65.87.180.99: /* History */

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.

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			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]]

en>Ysangkok: adding 256-bit because it works for ymm registers too

2012-07-28T15:37:28Z

adding 256-bit because it works for ymm registers too

New page

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FMA instruction set - Revision history

65.87.180.99: /* History */

en>Ysangkok: adding 256-bit because it works for ymm registers too