Stalagmometric method - Revision history

en>Magioladitis: Reverted to revision 616099049 by Reify-tech (talk): Rv. (TW)

2014-12-01T08:23:23Z

Reverted to revision 616099049 by Reify-tech (talk): Rv. (TW)

← Older revision		Revision as of 10:23, 1 December 2014
Line 1:		Line 1:
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	~~[[Image:Hesse normal form~~.~~png\|thumb\|Drawing of~~ the ~~normal~~ and the ~~distance calculated with the Hesse normal form]]~~
	~~The '''Hesse normal form''' named after [[Otto Hesse]],~~ is ~~an equation used in [[analytic geometry]]~~, and ~~describes~~ a ~~line in~~ <~~math~~>~~\mathbb{R}^2~~<~~/math~~> or a ~~plane in Euclidean space <math>\mathbb{R}^3</math> or~~ a ~~hyperplane in higher dimensions~~. It is ~~primarily used for calculating distances~~, ~~and is written in vector notation~~ as

	~~:<math>\vec r \cdot \vec n_0 - d = 0~~.~~\,</math>~~

	~~This equation~~ is ~~satisfied~~ by ~~all points~~ '~~'P'' described by the location vector~~ <~~math~~>~~\vec r~~<~~/math~~>~~, which lie precisely in the plane ''E'' (or in 2D,~~ on ~~the line ''g'')~~.

	The ~~vector <math>\vec n_0</math> represents~~ the ~~unit normal vector of ''E'' or ''g'', that points from~~ the ~~origin of~~ the ~~coordinate system to the plane (~~or ~~line~~, in ~~2D)~~. ~~The distance <math>d \ge 0</math> is~~ the ~~distance~~ from ~~the origin to the plane (~~or ~~line)~~. ~~The dot~~ <~~math~~>~~\cdot~~<~~/math~~> ~~indicates the scalar product or dot product~~.

	~~== Derivation/Calculation from the normal form ==~~

	~~Note: For simplicity~~, the ~~following derivation discusses the 3D case. However~~, it ~~is also applicable in 2D.~~

	~~In the normal form~~,

	~~:<math>(\vec r -\vec~~ a~~)\cdot \vec n = 0\~~,~~</math>~~

	~~a plane is given by a normal vector <math>\vec n</math> as well as an arbitrary position vector <math>\vec a</math>~~ of ~~a point <math>A \in E</math>. The direction~~ of ~~<math>\vec n</math> is chosen~~ to ~~satisfy the following inequality~~

	~~:<math>\vec a\cdot \vec n \geq 0\~~,~~</math>~~

	~~By dividing~~ the ~~normal vector <math>\vec n</math> by~~ its ~~[[Euclidean_vector#Length_of_a_vector\|Magnitude]] <math>\| \vec n \|</math>~~, ~~we obtain the unit (or normalized) normal vector~~

	~~:<math>\vec n_0 = {{\vec n} \over {\| \vec n \|}}\~~,~~</math>~~

	and ~~the above equation can be rewritten as~~

	~~:<math>(\vec r -\vec a)\cdot \vec n_0 = 0~~.\,<~~/math~~>

	~~Substituting~~

	:<~~math~~>~~d = \vec~~ a~~\cdot \vec n_0 \geq 0\,</math>~~

	~~we obtain the Hesse normal form~~

	~~:<math>\vec r \cdot \vec n_0 - d = 0~~.~~\,</math>~~

	~~<center>~~[~~[File~~:~~Ebene Hessesche Normalform.PNG]]<~~/~~center>~~

	~~In this diagram, ''d'' is the distance from the origin~~. ~~Because <math>\vec r \cdot \vec n_0 = d</math> holds for every point~~ in ~~the plane, it is also true at point ''Q'' (the point where the vector from the origin meets the plane E), with <math>\vec r = \vec r_s<~~/~~math>, per the definition of the [[Scalar product~~]]

	~~:<math>d = \vec r_s \cdot \vec n_0 = \|\vec r_s\| \cdot \|\vec n_0\| \cdot \cos(0^\circ) = \|\vec r_s\| \cdot 1 = \|\vec r_s\|~~.\,~~</math>~~

	~~The magnitude <math>\|\vec r_s\|</math> of <math>{\vec r_s}</math>~~ is the ~~shortest distance from the origin to the plane~~.

	~~[[Category:Analytic geometry]]~~

87.115.124.8: /* Stalagmometer */

2013-10-04T18:41:47Z

Stalagmometer

← Older revision		Revision as of 20:41, 4 October 2013
Line 1:		Line 1:
			{{Noref\|date=August 2009}}
			[[Image:Hesse normal form.png\|thumb\|Drawing of the normal and the distance calculated with the Hesse normal form]]
			The '''Hesse normal form''' named after [[Otto Hesse]], is an equation used in [[analytic geometry]], and describes a line in <math>\mathbb{R}^2</math> or a plane in Euclidean space <math>\mathbb{R}^3</math> or a hyperplane in higher dimensions. It is primarily used for calculating distances, and is written in vector notation as

