|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Featured article}}
| | == Nike Air Max 2014 You can find blogs == |
| [[Image:CircularCoordinates.svg|thumb|250px|Points in the polar coordinate system with pole ''O'' and polar axis ''L''. In green, the point with radial coordinate 3 and angular coordinate 60 degrees, or (3,60°). In blue, the point (4,210°).]]
| |
| In [[mathematics]], the '''polar coordinate system''' is a [[dimension|two-dimensional]] [[coordinate system]] in which each [[point (mathematics)|point]] on a [[plane (mathematics)|plane]] is determined by a [[distance]] from a fixed point and an [[angle]] from a fixed direction.
| |
|
| |
|
| The fixed point (analogous to the origin of a [[Cartesian coordinate system|Cartesian system]]) is called the ''pole'', and the [[ray (geometry)|ray]] from the pole in the fixed direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'' or ''radius'', and the angle is the ''angular coordinate'', ''polar angle'', or ''[[azimuth]]''.<ref name="brown">{{Cite book| last = Brown| first = Richard G.| editor = Andrew M. Gleason| year = 1997| title = Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis| publisher = McDougal Littell| location = Evanston, Illinois| isbn = 0-395-77114-5}}</ref>
| | You can find blogs, articles and finish profile about superb and best players of cricket. Provides all news, views, live cricket, fixtures, results, videos, teams, ICC Ranking with cricket related games and pools which allows its visitors to keep [http://www.stuman.com.au/myshop/captcha/frame.asp Nike Air Max 2014] them aware of the latest and hottest news in cricket world. <br><br>In 1905, Albert Einstein wrote his first paper on special relativity. In it, he established that light travels at the same speed no matter how fast the observer moves. Today, years later, she continues to have that original poster and she has become the confident, selfassured, addictionfree individual that she visualized and represented on this poster. Even though her environment was the total opposite of what she thought about being, that vision of who she wanted to be transformed her from the inside out. <br><br>Environmental risk factors are water shortage, flies, poor hygiene conditions, and crowded households. A prolonged exposure to infection throughout childhood and young adulthood appears to be necessary to produce the complications seen in later life. <br><br>CED is a stocking distributor of Allen Bradley, Hoffman, Hubbell, Panduit, Fluke, Ideal, T plus much more. We support pharmaceuticals, biotech manufacturers, custom machine builders, along with the food beverage and water / wastewater industries. We want to thank you for choosing Pediatric Adult Medicine, Inc for your child medical care and welcome you to our practice. We can guarantee that you feel confident in our ability to meet and exceed all of your expecations. <br><br>A: Saskatchewan is a trading province, and mineral exports represent a significant percentage of our exports. Key markets for potash exports include China, United States, Brazil, and India for enhanced crop production in corn, wheat, rice, (37 per cent) fruits and vegetables (22 per cent) and oilseeds (16 per cent). <br><br>Most people take off to the Caribbean in order to take a well deserved break in the usual day to day hassle and bustle. If you're on such a mission, then all you need is an allinclusive vacation of a life time. This is no longer a question of selling subscriptions. [http://www.stuman.com.au/myshop/captcha/frame.asp Nike Air Max 1] The 'highimpact' obsession is warping our scientific judgment, damaging careers, and putting things off and valuable work.". <br><br>Along the way she must confront scary truths about her family's past and perhaps even her destiny. She's also reached make sure her mother is protected and decide what kind of life she'll have after these traumatic events.. The curriculum needs to be wellrounded to incorporate various aspects of graphic designing bringing out unique creations through the merging of the right [http://www.myhobbycraft.com/FreeDL/media.asp Air Yeezy 2] materials with the technical proficiency. A comprehensive course extended by a design school covers every facet of the creative development procedure [http://www.ngs.com.sg/publish/backup.asp Abercrombie Singapore] that motivates the students to launch creations that possess depth and symbolism to produce maximum impact on the target audience..<ul> |
| | |
| | <li>[http://superherokale.com/activity/p/403290/ http://superherokale.com/activity/p/403290/]</li> |
| | |
