Search results
Results From The WOW.Com Content Network
Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
The solution of triangles is the principal purpose of spherical trigonometry: given three, four or five elements of the triangle, determine the others. The case of five given elements is trivial, requiring only a single application of the sine rule.
Special cases are right triangles (p q 2). Uniform solutions are constructed by a single generator point with 7 positions within the fundamental triangle, the 3 corners, along the 3 edges, and the triangle interior. All vertices exist at the generator, or a reflected copy of it. Edges exist between a generator point and its image across a mirror.
Spherical triangle solved by the law of cosines. Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points u , v , and w on the sphere (shown at right).
The book contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle. [5] The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe.
English: Spherical trigonometry: the basic construction for the solution of a spherical triangle by successive use of two right-angled triangles. Date 26 July 2013, 14:55:13
Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers (p q r), each representing the angle at a vertex. The value n ⁄ d means the vertex angle is d ⁄ n of the half-circle. "2" means a right triangle.
They therefore enclose a spherical triangle on the surface of any sphere centered on that point and repeated reflections produce a multitude of copies of the triangle. If the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. If one places a vertex at a suitable point inside the ...