 Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. This is of great importance for calculations in astronomy and earth-surface and orbital and space navigation. Image File history File links a right-angled spherical triangle Picture taken over from German Wikipedia. ...
Spherical geometry is the geometry of the two-dimensional surface of a sphere. ...
Look up Polygon in Wiktionary, the free dictionary. ...
A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ...
A sphere (< Greek σφαίÏα) is a perfectly symmetrical geometrical object. ...
This article is about angles in geometry. ...
A giant Hubble mosaic of the Crab Nebula, a supernova remnant. ...
Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia. ...
Al-Jayyani, an Arabic scholar in Moorish Spain, wrote the first treatise on spherical trigonometry in 1060 AD. Abu Abd Allah Muhammad ibn Muadh Al-Jayyani (Al-Jayyani; 989, Cordoba, Spain - 1079, Jaen, Spain) was an Arabic mathematician from present-day Spain. ...
Al-Andalus is the Arabic name given the Iberian Peninsula by its Muslim conquerors; it refers to both the Caliphate proper and the general period of Muslim rule (711–1492). ...
Lines on a Sphere On the surface of a sphere, the closest analogue to straight lines are great circles, i.e. circles whose center coincide with the center of the sphere (for example, meridians and the equator are great circles on the Earth). As lines on a plane, great circles on a sphere are the closest connection of two points (if you constrain yourself to lines on the sphere). (cf. geodesic) Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ...
For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...
On the earth, a meridian is a north-south line between the North Pole and the South Pole. ...
World map showing the equator in red The Equator is an imaginary circle drawn around a planet (or other astronomical object) at a distance halfway between the poles. ...
Earth (IPA: , often referred to as the Earth, Terra, the World or Planet Earth) is the third planet in the solar system in terms of distance from the Sun, and the fifth largest. ...
Two intersecting planes in three-dimensional space In mathematics, a plane is a fundamental two-dimensional object. ...
A spatial point is an entity with a location in space but no extent (volume, area or length). ...
In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ...
An area on the sphere which is bounded by arcs of great circles is called a spherical polygon. Note that, unlike the case on a plane, spherical "biangles" (two-sided analogs to triangle) are possible (think about peeling an orange). In Euclidean geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of a circle. ...
For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...
Look up Polygon in Wiktionary, the free dictionary. ...
The sides of these polygons are most conveniently specified not by their length but by the angle under which its endpoints appear when looked at from the sphere's center. Note that this arc angle, measured in radians, and multiplied by the sphere's radius, is the arc length. This article does not refer to the ancient city Side on the Black Sea coast of Turkey, or the ancient city of Side in Laconia, Greece. ...
The radian is a unit of plane angle. ...
In classical geometry, a radius of a circle or sphere is any line segment from its center to its boundary. ...
Hence, a spherical triangle is specified as usual by its corner angles and its sides, but the sides are given not by their length, but by their arc angle.
Remarkably, the sum of the vertex angles of a spherical triangle is always larger than the 180° found in every planar triangle. The amount by which the sum of the angles exceeds 180° is called the spherical excess E: E = α + β + γ − 180°. This surplus determines the surface area of any spherical triangle. To determine this, the spherical excess must be expressed in radians; the surface area A is then given in terms of the sphere's radius R by the expression:
- A = R2 · E. From this formula, which is an application of the Gauss-Bonnet theorem, it becomes obvious that there are no similar triangles (triangles with equal angles but different side lengths and area) on a sphere.
In the special case of a sphere of radius 1, the area simply equals the excess angle: A = E. The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ...
To solve a geometric problem on the sphere, one dissects the relevant figure into right spherical triangles (i.e.: one of the triangle's corner angles is 90°) because one can then use Napier's pentagon:
 Napier's Circle shows the relations of parts of a right spherical triangle Napier's pentagon (also known as Napier's circle) is a mnemonic aid to easily find all relations between the angles in a right spherical triangle: Image File history File links Nepers Circle AKA Nepers Pentagram is a guide to applying Napiers Rules in spherical geometry File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Image File history File links Nepers Circle AKA Nepers Pentagram is a guide to applying Napiers Rules in spherical geometry File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
John Napier For other people with the same name, see John Napier (disambiguation). ...
A mnemonic (pronounced in Received Pronunciation) is a memory aid, and most serve as an educational related purpose. ...
Write the six angles of the triangle in the form of a circle, sticking to the order as they appear in the triangle (i.e.: start with a corner angle, write the arc angle of an attached side next to it, proceed with the next corner angle, etc. and close the circle). Then cross out the 90° corner angle and replace the arc angles adjacent to it by their complement to 90° (i.e. replace, say, a by 90° − a). The five numbers that you now have on your paper form Napier's Pentagon (or Napier's Circle). For them, it holds that the cosine of each angle is equal to: In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
- the product of the cotangents of the angles written next to it
- the product of the sines of the two angles written opposed to it
See also the Haversine formula, which relates the lengths of sides and angles in spherical triangles in a numerically stable form for navigation. Trigonometry In trigonometry, the cotangent is a function (see trigonometric function) defined as: or An interpretation of the cotangent of an angle x is as follows. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
The haversine formula is an equation important in navigation, giving great-circle distances between two points on a sphere from their longitudes and latitudes. ...
Identities Spherical triangles satisfy a spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides) refers to a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. ...
 The identity may be derived by considering the triangles formed by the tangent lines to the spherical triangle subtending angle C and using the plane law of cosines. Moreover, it reduces to the plane law in the small angle limit. In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ...
They also satisfy an analogue of the law of sines In trigonometry, the law of sines (or sine law) is a statement about arbitrary triangles in the plane. ...
 A more thorough list of identities is available here
See also Spherical geometry is the geometry of the two-dimensional surface of a sphere. ...
The great-circle distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere (as opposed to going through the spheres interior). ...
Celestial Navigation is the 15th episode of The West Wing. ...
The haversine formula is an equation important in navigation, giving great-circle distances between two points on a sphere from their longitudes and latitudes. ...
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