Spectral radius - Revision history

150.203.160.15: /* Matrices */ don't need "for each lambda" when lambda has already been chosen.

2014-09-29T07:21:03Z

Matrices: don't need "for each lambda" when lambda has already been chosen.

← Older revision		Revision as of 09:21, 29 September 2014
Line 1:		Line 1:
	~~{{multiple issues\|~~		I enjoy bilingual [https://Www.Gov.uk/search?q=network network] marketing, spending time with family and friends and developing relationships via social media. I have been an avid network marketer for the past 5 years and believe that earning an annuity or recurring income is the key. This means that you need to invest time, money and effort in a good system in a lucrative industry. I am certain we have it here. This is best value for money system and tools system I have seen as yet! I will work with you if you want to make it happen for you.<br><br>Feel free to surf to my web-site: [https://www.youtube.com/watch?v=l3BHiL-vMzM&list=UUg3DMmhiHy97o2DJARM9TRQ affiliate marketing/facebook instant income system/facebook marketing/blogging system/social network marketing/pure leverage tools/network marketing/blogging system/video email marketing/email service software/social media marketing/an authority blog/]
	~~{{external links\|date=November 2011}}~~
	~~{{unreferenced\|date=November 2011}}~~
	}}

	[~~[File~~:~~Venn 0000 0001 0001 0110~~.~~png\|thumb\|275px\|right\|[[Venn diagram]] created with CSG<br>The source is [[commons:File:Venn 0000 0001 0001 0110~~.~~png#POV-Ray source\|on the description page]].]]~~

	'''Constructive solid geometry (CSG)''' is a technique used in [[solid modeling]]. Constructive solid geometry allows a modeler to create a complex surface or object by using [[Boolean data type\|Boolean]] [[Operator (programming)\|operator]]s to combine objects. Often CSG presents a model or surface that appears visually complex, but is actually little more than cleverly combined or decombined objects.

	~~In [[3D computer graphics]] and [[CAD]] CSG is often used in procedural modeling. CSG can also be performed on [[Polygon mesh\|polygonal meshes]~~], and ~~may or may not be procedural~~ and~~/or parametric~~.

	~~==Workings of CSG==~~
	~~The simplest solid objects used~~ for the representation are called '''primitives'''. Typically they are the objects of simple shape: [[cuboid]]s, [[cylinder (geometry)\|cylinder]]s, [[Prism (geometry)\|prism]]s, [[Pyramid (geometry)\|pyramids]], [[sphere]]s, [[cone (geometry)\|cone]]s. The set of allowable primitives is limited by each software package. Some software packages allow CSG on curved objects while other packages do not.

	~~It is said~~ that an ~~object~~ is ~~'''constructed''' from primitives by~~ means ~~of allowable '''operations'''~~, ~~which are typically [[Boolean logic\|Boolean]] [[operation (mathematics)\|operations]] on [[set theory\|sets]]: [[union (set theory)\|union]], [[intersection (set theory)\|intersection]]~~ and ~~[[complement (set theory)\|difference]].~~

	~~A primitive can typically be described by~~ a ~~[[Algorithm\|procedure]] which accepts some number of [[parameter]]s; for example,~~ a ~~sphere may be described by the coordinates of its center point, along with a radius value~~. ~~These primitives can be combined into compound objects using operations like these:~~

	{\|
	\|-
	~~\|[[Image:Boolean union~~.~~PNG\|thumb\|275px\|'''Union'''<br>Merger of two objects into one]]~~
	~~\|[[Image:Boolean difference.PNG\|thumb\|275px\|'''Difference'''<br>Subtraction of one object from another]]~~
	~~\|[[Image:Boolean intersect.PNG\|thumb\|'''Intersection'''<br>Portion common to both objects\|315x315px]]~~
	\|}

	~~Combining these elementary operations, it~~ is ~~possible to build up objects with high complexity starting from simple ones.~~

