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A quadrantal spherical triangle together with Napier's circle for use in his mnemonics. A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of π /2 radians at the centre of the sphere: on the unit sphere the side has length π /2.
A quadrantal spherical triangle together with Napier's circle for use in his mnemonics. A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of π /2 radians at the centre of the sphere: on the unit sphere the side has length π /2.
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
Because these are double angles, each of q, r, and s represents two applications of the rotation implied by an edge of the spherical triangle. From the definitions, it follows that srq = uw −1 wv −1 vu −1 = 1, which tells us that the composition of these rotations is the identity transformation. In particular, rq = s −1 gives us
The spherical octant itself is the intersection of the sphere with one octant of space. Uniquely among spherical triangles, the octant is its own polar triangle. [2] The octant can be parametrized using a rational quartic Bézier triangle. [3] The solid angle subtended by a spherical octant is π /2 sr, one-eight of the solid angle of a sphere. [4]
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, [1] orbifold notation, [2] and order.
A complete quadrangle (at left) and a complete quadrilateral (at right).. In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points.
The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes.Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, that relates the sides and angles of spherical triangles.