| | <li>[http://site.vhostapp.com/news/html/?1908812.html http://site.vhostapp.com/news/html/?1908812.html]</li> |
| | |
| | <li>[http://www.tianmizhidu.com/forum.php?mod=viewthread&tid=603257 http://www.tianmizhidu.com/forum.php?mod=viewthread&tid=603257]</li> |
| | |
| | <li>[http://globaldialogue2013.go.tz/gdforum/activity http://globaldialogue2013.go.tz/gdforum/activity]</li> |
| | |
| | <li>[http://lmusicradio.altervista.org/osclass/index.php?page=item&id=74659 http://lmusicradio.altervista.org/osclass/index.php?page=item&id=74659]</li> |
| | |
| | </ul> |
|
| |
|
| ==History== | | == Louis Vuitton Belt institutions == |
| {{See also|History of trigonometric functions}}
| |
| [[Image:Hipparchos 1.jpeg|thumb|190px|Hipparchus]]
| |
| The concepts of angle and radius were already used by ancient peoples of the 1st millennium [[Before Christ|BC]]. The [[Greek astronomy|Greek astronomer]] and [[astrologer]] [[Hipparchus]] (190–120 BC) created a table of [[chord (geometry)|chord]] functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.<ref name="milestones">{{Cite web| last = Friendly| first = Michael| title = Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization| url = http://www.math.yorku.ca/SCS/Gallery/milestone/sec4.html| accessdate = 2006-09-10}}</ref>
| |
| In ''[[On Spirals]]'', [[Archimedes]] describes the [[Archimedean spiral]], a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.
| |
|
| |
|
| From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to [[Makkah]] ([[qibla]])—and its distance—from any location on the Earth.<ref>{{Cite book
| | "It's amazing. He was the backup a few months ago, and now he's the guy that's rising on the walls in stadiums. Out of work since Roque bounced him from city hall, "My client is financially suffering because of the defendants' actions in the complaint," said Gomez attorney Victor [http://www.municipalitegrenville.qc.ca/identification/base.asp Louis Vuitton Belt] Afanador. "It was the mayor's way or the highway. <br><br>Additionally, various states, institutions, and scholarships also use the FAFSA to grant private financial aid, so it is really important to complete this application. Here's what you need to fill out your FAFSA.. We understand the importance of privacy to our customers and visitors to our website. GMP Fitness collects just the personal information you provide to us when you purchase products from us, sign up for one of our courses or workshops, submit a request for a catalog or request to become put on our email or newsletter list. <br><br>Your access to and use of the Site is subject to the following terms and conditions ("Terms and Conditions"). WBP may revise these Terms and Conditions at any time by updating this posting. Select a companyFirst patient enrolled in Phase III medical trial for moxetumomab pasudotox as a treatment for unresponsive or relapsed hairy cell leukaemia patients. Olaparib planned to progress to Phase III for platinumsensitive relapsed ovarian cancer patients with BRCA mutations in 2013. <br><br>That [http://www.ace-men.com/japan/gucci/file.php Gucci Belt] doesn't, of course, negate McQ underlying point that Pelosi ineffectiveness at governing and inability to get anything done (especially any of the pet causes of the nutroots brigade). But if you think that spin/bias from the media (in whatever form) accounts for. <br><br>If at any time you do not agree with the terms of this Privacy Policy or any of the Terms of Utilization of Inspired Fitness, you must immediately discontinue use of the Sites and Services. Continued access to or use of the Sites and Services by any means affirmatively indicates your consent to this Online privacy policy and the Terms of Use.. <br><br>Apparently, the party is trying to attract more centrist and may be the right wing [http://www.gear4girls.com.au/generator/pages/page.html Nike Free Run] votes to outdo the Liberals and thus mimicking their political rivals. However, this silence reflects very badly on a party that is supposed to be leftist and it is [http://www.rsaturangi.co.nz/e107_themes/reline/style.html Nike Free 5.0 Nz] expected to respect social justice greater than any other established political party in BC.. <br><br>Hillbillies: The way they Got ThereLike many common terms, the word 'hillbilly' is in dispute. Most who use the term have no idea where it comes down from. The move is extremely similar to what they did last season with Randy Moss. When the 49ers are right, it could provide them with the big, physical corner they lacked last season.<ul> |
| | title= The Sacred Geography of Islam
| | |
| | first= David A.
