Random spherical triangles II: Shape densities

1989 ◽  
Vol 21 (3) ◽  
pp. 581-594 ◽  
Author(s):  
Huiling Le
Keyword(s):  
Shape Space ◽  
Spherical Triangle ◽  
Small Circle ◽  
Exact Evaluation ◽  
Spherical Triangles ◽  
Angular Radius

This paper gives the exact evaluation of the shape density on the shape space Σ(S2, 3) for a labelled random spherical triangle whose vertices are i.i.d.-uniform in a ‘cap' of S2 bounded by a ‘small' circle of angular radius ρ0.

Random spherical triangles II: Shape densities

1989 ◽  
Vol 21 (03) ◽  
pp. 581-594 ◽  
Author(s):  
Huiling Le
Keyword(s):  
Shape Space ◽  
Spherical Triangle ◽  
Small Circle ◽  
Exact Evaluation ◽  
Spherical Triangles ◽  
Angular Radius

This paper gives the exact evaluation of the shape density on the shape space Σ(S 2, 3) for a labelled random spherical triangle whose vertices are i.i.d.-uniform in a ‘cap' of S2 bounded by a ‘small' circle of angular radius ρ 0 .

Areas, Angles, and Polyhedra

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Eighteenth Century ◽  
Unit Sphere ◽  
Regular Polyhedron ◽  
Spherical Triangle ◽  
French Mathematician ◽  
Regular Polyhedra ◽  
Leonhard Euler ◽  
Spherical Triangles

This chapter explains how to find the area of an angle or polyhedron. It begins with a discussion of how to determine the area of a spherical triangle or polygon. The formula for the area of a spherical triangle is named after Albert Girard, a French mathematician who developed a theorem on the areas of spherical triangles, found in his Invention nouvelle. The chapter goes on to consider Euler's polyhedral formula, named after the eighteenth-century mathematician Leonhard Euler, and the geometry of a regular polyhedron. Finally, it describes an approach to finding the proportion of the volume of the unit sphere that the various regular polyhedra occupy.

Random spherical triangles I: Geometrical background

1989 ◽  
Vol 21 (3) ◽  
pp. 570-580
Author(s):  
Huiling Le
Keyword(s):  
Mathematical Model ◽  
Shape Space ◽  
Companion Paper ◽  
Statistical Properties ◽  
Data Sets ◽  
Riemannian Structure ◽  
Geometric Structures ◽  
Fundamental Properties ◽  
Random Mechanism ◽  
Spherical Triangles

In this paper we identify the shape space Σ(S2, k) for k labelled points on the sphere S2 that gives a mathematical model applicable to data sets whose elements are, or can be represented by, configurations of labelled sequences of points on S2 and for which the fundamental properties of interest are the shapes of these configurations, and we examine the geometric structures on the space, especially the riemannian structure on Σ(S2, 3). In a companion paper (pp. 581–594) we investigate the statistical properties of such shapes when the k points are generated by a random mechanism.

scholarly journals On the special spherical triangles for physical and cosmological applications

2021 ◽  
pp. 112-114
Author(s):  
Kalimuthu S
Keyword(s):  
Spherical Triangle ◽  
Spherical Triangles ◽  
The Way

It is well known that a spherical triangle of 270 degree triangle is constructible on the surface of a sphere; a globe is a good example. Take a point (A) on the equator, draw a line 1/4 the way around (90 degrees of longitude) on the equator to a new point (B).

scholarly journals Dihedral f-tilings of the sphere by rhombi and triangles

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Ana Breda ◽  
Altino F. Santos
Keyword(s):  
Spherical Triangle ◽  
International Audience ◽  
Spherical Triangles

International audience We classify, up to an isomorphism, the class of all dihedral f-tilings of S^2, whose prototiles are a spherical triangle and a spherical rhombus. The equiangular case was considered and classified in Ana M. Breda and Altino F. Santos, Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles. Here we complete the classification considering the case of non-equiangular rhombi.

New Formulae for Combined Spherical Triangles

2018 ◽  
Vol 72 (2) ◽  
pp. 503-512
Author(s):  
Tsung-Hsuan Hsieh ◽  
Shengzheng Wang ◽  
Wei Liu ◽  
Jiansen Zhao
Keyword(s):  
Research Field ◽  
Great Circle ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
Calculation Process ◽  
Different Types ◽  
Celestial Bodies ◽  
Spherical Triangles

Spherical trigonometry formulae are widely adopted to solve various navigation problems. However, these formulae only express the relationships between the sides and angles of a single spherical triangle. In fact, many problems may involve different types of spherical shapes. If we can develop the different formulae for specific spherical shapes, it will help us solve these problems directly. Thus, we propose two types of formulae for combined spherical triangles. The first set are the formulae of the divided spherical triangle, and the second set are the formulae of the spherical quadrilateral. By applying the formulae of the divided spherical triangle, waypoints on a great circle track can be obtained directly without finding the initial great circle course angle in advance. By applying the formulae of the spherical quadrilateral, the astronomical vessel position can be yielded directly from two celestial bodies, and the calculation process concept is easier to comprehend. The formulae we propose can not only be directly used to solve corresponding problems, but also expand the spherical trigonometry research field.