			:<math>\vec r \cdot \vec n_0 - d = 0.\,</math>

	~~e - Shop Word - Press is a excellent cart for your on the web shopping organization.~~ This ~~one~~ is ~~one of~~ the ~~most beneficial features of Word - Press as this feature allows users to define~~ the ~~user roles. Change the site~~'~~s theme and you have essentially changed the site~~'~~s personality. Dead links are listed out simply because it will negatively have an influence~~ on the ~~website~~'~~s search engine rating~~. ~~It is found that most~~ of ~~the visitors only look for the results~~ that ~~are displayed on~~ the ~~first page~~ of the ~~search engines and so if you get~~ the ~~service~~ from ~~professional service providers then they strive for~~ the ~~first page ranking of your site and improve~~ the ~~online visibility~~. <br><br>~~The Internet is a vast open market where businesses and consumers congregate~~. ~~If you are a positive thinker businessman then today you have to put your business online~~. ~~This~~ is ~~the reason for the increased risk of Down Syndrome babies~~ in ~~women over age 35~~. ~~Being able to help with your customers can make~~ a ~~change in how~~ a ~~great deal work, repeat online business, and referrals you'll be~~ given~~. Akismet is really~~ a ~~sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites~~. <br><br>~~If you loved this posting and you would like~~ to ~~obtain additional facts with regards to [http~~://~~mmservice.dk~~/~~backup_plugin_182444 wordpress backup plugin] kindly visit our web site. Saying that~~, ~~despite~~ the ~~launch of Wordpress Express many months ago~~, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. After sending these details, your Word - Press blog will be setup within a few days. Use this section to change many formatting elements. Newer programs allow website owners and internet marketers to automatically and ~~dynamically change words in their content to match~~ the ~~keywords entered by their web visitors in their search queries'~~a feat that they cannot easily achieve with older software. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. <br><br>~~There has been~~ a ~~huge increase in~~ the ~~number of developers releasing free premium Word~~ - ~~Press themes over the years~~. ~~Cameras with a pentaprism (as in comparison to pentamirror) ensure that little mild is lost before it strikes your eye~~, ~~however these often increase the cost of the digital camera considerably~~. ~~However~~, ~~you may not be able to find a theme that~~ is ~~in sync with your business. It supports backup scheduling and allows you to either download~~ the ~~backup file or email it to you. Fortunately, Word - Press Customization Service is available these days, right~~ from ~~custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project~~. <br><br>~~Someone with a basic knowledge of setting up a website should be able to complete the process in a couple of minutes however even basic users should find they are able to complete the installation~~ in ~~around 20 minutes by following~~ the ~~step by step guide online. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. Must being~~, it'~~s beneficial because I don~~'~~t know about you, but loading an old website on a mobile, having to scroll down, up, and sideways~~' ~~I find links being clicked and bounced around like I~~'~~m on a freaking trampoline.~~ with ~~posts or testimonials updated as they are uploaded to a particular section~~ of the ~~website~~. ~~Verify whether your company has a team~~ of ~~developers or programmers having hands-on experience and knowledge about all Word - Press concepts~~.		This equation is satisfied by all points ''P'' described by the location vector <math>\vec r</math>, which lie precisely in the plane ''E'' (or in 2D, on the line ''g'').

			The vector <math>\vec n_0</math> represents the unit normal vector of ''E'' or ''g'', that points from the origin of the coordinate system to the plane (or line, in 2D). The distance <math>d \ge 0</math> is the distance from the origin to the plane (or line). The dot <math>\cdot</math> indicates the scalar product or dot product.

			== Derivation/Calculation from the normal form ==

			Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.

			In the normal form,

			:<math>(\vec r -\vec a)\cdot \vec n = 0\,</math>

			a plane is given by a normal vector <math>\vec n</math> as well as an arbitrary position vector <math>\vec a</math> of a point <math>A \in E</math>. The direction of <math>\vec n</math> is chosen to satisfy the following inequality

			:<math>\vec a\cdot \vec n \geq 0\,</math>

			By dividing the normal vector <math>\vec n</math> by its [[Euclidean_vector#Length_of_a_vector\|Magnitude]] <math>\| \vec n \|</math>, we obtain the unit (or normalized) normal vector

			:<math>\vec n_0 = {{\vec n} \over {\| \vec n \|}}\,</math>

			and the above equation can be rewritten as

			:<math>(\vec r -\vec a)\cdot \vec n_0 = 0.\,</math>

			Substituting

			:<math>d = \vec a\cdot \vec n_0 \geq 0\,</math>

			we obtain the Hesse normal form

			:<math>\vec r \cdot \vec n_0 - d = 0.\,</math>

			<center>[[File:Ebene Hessesche Normalform.PNG]]</center>

			In this diagram, ''d'' is the distance from the origin. Because <math>\vec r \cdot \vec n_0 = d</math> holds for every point in the plane, it is also true at point ''Q'' (the point where the vector from the origin meets the plane E), with <math>\vec r = \vec r_s</math>, per the definition of the [[Scalar product]]

			:<math>d = \vec r_s \cdot \vec n_0 = \|\vec r_s\| \cdot \|\vec n_0\| \cdot \cos(0^\circ) = \|\vec r_s\| \cdot 1 = \|\vec r_s\|.\,</math>

			The magnitude <math>\|\vec r_s\|</math> of <math>{\vec r_s}</math> is the shortest distance from the origin to the plane.

			[[Category:Analytic geometry]]

en>ZéroBot: r2.7.1) (Robot: Adding ca:Estalagmòmetre

2012-07-23T09:34:05Z

r2.7.1) (Robot: Adding ca:Estalagmòmetre

New page

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