Codex Bezae - Revision history

en>EastTN: /* References */ No more bare URLs

2014-09-09T16:03:37Z

References: No more bare URLs

en>Breezeboy: /* Acts of the Apostles */

2014-02-13T08:30:15Z

Acts of the Apostles

← Older revision		Revision as of 10:30, 13 February 2014
Line 1:		Line 1:
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	~~{{Merge~~ from~~\|Blackburn pendulum\|discuss=Talk:Harmonograph#Merge_proposal\|date=October 2011}}~~
	~~{{oneref\|date=January 2010}}~~
	}}
	~~[[Image:Harmonograph 2.jpg\|thumb\|200px\|A harmonograph output]]~~
	~~A '''harmonograph''' is~~ a ~~mechanical apparatus that employs [[pendulum]]s to create a geometric image~~. ~~The drawings created typically are [[Lissajous curve]]s~~, ~~or related drawings of greater complexity. The devices~~, ~~which began to appear in the mid-19th century and peaked in popularity in the 1890s~~, ~~cannot be conclusively attributed to a single person~~, ~~although [[Hugh Blackburn]]~~, ~~a professor of [[mathematics]] at the [[University of Glasgow]]~~, ~~is commonly believed~~ to ~~be the official inventor~~.<~~ref~~>{{cite journal \| last = Turner \|first = Steven \| title = Demonstrating Harmony: Some of the Many Devices Used To Produce Lissajous Curves Before the Oscilloscope \| journal = Rittenhouse \| volume = 11 \| issue = 42 \| page = 41 \|date=February 1997}}<~~/ref~~>

	~~A simple~~, ~~so-called "lateral" harmonograph uses two pendulums to~~ control the ~~movement of a [[pen]] relative to a drawing [[surface]]~~. One pendulum moves the pen back and forth along one axis and the other pendulum moves the drawing surface back and forth along a [[perpendicular]] axis. By varying the [[frequency]] and phase of the pendulums relative to one another, different patterns are created~~. Even a simple harmonograph as described can create [[ellipse]]s~~, ~~[[spiral]]s~~, ~~[[wikt:figure eight\| figure eights]] and other Lissajous figures.~~

	More complex harmonographs incorporate three or more pendulums or linked pendulums together (for example hanging one pendulum off another), or involve rotary motion in which one or more pendulums is mounted on [[gimbal]]s ~~to allow movement in any direction~~.

	~~A particular type of harmonograph,~~ a ~~[[pintograph]], is based on the relative motion of two rotating disks, as illustrated in the links below.~~

	~~==Computer~~-~~generated harmonograph figure==~~
	~~A harmonograph creates its figures using the movements~~ of ~~damped pendulums~~. ~~The movement of a damped pendulum is described by the equation~~

	:<~~math~~>~~x(t) = A \sin (tf + p) e^{-dt}, \,\!~~<~~/math>~~

	~~in which <math>f</math> represents frequency, <math>p</math> represent phase, <math~~>A~~</math> represent amplitude, <math>d</math> represents damping and <math>t</math> represents time. If that pendulum can move about two axes (~~in ~~a circular or elliptical shape), due to~~ the ~~principle of superposition, the motion of~~ a ~~rod connected to the bottom~~ of the ~~pendulum along one axes will be described~~ by the ~~equation~~

	:<~~math~~>~~x(t) = A_1 \sin (tf_1 + p_1) e^{-d_1t} + A_2 \sin (tf_2 + p_2) e^{-d_2t}. \,\!~~<~~/math~~>

	~~A typical harmonograph has two pendulums that move in such a fashion,~~ and ~~a pen that~~ is ~~moved by two perpendicular rods connected~~ to ~~these pendulums~~. ~~Therefore the path of the harmonograph figure is described by the parametric equations~~

	~~:<math>x(t) = A_1 \sin (tf_1 + p_1) e^{-d_1t} + A_2 \sin (tf_2 + p_2) e^{-d_2t}, \,\!~~<~~/math~~>
	:<~~math~~>~~y(t) = A_3 \sin (tf_3 + p_3) e^{~~-~~d_3t} + A_4 \sin (tf_4 + p_4) e^{~~-~~d_4t}~~. ~~\,\!</math>~~

	~~An appropriate computer program can translate those equations into~~ a ~~graph that emulates a harmonograph~~. ~~Applying the first equation~~ a ~~second~~ time to ~~each equation can emulate a moving piece~~ of ~~paper (see the figure below)~~.