	~~== Applications of CSG ==~~
	[[Image:Csg tree.png\|thumb\|right\|275px\|CSG objects can be represented by binary trees, where leaves represent primitives, and nodes represent operations. In this figure, the nodes are labeled <math>\cap</math> for ~~intersection, <math>\cup</math> for union, and <math>-\!</math> for difference.]]~~

	~~Constructive solid geometry has a number of practical uses. It is used in cases where simple geometric objects are desired, or where mathematical accuracy is important. The [[Quake engine]]~~ and ~~[[Unreal engine]] both use this~~ system, as does [[Valve Hammer Editor\|Hammer]] (the native [[Source engine]] level editor), and [[Torque Game Engine]]/[[Torque Game Engine Advanced]]. CSG is popular because a modeler can use a set of relatively simple objects to create very complicated geometry. When CSG is procedural or parametric, the user can revise their complex geometry by changing the position of objects or by changing the Boolean operation used to combine those objects.

	~~One of the advantages of CSG is that~~ it ~~can easily assure that objects are "solid" or water-tight if all of the primitive shapes are water-tight. This can be important~~ for ~~some manufacturing or engineering computation applications~~. ~~By comparison, when creating geometry based upon [[boundary representation]]s, additional topological data is required, or consistency checks must be performed~~ to ~~assure that the given boundary description specifies a valid solid object.~~

	~~A convenient property of CSG shapes is that it is easy~~ to classify arbitrary points as being either inside or outside the shape created by CSG. The point is simply classified against all the underlying primitives and the resulting boolean expression is evaluated. This is a desirable quality for some applications such as [[collision detection]].

	~~== Applications with CSG support ==~~

	* [[3Delight]]
	* [[Blender (software)\|Blender (provides meta objects)]]
	* [[BRL-~~CAD]]~~
	* [[Clara.io]]
	* [http:~~//sourceforge.net/projects/netgen-mesher NETGEN] - an automatic 3d tetrahedral mesh generator. It accepts input from constructive solid geometry (CSG) or boundary representation (BRep)~~
	* [[Feature Manipulation Engine]]
	* [[FreeCAD]]
	* [[GtkRadiant]]
	* [[HyperFun]]
	* [[OpenSCAD]]
	* [~~[PhotoRealistic RenderMan]]~~
	* [[PLaSM]] - Programming Language of Solid Modeling
	* [[POV-Ray]]
	* [http://www.~~cern.ch/theis/simplegeo SimpleGeo] A solid modeling for particle transport Monte Carlo simulations~~
	* [[SolidWorks]] mechanical [[CAD]] suite
	* [http://unbboolean.sourceforge.net/ UnBBoolean] a [[Java3D]] implementation
	* [[Vectorworks]]
	* [http://www.inevo.pt/portfolio/gides/ GiDES++] a gesture-based CSG CAD

	~~=== Gaming ===~~
	* [[3D World Studio]]
	* [[UnrealEd]]
	* [[Valve Hammer Editor]]
	* [http://www.leadwerks.com/ ~~Leadwerks]~~

	==~~= Libraries ===~~
	* [http://~~code.google.com~~/p/~~carve~~/ ~~Carve CSG] - a fast and robust constructive solid geometry library~~
	* [http://~~evanw.github.com~~/~~csg.js~~/ ~~CSG.js] A [[JavaScript]] implementation using [[WebGL]]~~
	* [http://~~gts.sourceforge.net/index.html GTS] -~~ an ~~Open Source Free Software Library intended to provide a set of useful functions to deal with 3D surfaces meshed with interconnected triangles~~
	* [http://~~www.geometros.com sgCore C++/C# library]~~

	~~== External links ==~~
	* [http://www.leadwerks.com/files/csg.pdf Leadwerks Software 'What is Constructive Solid Geometry?'] - explanation of CSG definitions, equations, techniques, and uses.