| | <li>[http://www.eventbuddie.com/activity/p/542741/ http://www.eventbuddie.com/activity/p/542741/]</li> |
| | last= King
| | |
| | others = p.166. In {{Cite book
| | <li>[http://www.jyj578.com/news/html/?72330.html http://www.jyj578.com/news/html/?72330.html]</li> |
| | title= Mathematics and the Divine: A Historical Study
| | |
| | editor1-last= Koetsier
| | <li>[http://www.121e.cn/forum.php?mod=viewthread&tid=2185631 http://www.121e.cn/forum.php?mod=viewthread&tid=2185631]</li> |
| | editor1-first= Teun
| | |
| | editor2-first= Bergmans
| | <li>[http://www.yinmao.org/forum.php?mod=viewthread&tid=133864&fromuid=22455 http://www.yinmao.org/forum.php?mod=viewthread&tid=133864&fromuid=22455]</li> |
| | editor2-last= Luc
| | |
| | publisher= Elsevier
| | <li>[http://www.thomassankara.net/spip.php?article605&date=2008-05/ http://www.thomassankara.net/spip.php?article605&date=2008-05/]</li> |
| | location= Amsterdam
| | |
| | year= 2005
| | </ul> |
| | pages= 162–78
| |
| | isbn= 0-444-50328-5
| |
| | url= http://books.google.com/?id=AMOQZfrZq-EC&printsec=frontcover#v=onepage
| |
| |ref=harv}}
| |
| | year= 2005
| |
| | url= http://books.google.com.au/books?id=AMOQZfrZq-EC&pg=PA161#v=onepage&f=false
| |
| | ref=harv}}
| |
| </ref> From the 9th century onward they were using [[spherical trigonometry]] and [[map projection]] methods to determine these quantities accurately. The calculation is essentially the conversion of the [[Geodetic_coordinates#Coordinates|equatorial polar coordinates]] of Mecca (i.e. its [[longitude]] and [[latitude]]) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its [[antipodal point]].<ref>King ([[#CITEREFKing2005|2005]], [http://books.google.com.au/books?id=AMOQZfrZq-EC&pg=PA169#v=onepage&f=false p. 169]). The calculations were as accurate as could be achieved under the limitations imposed by their assumption that the Earth was a perfect sphere.</ref> | |
|
| |
|
| There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in [[Harvard University|Harvard]] professor [[Julian Lowell Coolidge]]'s ''Origin of Polar Coordinates.''<ref name="coolidge">{{Cite journal| last = Coolidge| first = Julian| authorlink = Julian Lowell Coolidge| title = The Origin of Polar Coordinates| journal = American Mathematical Monthly| volume = 59| pages = 78–85| year = 1952| url = http://www-history.mcs.st-and.ac.uk/Extras/Coolidge_Polars.html| doi = 10.2307/2307104| issue = 2| publisher = Mathematical Association of America| jstor = 2307104}}</ref> [[Grégoire de Saint-Vincent]] and [[Bonaventura Cavalieri]] independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an [[Archimedean spiral]]. [[Blaise Pascal]] subsequently used polar coordinates to calculate the length of [[parabola|parabolic arcs]].
| | == Nike Air Max 90 Singapore 7130 == |
|
| |
|
| In ''[[Method of Fluxions]]'' (written 1671, published 1736), Sir [[Isaac Newton]] examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems.<ref>{{Cite journal| last = Boyer| first = C. B.| title = Newton as an Originator of Polar Coordinates| journal = American Mathematical Monthly| volume = 56| pages = 73–78| year = 1949| doi = 10.2307/2306162| issue = 2| publisher = Mathematical Association of America| jstor = 2306162}}</ref> In the journal ''[[Acta Eruditorum]]'' (1691), [[Jacob Bernoulli]] used a system with a point on a line, called the ''pole'' and ''polar axis'' respectively. Coordinates were specified by the distance from the pole and the angle from the ''polar axis''. Bernoulli's work extended to finding the [[Radius of curvature (mathematics)|radius of curvature]] of curves expressed in these coordinates.
