spherical trigonometry the line-element ds of the surface of the sphere, i.e. the differential of the arc length. He constructed the spherical trigonometry as an inner geometry on the surface of the sphere. He replaced the trigonometry lines, which were in use befbre him, by trigonometric functions.展开更多
The spherical model of time and location calculation of the lightning discharge is given. The calculations are made by means of radio signals detection by sensors of the distributed network. The full solution of a pro...The spherical model of time and location calculation of the lightning discharge is given. The calculations are made by means of radio signals detection by sensors of the distributed network. The full solution of a problem of lightning discharge cloud-ground type location for three sensors is given. Based on this task the lightning location method for a network of sensors was developed. By means of computational experiments, the analysis of accuracy of the model depending on radio signals detection accuracy at observing stations was done.展开更多
On the unit sphere, the geometric problem of calculating the position of a point relative to three given points is considered. We know the length of three spherical segments that go out from the given points in the di...On the unit sphere, the geometric problem of calculating the position of a point relative to three given points is considered. We know the length of three spherical segments that go out from the given points in the direction of the unknown point. The requirement must be fulfilled: the distance from each point to an unknown point must be equal to the sum of the length of the segment outgoing from this point, and some increment, the same for all three segments. In the article, the conditions for the solvability of a geometric problem are established by the methods of spherical trigonometry and vector algebra. It is proved that when they are fulfilled, the problem is always solvable. The number of solutions is two, except in rare cases where there is only one solution. A solution method is presented. One of the practical applications is the problem of determining the time and location of a cloud-to-ground lightning discharge, which is directly reduced to this problem.展开更多
The study of the dihedral f-tilings of the sphere S2 whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in a previous work. In this paper we continue this classification presenting the st...The study of the dihedral f-tilings of the sphere S2 whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in a previous work. In this paper we continue this classification presenting the study of all dihedral spherical f-tilings by scalene triangles and isosceles trapezoids in some cases of adjacency.展开更多
In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by ...In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by two discrete families denoted by ε^m,m ≥ 2,m ∈ N,F^k,k ≥ 4,k ∈ N and two sporadic tilings denoted by G and H.展开更多
文摘In one of his astronomical works the prominent arabic medieval scientists Thabit ibn Qurra （836-901） studied the visible motion of the Sun and found the points, where its velocity is maximum or minimum. He also lbund the points on the ecliptic, where this velocity is equal to the average velocity of the Sun over all the ecliptic. For this purpose he used the idea of infinitely small arcs and their ratios in different points of the circle. The great scientist Leonard Euler （1707-1783） introduced in his works on spherical trigonometry the line-element ds of the surface of the sphere, i.e. the differential of the arc length. He constructed the spherical trigonometry as an inner geometry on the surface of the sphere. He replaced the trigonometry lines, which were in use befbre him, by trigonometric functions.
文摘The spherical model of time and location calculation of the lightning discharge is given. The calculations are made by means of radio signals detection by sensors of the distributed network. The full solution of a problem of lightning discharge cloud-ground type location for three sensors is given. Based on this task the lightning location method for a network of sensors was developed. By means of computational experiments, the analysis of accuracy of the model depending on radio signals detection accuracy at observing stations was done.
文摘On the unit sphere, the geometric problem of calculating the position of a point relative to three given points is considered. We know the length of three spherical segments that go out from the given points in the direction of the unknown point. The requirement must be fulfilled: the distance from each point to an unknown point must be equal to the sum of the length of the segment outgoing from this point, and some increment, the same for all three segments. In the article, the conditions for the solvability of a geometric problem are established by the methods of spherical trigonometry and vector algebra. It is proved that when they are fulfilled, the problem is always solvable. The number of solutions is two, except in rare cases where there is only one solution. A solution method is presented. One of the practical applications is the problem of determining the time and location of a cloud-to-ground lightning discharge, which is directly reduced to this problem.
基金Research funded by the Portuguese Government through the FCT-Fundaao para a Ciencia e a Tecnologia-under the project PEst-OE/MAT/UI4080/2011
文摘The study of the dihedral f-tilings of the sphere S2 whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in a previous work. In this paper we continue this classification presenting the study of all dihedral spherical f-tilings by scalene triangles and isosceles trapezoids in some cases of adjacency.
基金Supported by FEDER funds through COMPETE Operational Programme Factors of Competitiveness(Programa Operacional Factores de Competitividade)Supported by FSE+3 种基金Supported by Portuguese funds through the Center for Researchand Development in Mathematics and Applications(University of Aveiro)the Portuguese Foundation for Science and Technology(FCT Fundao para a Ciência e a Tecnologia)project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690supported partially by an NSERC Canada Discovery Grant
文摘In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by two discrete families denoted by ε^m,m ≥ 2,m ∈ N,F^k,k ≥ 4,k ∈ N and two sporadic tilings denoted by G and H.
