Angle excess


Angle excess

Angle excess is the amount by which the sum of the angles of a polygon on a sphere exceeds the sum of the angles of a polygon with the same number of sides in a plane. For instance, a plane triangle has an angle sum of 180°; an octant is a spherical triangle with three right angles, so its angle sum is 270°, and its angle excess is 90°. The angle excess of any polygon on a sphere is proportional to the polygon's area, with the proportionality constant being the reciprocal of the square of the sphere's radius.

In surveying, one checks whether the angles and distances form a closed polygon, and by how much it is off. If the area is sufficiently large, the polygon will not close no matter how accurately measured if it is calculated on a plane. The area of a polygon whose angle excess is 1 second of arc, which is the precision (though not necessarily the accuracy) of surveying, is 393 square kilometres, or about 20 kilometres square.

Angle deficit is defined similarly on a pseudosphere, and is likewise proportional to area in hyperbolic geometry.

ee also

*defect (geometry)
*spherical trigonometry
*steradian


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Angle of Climb — In aerodynamics, angle of climb is the ratio between distance travelled over the ground and altitude gained. The angle of climb can be expressed as the angle between a plane horizontal to the earth s surface and the actual flight path followed by …   Wikipedia

  • Angle grinder — An angle grinder is a handheld power tool used for cutting, grinding and polishing.Angle grinders can be powered by an electric motor, petrol engine or compressed air. The motor drives a geared head at a right angle on which is mounted an… …   Wikipedia

  • Solid angle — The solid angle, Ω , is the angle in three dimensional space that an object subtends at a point. It is a measure of how big that object appears to an observer looking from that point. For instance, a small object nearby could subtend the same… …   Wikipedia

  • Spherical excess — Spherical Spher ic*al, Spheric Spher ic, a. [L. sphaericus, Gr. ???: cf. F. sph[ e]rique.] 1. Having the form of a sphere; like a sphere; globular; orbicular; as, a spherical body. [1913 Webster] 2. Of or pertaining to a sphere. [1913 Webster] 3 …   The Collaborative International Dictionary of English

  • Lead angle — Lead Lead, n. 1. The act of leading or conducting; guidance; direction; as, to take the lead; to be under the lead of another. [1913 Webster] At the time I speak of, and having a momentary lead, . . . I am sure I did my country important service …   The Collaborative International Dictionary of English

  • Spherical angle — Spherical Spher ic*al, Spheric Spher ic, a. [L. sphaericus, Gr. ???: cf. F. sph[ e]rique.] 1. Having the form of a sphere; like a sphere; globular; orbicular; as, a spherical body. [1913 Webster] 2. Of or pertaining to a sphere. [1913 Webster] 3 …   The Collaborative International Dictionary of English

  • The Zenith Angle — Infobox Book name = The Zenith Angle title orig = translator = image caption = author = Bruce Sterling illustrator = cover artist = country = United States language = English series = genre = Science fiction publisher = Del Ray release date = May …   Wikipedia

  • Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… …   Wikipedia

  • Solide de Platon — En géométrie euclidienne, un solide de Platon est un polyèdre régulier et convexe. Entre les polygones réguliers et convexes de la géométrie plane, et les polyèdres réguliers convexes de l’espace à trois dimensions, il y a une analogie, mais… …   Wikipédia en Français

  • Steradian — The steradian (symbol: sr) is the SI unit of solid angle. It is used to describe two dimensional angular spans in three dimensional space, analogous to the way in which the radian describes angles in a plane. The name is derived from the Greek… …   Wikipedia