	~~[[Category:Computer-aided design]]~~
	~~[[Category:3D computer graphics]]~~
	~~[[Category:Euclidean solid geometry]~~]

en>Mgkrupa: /* See also */ Added {{Functional Analysis}} footer

2014-01-10T03:28:38Z

See also: Added {{Functional Analysis}} footer

← Older revision		Revision as of 05:28, 10 January 2014
Line 1:		Line 1:
	~~Catrina Le~~ is ~~what~~'s constructed on ~~her birth card though she doesn~~'~~t essentially like being called prefer~~ this. ~~Software improving~~ is where ~~her main income comes from remember~~, ~~though~~ , ~~soon her husband but also her will start~~ their ~~personal own business~~. ~~What your sweetheart loves doing~~ is to ~~get~~ to ~~karaoke but this woman~~ is ~~thinking on~~ [~~https~~://~~Www~~.~~Google~~.~~com~~/~~search?hl~~=~~en&gl~~=~~us&tbm~~=~~nws&q~~=~~starting starting~~] ~~a new challenge. For years she's been living on the~~ [~~https~~://~~Www~~.~~google~~.com/~~search?hl~~=~~en&gl~~=~~us&tbm~~=~~nws&q~~=~~inside+Vermont inside Vermont~~]~~. She is running~~ and ~~always keeping a blog here~~: http://~~prometeu~~.net~~<br><br>Feel free~~ to ~~visit my blog post -~~ [http://~~prometeu~~.~~net clash~~ of ~~clans hack without survey~~]		{{multiple issues\|
			{{external links\|date=November 2011}}
			{{unreferenced\|date=November 2011}}
			}}

			[[File:Venn 0000 0001 0001 0110.png\|thumb\|275px\|right\|[[Venn diagram]] created with CSG<br>The source is [[commons:File:Venn 0000 0001 0001 0110.png#POV-Ray source\|on the description page]].]]

			'''Constructive solid geometry (CSG)''' is a technique used in [[solid modeling]]. Constructive solid geometry allows a modeler to create a complex surface or object by using [[Boolean data type\|Boolean]] [[Operator (programming)\|operator]]s to combine objects. Often CSG presents a model or surface that appears visually complex, but is actually little more than cleverly combined or decombined objects.

			In [[3D computer graphics]] and [[CAD]] CSG is often used in procedural modeling. CSG can also be performed on [[Polygon mesh\|polygonal meshes]], and may or may not be procedural and/or parametric.

			==Workings of CSG==
			The simplest solid objects used for the representation are called '''primitives'''. Typically they are the objects of simple shape: [[cuboid]]s, [[cylinder (geometry)\|cylinder]]s, [[Prism (geometry)\|prism]]s, [[Pyramid (geometry)\|pyramids]], [[sphere]]s, [[cone (geometry)\|cone]]s. The set of allowable primitives is limited by each software package. Some software packages allow CSG on curved objects while other packages do not.

			It is said that an object is '''constructed''' from primitives by means of allowable '''operations''', which are typically [[Boolean logic\|Boolean]] [[operation (mathematics)\|operations]] on [[set theory\|sets]]: [[union (set theory)\|union]], [[intersection (set theory)\|intersection]] and [[complement (set theory)\|difference]].

			A primitive can typically be described by a [[Algorithm\|procedure]] which accepts some number of [[parameter]]s; for example, a sphere may be described by the coordinates of its center point, along with a radius value. These primitives can be combined into compound objects using operations like these:

			{\|
			\|-
			\|[[Image:Boolean union.PNG\|thumb\|275px\|'''Union'''<br>Merger of two objects into one]]
			\|[[Image:Boolean difference.PNG\|thumb\|275px\|'''Difference'''<br>Subtraction of one object from another]]
			\|[[Image:Boolean intersect.PNG\|thumb\|'''Intersection'''<br>Portion common to both objects\|315x315px]]
			\|}

			Combining these elementary operations, it is possible to build up objects with high complexity starting from simple ones.