scholarly journals THE PROBLEM OF EMPLACEMENT OF TRIANGULAR GEOMETRIC NET ON THE SPHERE WITH NODES ON THE SAME LEVEL

2017 ◽  
Vol 13 (2) ◽  
pp. 154-160 ◽  
Author(s):  
Vasilij D. Antoshkin
Keyword(s):  
Spherical Triangle ◽  
Minimum Size ◽  
Effective Solution ◽  
Shortest Distance ◽  
Minimum Number ◽  
Triangular Network ◽  
Spherical Triangles

One of the methods of formation of triangular networks in the field is investigated. Conditions of the problem of locating a triangular network in the area are delivered. The criterion for assessing the effectiveness of the solution of the problem is the minimum number of sizes of the dome panels, the possibility of pre-assembly and pre-stressing. The solution of the problem of one embodiment of a triangular network of accommodation in a compatible spherical triangle and, accordingly, on the sphere. Placing on the area of regular and irregular hexagon inscribed in a circle, ie, flat figures or composed in turn of spherical triangles with minimum dimensions of the ribs, is an effective solution in the form of a network formed by circles of minimum radii, ie, circles on a sphere obtained at the touch of three adjacent circles whose centers are at the shortest distance from each other. Task align the supports at one level can be resolved by placement in the regular hexagons and irregular pentagons hexagonsinscribed in a circle of minimum size.

scholarly journals A new instrument for solving spherical triangles

1946 ◽  
Vol 185 (1002) ◽  
pp. 241-249
Keyword(s):  
Spherical Triangle ◽  
Zenith Distance ◽  
Ohm’S Law ◽  
Sources Of Error ◽  
Celestial Navigation ◽  
New Instrument ◽  
Ohm's Law ◽  
Spherical Triangles

A simple calculator for solving the equation of a spherical triangle is described. This was designed for use in celestial navigation by amateur yachtsmen. It is much more rapid to use than the tables commonly employed. It can be made to be accurate to ± 1/2´of arc in calculating zenith distance, and so is suitable for any navigation to the usual standards of accuracy. Computation is based on Ohm’s law and effected by two potentiometers, each having two moving contacts. These are set by dials marked in angle to make resistances represent the appropriate derivations of latitude, declination, hour angle and zenith distance. When one of these is unknown, if the potentiometers are balanced the fourth resistance gives the solution. Sources of error and their avoidance in practical construction are discussed. It can solve any spherical triangle for a body above the observer’s horizon for declinations and latitudes up to about 80°, but for zenith distances less than 30° its accuracy is limited. The instrument is robust, portable and about the same weight as a book of nautical fivefigure tables.

Spherical trigonometry constrained kinematics for a dexterous robotic hand with an articulated palm

Robotica ◽  
2015 ◽  
Vol 34 (12) ◽  
pp. 2788-2805 ◽  
Author(s):  
Evangelos Emmanouil ◽  
Guowu Wei ◽  
Jian S. Dai
Keyword(s):  
Point Clouds ◽  
Joint Space ◽  
Singularity Analysis ◽  
Spherical Triangle ◽  
Design Criteria ◽  
Robotic Hand ◽  
Spherical Trigonometry ◽  
Joint Angles ◽  
Singularity Avoidance ◽  
Spherical Triangles

SUMMARYThis work presents a method based on spherical trigonometry for computing all joint angles of the spherical metamorphic palm. The spherical palm is segmented into spherical triangles which are then solved and combined to fully solve the palm configuration. Further, singularity analysis is investigated with the analysis of each spherical triangle the palm is decomposed. Singularity-avoidance-based design criteria are then presented. Finally, point clouds are generated that represent the joint space of the palm as well as the workspace of the hand with the advantage of an articulated palm is shown.

Random spherical triangles I: Geometrical background

1989 ◽  
Vol 21 (03) ◽  
pp. 570-580 ◽  
Author(s):  
Huiling Le
Keyword(s):  
Mathematical Model ◽  
Shape Space ◽  
Companion Paper ◽  
Statistical Properties ◽  
Data Sets ◽  
Riemannian Structure ◽  
Geometric Structures ◽  
Fundamental Properties ◽  
Random Mechanism ◽  
Spherical Triangles

In this paper we identify the shape space Σ(S 2, k) for k labelled points on the sphere S 2 that gives a mathematical model applicable to data sets whose elements are, or can be represented by, configurations of labelled sequences of points on S2 and for which the fundamental properties of interest are the shapes of these configurations, and we examine the geometric structures on the space, especially the riemannian structure on Σ(S 2, 3). In a companion paper (pp. 581–594) we investigate the statistical properties of such shapes when the k points are generated by a random mechanism.

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