| | Many careers require a minimum degree for employment. Online colleges, vocational [http://www.icscancer.com/test/test/img/test.asp Nike Air Max 90 Singapore] schools, twoyear colleges, and fouryear universities all offer some kind of degree program.. In 2006, following a year of confusion and frustration, I applied everything I'd learned from other online marketing experts and opened a brand new store. Today, that store receives over 20,000 visitors per month, shows up as the number one lead [http://www.riverwalkcottage.com.au/site/about.html Longchamp Australia] to Google for decor (can you figure out which themetastic store is mine?) and earns over $12,000 a month in [http://www.wentworth.ca/accueil/catalogue/file.asp Abercrombie Canada] sales. <br><br>His paintings contained unreal dream space by strange characters of his imagery. He described his pictures as handpainted dream photographs'.. 1998 Feb 14;316(7130):492492. [PMC free article] [Articles from Quality Safety in Healthcare are provided here courtesy of BMJ Group. <br><br>You'll be able to get daily and weekly statistics on the number of readers you are receiving along with the revenue that you have generated. It is really rewarding to display your work in this way and many people feel that it gets addicting.. But when you going to buy into the Mayan calendar theory, why don't you couple it with some faith within the Norse myth of Ragnarok? According to this Teutonic legend, Ragnarok, while heralding no more the world, also marks its new beginning. The Norse gods will be unseated, the story goes, and the world can come to an end, but a new and better world will rise from the ashes.. <br><br>One of reasons Royal Dutch Shell gives for mothballing the last of the city refineries is that it now has a worldwide strategy of focusing on crude oil rather than refined oil. But one more reason is that the Montreal facility was among the least competitive of Shell installations to assist do the refining that the company still needs to do.. <br><br>BBC NewsNewspaper review: Press set of Evans arrestBBC NewsThe arrest of Conservative MP Nigel Evans, on suspicion of sexual offences, is on the front page of many Sunday papers. The sun's rays says the arrest has "stunned" Westminster, and it quotes friends as saying it was "a bolt from the blue". <br><br>Almost anywhere that you can go are great places to complete some exercises even if you have only 15 minutes or so out of your busy day. To obtain a full effect of the exercises that you simply do, just make sure you also fuel yourself up to have an even better result that will come [http://www.gsar.ie/content/finder.asp Mbt Shop Dublin] out of it. <br><br>This is important because it gets individuals to connect with you and your company and it builds your reputation. For example: When you see a little white apple having a bite taken out of it, what company comes to mind? Simple graphics and images can perform a lot for your business.<ul> |
| | | |
| The actual term ''polar coordinates'' has been attributed to [[Gregorio Fontana]] and was used by 18th-century Italian writers. The term appeared in [[English language|English]] in [[George Peacock]]'s 1816 translation of [[Sylvestre François Lacroix|Lacroix]]'s ''Differential and Integral Calculus''.<ref>{{Cite web| last = Miller| first = Jeff| title = Earliest Known Uses of Some of the Words of Mathematics| url = http://members.aol.com/jeff570/p.html| accessdate = 2006-09-10}}</ref><ref>{{Cite book| last = Smith| first = David Eugene| title = History of Mathematics, Vol II| publisher = Ginn and Co.| year = 1925| location = Boston| pages = 324}}</ref> [[Alexis Clairaut]] was the first to think of polar coordinates in three dimensions, and [[Leonhard Euler]] was the first to actually develop them.<ref name="coolidge" />
| | <li>[http://www.tianwaitianrihua.com/news/html/?680897.html http://www.tianwaitianrihua.com/news/html/?680897.html]</li> |
| | | |
| ==Conventions==
| | <li>[https://blog-relay.rhcloud.com/bbs/forum.php?mod=viewthread&tid=2132 https://blog-relay.rhcloud.com/bbs/forum.php?mod=viewthread&tid=2132]</li> |
| [[Image:Polar graph paper.svg|thumb|right|300px|A polar grid with several angles labeled in degrees]]
| | |
| The radial coordinate is often denoted by ''r'', and the angular coordinate by ''φ'', ''θ'' or ''t''. The angular coordinate is specified as ''φ'' by [[International Organisation for Standardisation|ISO]] standard [[ISO 31-11|31-11]]. | | <li>[http://www.scientists.unas.cz/index.php?site=forum_topic&topic=13249 http://www.scientists.unas.cz/index.php?site=forum_topic&topic=13249]</li> |
| | | |
| Angles in polar notation are generally expressed in either [[degree (angle)|degree]]s or [[radian]]s (2[[pi|π]] rad being equal to 360°). Degrees are traditionally used in [[navigation]], [[surveying]], and many applied disciplines, while radians are more common in mathematics and mathematical [[physics]].<ref>{{Cite book| last = Serway| first = Raymond A.| coauthors = Jewett, Jr., John W.| title = Principles of Physics| publisher = Brooks/Cole—Thomson Learning| year = 2005| isbn = 0-534-49143-X}}</ref>
| | <li>[http://www.100wwww.com/space.php?uid=11244&do=blog&id=169676 http://www.100wwww.com/space.php?uid=11244&do=blog&id=169676]</li> |
| | | |
| In many contexts, a positive angular coordinate means that the angle ''φ'' is measured [[clockwise|counterclockwise]] from the axis.
