FMA instruction set

2014-01-16T23:38:04Z

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{{Redirect|Ungula|the nails of platyrhinne primates|ungula (primate anatomy)}}
[[File:Spherical Wedge.svg|thumb|right|250px|A spherical wedge with radius ''r'' and angle of the wedge ''α'']]

In [[geometry]], a '''spherical wedge''' or '''ungula''' is a portion of a [[ball (mathematics)|ball]] bounded by two plane [[semidisk]]s and a [[Lune (mathematics)#Spherical geometry|spherical lune]] (termed the wedge's ''base''). The angle between the [[radii]] lying within the bounding semidisks is the [[Dihedral angle|dihedral]] ''angle of the wedge'' ''α''. If ''AB'' is a semidisk that forms a ball when completely revolved about the ''z''-axis, revolving ''AB'' only through a given ''α'' produces a spherical wedge of the same angle ''α''.<ref>{{cite book|title=Geometry, plane, solid, and spherical, in six books|author=P. Morton|publisher=Baldwin and Cradock|year=1830|page=180}}</ref> Beman (2008)<ref>{{cite book|title=New Plane and Solid Geometry|page=338|author=D. W. Beman|publisher=BiblioBazaar, LLC|year=2008|isbn=0-554-44701-0}}</ref> remarks that "a spherical wedge is to the sphere of which it is a part as the angle of the wedge is to a perigon."{{ref|a|[A]}} A spherical wedge of ''α'' = π [[radian]]s (180°) is called a ''[[Sphere|hemisphere]]'', while a spherical wedge of ''α'' = 2π radians (360°) constitutes a complete ball.

The [[volume]] of a spherical wedge can be intuitively related to the ''AB'' definition in that while the volume of a ball of radius ''r'' is given by <math>\tfrac{4}{3} \pi r^3</math>,
the volume a spherical wedge of the same radius ''r'' is given by<ref name="Hart" />
:<math>V = \frac{\alpha}{2\pi} \cdot \frac{4}{3} \pi r^3 = \frac{2}{3} \alpha r^3.</math>

Extrapolating the same principle and considering that the surface area of a sphere is given by <math>4\pi r^2</math>, it can be seen that the surface area of the lune corresponding to the same wedge is given by{{Ref|a|[A]}}
:<math>A = \frac{\alpha}{2\pi} \cdot 4 \pi r^2 = 2 \alpha r^2</math>

Hart (2009)<ref name="Hart">{{cite book|title=Solid Geometry|author=C. A. Hart|page=465|publisher=BiblioBazaar, LLC|year=2009|isbn=1-103-11804-8}}</ref> states that the "volume of a spherical wedge is to the volume of the sphere as the number of [[degree (angle)|degree]]s in the [angle of the wedge] is to 360".{{ref|a|[A]}} Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if <math>V_s</math> is the volume of the sphere and <math>V_w</math> is the volume of a given spherical wedge,

:<math>\frac{V_w}{V_s} = \frac{\alpha}{2\pi}</math>

Also, if ''S''<sub>''l''</sub> is the [[surface area|area]] of a given wedge's lune, and ''S''<sub>''s''</sub> is the area of the wedge's sphere,<ref>{{cite book|title=Marks' standard handbook for mechanical engineers|page=43|author=E. A. Avallone, T. Baumeister, A. Sadegh, L. S. Marks|publisher=McGraw-Hill Professional|year=2006|isbn=0-07-142867-4}}</ref>{{Ref|a|[A]}}

:<math>\frac{S_l}{S_s} = \frac{\alpha}{2\pi}</math>

== See also ==
*[[Spherical cap]]

== Notes ==
:A. {{note|a}} A distinction is sometimes drawn between the terms "[[sphere]]" and "[[ball (mathematics)|ball]]", where a sphere is regarded as being merely the outer surface of a solid ball. It is common to use the terms interchangeably, as the commentaries of both Beman (2008) and Hart (2008) do.

== References ==
{{reflist}}

[[Category:Spherical geometry]]

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FMA instruction set