	~~==Gallery==~~
	<~~gallery~~>
	~~File:Harmonograph~~.~~jpg\|A harmonograph at [[Questacon]] in [[Canberra]]~~, ~~[[Australia]]~~.
	~~Image:lateral2~~.~~jpg\|A figure produced by a simple lateral harmonograph~~.
	~~Image:lateral3.jpg\|A figure produced by a simple lateral harmonograph.~~
	~~Image:pgraph1.jpg\|A figure produced by a pintograph.~~
	~~Image:Harmonograph 9998.png\|Computer generated harmonograph figure.~~
	<~~/gallery~~>

	~~==See also==~~
	*[[Spirograph]]

	~~==Notes==~~
	~~{{reflist}}~~

	~~==External links==~~
	*[http://www.jonathan.lansey.net/pastimes/pendulum/index.html A complex harmonograph with a ~~unique single pendulum design]~~
	*[http://www.~~sequences.org.uk/harmonograph/ Online interactive Harmonograph]~~
	*[~~http://local.wasp.uwa.edu.au/~pbourke/geometry/harmonograph/ Harmonograph background, equations, and illustrations]~~
	*[http://www.~~karlsims~~.com/~~harmonograph/ How~~ to ~~build a 3-pendulum rotary harmonograph]~~
	*[http://~~hernan~~.~~amiune.com/labs/harmonograph/animated-harmonograph.html HTML5 Animated Harmonograph]~~
	*[http://~~www~~.~~worldtreesoftware~~.~~com/apps/harmonograph/harmonograph.html Virtual Harmonograph web application]~~
	*[http://excelunusual.com/archive/2011/~~07/harmonograph-video-demo/ An Animated Harmonogharph Model in MS Excel]~~
	*[https://itunes.apple.com/us/app/pintograph-mesmerizing-line/id548056488?mt=8&uo=~~4 An interactive Pintograph for iOS]~~

	~~[[Category:Curves]]~~
	~~[[Category:Pendulums]~~]

en>EastTN: /* History */ Let's use a more specific link

2014-01-13T17:02:36Z

History: Let's use a more specific link

← Older revision		Revision as of 19:02, 13 January 2014
Line 1:		Line 1:
	~~Gabrielle is what her spouse loves to call your wife's though she~~ [~~http~~:~~//Www~~.~~twitpic~~.~~com/tag/doesn%27t doesn't~~] ~~very like being called~~ of ~~that ranking. As a woman what this lady really likes is mah jongg but she hasn't ever made a dime by using it~~. ~~Software improving is where her main income comes from still~~ , ~~soon her husband~~ and ~~consequently her will start their own business. For a while she's yet been~~ in ~~Massachusetts. Go~~ to ~~her~~ [~~http~~:~~//Www.Google.Co.uk/search?hl~~=~~en&gl~~=~~us&tbm~~=~~nws&q~~=~~website&gs_l~~=~~news website] to seek out more: http:~~/~~/prometeu.net<br><br~~>		{{multiple issues\|
			{{Merge from\|Blackburn pendulum\|discuss=Talk:Harmonograph#Merge_proposal\|date=October 2011}}
			{{oneref\|date=January 2010}}
			}}
			[[Image:Harmonograph 2.jpg\|thumb\|200px\|A harmonograph output]]
			A '''harmonograph''' is a mechanical apparatus that employs [[pendulum]]s to create a geometric image. The drawings created typically are [[Lissajous curve]]s, or related drawings of greater complexity. The devices, which began to appear in the mid-19th century and peaked in popularity in the 1890s, cannot be conclusively attributed to a single person, although [[Hugh Blackburn]], a professor of [[mathematics]] at the [[University of Glasgow]], is commonly believed to be the official inventor.<ref>{{cite journal \| last = Turner \|first = Steven \| title = Demonstrating Harmony: Some of the Many Devices Used To Produce Lissajous Curves Before the Oscilloscope \| journal = Rittenhouse \| volume = 11 \| issue = 42 \| page = 41 \|date=February 1997}}</ref>