| | <li>[http://www.cnljjf.com/news/html/?22443.html http://www.cnljjf.com/news/html/?22443.html]</li> |
| | | |
| In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.
| | </ul> |
| | |
| ===Uniqueness of polar coordinates=== | |
| Adding any number of full [[turn (geometry)|turn]]s (360°) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates {{nowrap|(''r'', ''φ'' ± ''n''×360°)}} or {{nowrap|(−''r'', ''φ'' ± (2''n'' + 1)180°)}}, where ''n'' is any [[integer]].<ref>{{Cite web| url = http://www.fortbendisd.com/campuses/documents/Teacher/2006%5Cteacher_20060413_0948.pdf| title = Polar Coordinates and Graphing| accessdate = 2006-09-22| date = 2006-04-13| format = PDF}}</ref> Moreover, the pole itself can be expressed as (0, ''φ'') for any angle ''φ''.<ref>{{Cite book|title=Precalculus: With Unit-Circle Trigonometry|last=Lee|first=Theodore|coauthors=David Cohen, David Sklar|year=2005|publisher=Thomson Brooks/Cole|edition=Fourth|isbn=0-534-40230-5}}</ref>
| |
| | |
| Where a unique representation is needed for any point, it is usual to limit ''r'' to [[non-negative number]]s ({{nowrap|''r'' ≥ 0}}) and ''φ'' to the [[interval (mathematics)|interval]] [0, 360°) or (−180°, 180°] (in radians, [0, 2π) or (−π, π]).<ref>{{Cite book|title=Complex Analysis (the Hitchhiker's Guide to the Plane)|first=Ian|last=Stewart|coauthors=David Tall|year=1983|publisher=Cambridge University Press|isbn=0-521-28763-4}}</ref> One must also choose a unique azimuth for the pole, e.g., ''φ'' = 0.
| |
| | |
| ==Converting between polar and Cartesian coordinates==
| |
| [[Image:Polar to cartesian.svg|right|thumb|250px|A diagram illustrating the relationship between polar and Cartesian coordinates.]]
| |
| [[File:Cartesian to polar.gif|right|thumb|251px|A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, <math>y = \sin (6x) + 2</math> is mapped onto <math>r = \sin (6 \varphi) + 2</math>. Click on image for details.]]
| |
| | |
| The polar coordinates ''r'' and ''φ'' can be converted to the [[Cartesian coordinate system|Cartesian coordinates]] ''x'' and ''y'' by using the [[trigonometric function]]s sine and cosine:
| |
| :<math>x = r \cos \varphi \,</math>
| |
| :<math>y = r \sin \varphi \,</math>
| |
| | |
| The Cartesian coordinates ''x'' and ''y'' can be converted to polar coordinates ''r'' and ''φ'' with ''r'' ≥ 0 and ''φ'' in the interval (−π, π] by:<ref>{{Cite book|first=Bruce Follett|last=Torrence|coauthors=Eve Torrence|title=The Student's Introduction to Mathematica|year=1999|publisher=Cambridge University Press|isbn=0-521-59461-8}}</ref>
| |
| | |
| :<math>r = \sqrt{x^2 + y^2} \quad</math> (as in the [[Pythagorean theorem]] or the [[Euclidean norm]]), and
| |
| :<math>\varphi = \operatorname{atan2}(y, x) \quad</math>,
| |
| where [[atan2]] is a common variation on the [[arctangent]] function defined as
| |
| :<math>\operatorname{atan2}(y, x) =
| |
| \begin{cases}
| |
| \arctan(\frac{y}{x}) & \mbox{if } x > 0\\
| |
| \arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\
| |
| \arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\
| |
| \frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\
| |
| -\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\
| |
| \text{undefined} & \mbox{if } x = 0 \mbox{ and } y = 0
| |
| \end{cases}</math>
| |
| | |
| The value of ''φ'' above is the [[principal value]] of the complex number function [[Argument (complex analysis)|arg]] applied to ''x''+''iy''. An angle in the range [0, 2π) may be obtained by adding 2π to the value in case it is negative.