			== Applications of CSG ==
			[[Image:Csg tree.png\|thumb\|right\|275px\|CSG objects can be represented by binary trees, where leaves represent primitives, and nodes represent operations. In this figure, the nodes are labeled <math>\cap</math> for intersection, <math>\cup</math> for union, and <math>-\!</math> for difference.]]

			Constructive solid geometry has a number of practical uses. It is used in cases where simple geometric objects are desired, or where mathematical accuracy is important. The [[Quake engine]] and [[Unreal engine]] both use this system, as does [[Valve Hammer Editor\|Hammer]] (the native [[Source engine]] level editor), and [[Torque Game Engine]]/[[Torque Game Engine Advanced]]. CSG is popular because a modeler can use a set of relatively simple objects to create very complicated geometry. When CSG is procedural or parametric, the user can revise their complex geometry by changing the position of objects or by changing the Boolean operation used to combine those objects.

			One of the advantages of CSG is that it can easily assure that objects are "solid" or water-tight if all of the primitive shapes are water-tight. This can be important for some manufacturing or engineering computation applications. By comparison, when creating geometry based upon [[boundary representation]]s, additional topological data is required, or consistency checks must be performed to assure that the given boundary description specifies a valid solid object.

			A convenient property of CSG shapes is that it is easy to classify arbitrary points as being either inside or outside the shape created by CSG. The point is simply classified against all the underlying primitives and the resulting boolean expression is evaluated. This is a desirable quality for some applications such as [[collision detection]].

			== Applications with CSG support ==

			* [[3Delight]]
			* [[Blender (software)\|Blender (provides meta objects)]]
			* [[BRL-CAD]]
			* [[Clara.io]]
			* [http://sourceforge.net/projects/netgen-mesher NETGEN] - an automatic 3d tetrahedral mesh generator. It accepts input from constructive solid geometry (CSG) or boundary representation (BRep)
			* [[Feature Manipulation Engine]]
			* [[FreeCAD]]
			* [[GtkRadiant]]
			* [[HyperFun]]
			* [[OpenSCAD]]
			* [[PhotoRealistic RenderMan]]
			* [[PLaSM]] - Programming Language of Solid Modeling
			* [[POV-Ray]]
			* [http://www.cern.ch/theis/simplegeo SimpleGeo] A solid modeling for particle transport Monte Carlo simulations
			* [[SolidWorks]] mechanical [[CAD]] suite
			* [http://unbboolean.sourceforge.net/ UnBBoolean] a [[Java3D]] implementation
			* [[Vectorworks]]
			* [http://www.inevo.pt/portfolio/gides/ GiDES++] a gesture-based CSG CAD

			=== Gaming ===
			* [[3D World Studio]]
			* [[UnrealEd]]
			* [[Valve Hammer Editor]]
			* [http://www.leadwerks.com/ Leadwerks]

			=== Libraries ===
			* [http://code.google.com/p/carve/ Carve CSG] - a fast and robust constructive solid geometry library
			* [http://evanw.github.com/csg.js/ CSG.js] A [[JavaScript]] implementation using [[WebGL]]
			* [http://gts.sourceforge.net/index.html GTS] - an Open Source Free Software Library intended to provide a set of useful functions to deal with 3D surfaces meshed with interconnected triangles
			* [http://www.geometros.com sgCore C++/C# library]

			== External links ==
			* [http://www.leadwerks.com/files/csg.pdf Leadwerks Software 'What is Constructive Solid Geometry?'] - explanation of CSG definitions, equations, techniques, and uses.

			[[Category:Computer-aided design]]
			[[Category:3D computer graphics]]
			[[Category:Euclidean solid geometry]]

en>Slawekb: It's better to write it out in words.

2012-05-08T19:39:16Z

It's better to write it out in words.

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