	~~Review my blog post how~~ to ~~hack clash~~ of ~~clans~~ - [http://~~prometeu~~.net ~~simply click the following internet page~~] -		A simple, so-called "lateral" harmonograph uses two pendulums to control the movement of a [[pen]] relative to a drawing [[surface]]. One pendulum moves the pen back and forth along one axis and the other pendulum moves the drawing surface back and forth along a [[perpendicular]] axis. By varying the [[frequency]] and phase of the pendulums relative to one another, different patterns are created. Even a simple harmonograph as described can create [[ellipse]]s, [[spiral]]s, [[wikt:figure eight\| figure eights]] and other Lissajous figures.

			More complex harmonographs incorporate three or more pendulums or linked pendulums together (for example hanging one pendulum off another), or involve rotary motion in which one or more pendulums is mounted on [[gimbal]]s to allow movement in any direction.

			A particular type of harmonograph, a [[pintograph]], is based on the relative motion of two rotating disks, as illustrated in the links below.

			==Computer-generated harmonograph figure==
			A harmonograph creates its figures using the movements of damped pendulums. The movement of a damped pendulum is described by the equation

			:<math>x(t) = A \sin (tf + p) e^{-dt}, \,\!</math>

			in which <math>f</math> represents frequency, <math>p</math> represent phase, <math>A</math> represent amplitude, <math>d</math> represents damping and <math>t</math> represents time. If that pendulum can move about two axes (in a circular or elliptical shape), due to the principle of superposition, the motion of a rod connected to the bottom of the pendulum along one axes will be described by the equation

			:<math>x(t) = A_1 \sin (tf_1 + p_1) e^{-d_1t} + A_2 \sin (tf_2 + p_2) e^{-d_2t}. \,\!</math>

			A typical harmonograph has two pendulums that move in such a fashion, and a pen that is moved by two perpendicular rods connected to these pendulums. Therefore the path of the harmonograph figure is described by the parametric equations

			:<math>x(t) = A_1 \sin (tf_1 + p_1) e^{-d_1t} + A_2 \sin (tf_2 + p_2) e^{-d_2t}, \,\!</math>
			:<math>y(t) = A_3 \sin (tf_3 + p_3) e^{-d_3t} + A_4 \sin (tf_4 + p_4) e^{-d_4t}. \,\!</math>

			An appropriate computer program can translate those equations into a graph that emulates a harmonograph. Applying the first equation a second time to each equation can emulate a moving piece of paper (see the figure below).

			==Gallery==
			<gallery>
			File:Harmonograph.jpg\|A harmonograph at [[Questacon]] in [[Canberra]], [[Australia]].
			Image:lateral2.jpg\|A figure produced by a simple lateral harmonograph.
			Image:lateral3.jpg\|A figure produced by a simple lateral harmonograph.
			Image:pgraph1.jpg\|A figure produced by a pintograph.
			Image:Harmonograph 9998.png\|Computer generated harmonograph figure.
			</gallery>

			==See also==
			*[[Spirograph]]

			==Notes==
			{{reflist}}

			==External links==
			*[http://www.jonathan.lansey.net/pastimes/pendulum/index.html A complex harmonograph with a unique single pendulum design]
			*[http://www.sequences.org.uk/harmonograph/ Online interactive Harmonograph]
			*[http://local.wasp.uwa.edu.au/~pbourke/geometry/harmonograph/ Harmonograph background, equations, and illustrations]
			*[http://www.karlsims.com/harmonograph/ How to build a 3-pendulum rotary harmonograph]
			*[http://hernan.amiune.com/labs/harmonograph/animated-harmonograph.html HTML5 Animated Harmonograph]
			*[http://www.worldtreesoftware.com/apps/harmonograph/harmonograph.html Virtual Harmonograph web application]
			*[http://excelunusual.com/archive/2011/07/harmonograph-video-demo/ An Animated Harmonogharph Model in MS Excel]
			*[https://itunes.apple.com/us/app/pintograph-mesmerizing-line/id548056488?mt=8&uo=4 An interactive Pintograph for iOS]

			[[Category:Curves]]
			[[Category:Pendulums]]

67.169.167.8 at 09:09, 23 August 2012

2012-08-23T09:09:40Z

New page

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