| |
| | |
| ==Polar equation of a curve==
| |
| The equation defining an [[algebraic curve]] expressed in polar coordinates is known as a ''polar equation''. In many cases, such an equation can simply be specified by defining ''r'' as a [[function (mathematics)|function]] of ''φ''. The resulting curve then consists of points of the form (''r''(''φ''), ''φ'') and can be regarded as the [[graph of a function|graph]] of the polar function ''r''.
| |
| | |
| Different forms of [[symmetry]] can be deduced from the equation of a polar function ''r''. If {{nowrap|''r''(−''φ'') {{=}} ''r''(''φ'')}} the curve will be symmetrical about the horizontal (0°/180°) ray, if {{nowrap|''r''(π − ''φ'') {{=}} ''r''(''φ'')}} it will be symmetric about the vertical (90°/270°) ray, and if {{nowrap|''r''(''φ'' − α) {{=}} ''r''(''φ'')}} it will be [[rotational symmetry|rotationally symmetric]] α [[clockwise|counterclockwise]] about the pole.
| |
| | |
| Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the [[Rose (mathematics)|polar rose]], [[Archimedean spiral]], [[Lemniscate of Bernoulli|lemniscate]], [[limaçon]], and [[cardioid]].
| |
| | |
| For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.
| |
| | |
| ===Circle===
| |
| [[Image:circle r=1.svg|thumb|right|A circle with equation {{nowrap|''r''(''φ'') {{=}} 1}}]]
| |
| The general equation for a circle with a center at {{nowrap|(''r''<sub>0</sub>, <math>\gamma</math>)}} and radius ''a'' is
| |
| :<math>r^2 - 2 r r_0 \cos(\varphi - \gamma) + r_0^2 = a^2.\, </math>
| |
| | |
| This can be simplified in various ways, to conform to more specific cases, such as the equation
| |
| :<math>r(\varphi)=a \,</math>
| |
| for a circle with a center at the pole and radius ''a''.<ref name="ping">{{Cite web| first=Johan| last=Claeys| url=http://www.ping.be/~ping1339/polar.htm| title=Polar coordinates| accessdate=2006-05-25}}</ref>
| |
| | |
| When {{math|''r''}}<sub>0</sub> = {{math|a}}, or when the origin lies on the circle, the equation becomes
| |
| :<math>r = 2 a\cos(\varphi - \gamma)</math>.
| |
| | |
| In the general case, the equation can be solved for {{math|''r''}}, giving
| |
| :<math>r = r_0 \cos(\varphi - \gamma) + \sqrt{a^2 - r_0^2 \sin^2(\varphi - \gamma)}</math>,
| |
| the solution with a minus sign in front of the square root gives the same curve.
| |
| | |
| ===Line===
| |
| [[Image:Rose 2sin(4theta).svg|thumb|right|A polar rose with equation {{nowrap|''r''(''φ'') {{=}} 2 sin 4''φ''}}]]
| |
| ''Radial'' lines (those running through the pole) are represented by the equation
| |
| :<math>\varphi = \gamma \,</math>,
| |
| where ɣ is the angle of elevation of the line; that is, {{nowrap|ɣ {{=}} arctan ''m''}} where ''m'' is the [[slope]] of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line {{nowrap|''φ'' {{=}} ɣ}} [[perpendicular]]ly at the point (''r<sub>0</sub>'', ɣ) has the equation
| |
| :<math>r(\varphi) = {r_0}\sec(\varphi-\gamma). \,</math>
| |
| | |
| Otherwise stated (''r<sub>0</sub>'', ɣ) is the point in which the tangent intersects the imaginary circle of radius ''r<sub>0</sub>''.
| |
| | |
| ===Polar rose===
| |
| A [[rose (mathematics)|polar rose]] is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
| |
| :<math>r(\varphi) = a \cos (k\varphi + \gamma_0)\,</math>
| |
| for any constant ɣ<sub>0</sub> (including 0). If ''k'' is an integer, these equations will produce a ''k''-petaled rose if ''k'' is [[even and odd numbers|odd]], or a 2''k''-petaled rose if ''k'' is even. If ''k'' is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The [[variable (math)|variable]] ''a'' represents the length of the petals of the rose.
| |
| {{-}}
| |
| | |
| ===Archimedean spiral===
| |
| [[Image:Archimedian spiral.svg|thumb|right|One arm of an Archimedean spiral with equation {{nowrap|''r''(''φ'') {{=}} ''φ'' / 2π }} for {{nowrap|0 < ''φ'' < 6π}}]]
| |
| The [[Archimedean spiral]] is a famous spiral that was discovered by [[Archimedes]], which also can be expressed as a simple polar equation. It is represented by the equation
| |
| :<math>r(\varphi) = a+b\varphi. \,</math>
| |
| Changing the parameter ''a'' will turn the spiral, while ''b'' controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for {{nowrap|''φ'' > 0}} and one for {{nowrap|''φ'' < 0}}. The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the [[conic section]]s, to be described in a mathematical treatise, and as being a prime example of a curve that is best defined by a polar equation.
| |
| {{-}}
| |
| | |
| ===Conic sections===
| |
| [[Image:Elps-slr.svg|thumb|right|250px|Ellipse, showing semi-latus rectum]]
| |
| A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's [[semi-major axis|major axis]] lies along the polar axis) is given by:
| |
| | |
| : <math>r = { \ell\over {1 + e \cos \varphi}}</math>
| |
| | |
| where ''e'' is the [[eccentricity (mathematics)|eccentricity]] and <math>\ell</math> is the [[semi-latus rectum]] (the perpendicular distance at a focus from the major axis to the curve). If {{nowrap|''e'' > 1}}, this equation defines a [[hyperbola]]; if {{nowrap|''e'' {{=}} 1}}, it defines a [[parabola]]; and if {{nowrap|''e'' < 1}}, it defines an [[ellipse]]. The special case {{nowrap|''e'' {{=}} 0}} of the latter results in a circle of radius <math>\ell</math>.
| |
| {{-}}
| |
| | |
| ==Intersection of two polar curves==
| |
| The graphs of two polar functions <math>r=f(\theta)</math> and <math>r=g(\theta)</math> have possible intersections in 3 cases:
| |
| # In the origin if the equations <math>f(\theta)=0</math> and <math>g(\theta)=0</math> have at least one solution each.
| |
| # All the points <math>[g(\theta_i),\theta_i]</math> where <math>\theta_i</math> are the solutions to the equation <math>f(\theta)=g(\theta)</math>.
| |
| # All the points <math>[g(\theta_i),\theta_i]</math> where <math>\theta_i</math> are the solutions to the equation <math>f(\theta+(2k+1)\pi)=-g(\theta)</math> where <math>k</math> is an integer.
| |
| {{-}}
| |
| | |
| ==Complex numbers==
| |
| [[Image:Imaginarynumber2.svg|thumb|right|265px|An illustration of a complex number ''z'' plotted on the complex plane]]
| |
| [[Image:Euler's formula.svg|thumb|right|265px|An illustration of a complex number plotted on the complex plane using [[Euler's formula]]]]
| |
| Every [[complex number]] can be represented as a point in the [[complex plane]], and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form). The complex number ''z'' can be represented in rectangular form as
| |
| : <math>z = x + iy\,</math>
| |
| where ''i'' is the [[imaginary unit]], or can alternatively be written in polar form (via the conversion formulae given [[#Converting between polar and Cartesian coordinates|above]]) as
| |
| :<math>z = r\cdot(\cos\varphi+i\sin\varphi)</math>
| |
| and from there as
| |
| : <math>z = re^{i\varphi} \,</math>
| |
| where ''e'' is [[e (mathematical constant)|Euler's number]], which are equivalent as shown by [[Euler's formula]].<ref>
| |
| {{Cite book| last = Smith| first = Julius O.| title = Mathematics of the Discrete Fourier Transform (DFT)| accessdate = 2006-09-22| year = 2003| publisher = W3K Publishing| isbn = 0-9745607-0-7| chapter = Euler's Identity| chapterurl = http://ccrma-www.stanford.edu/~jos/mdft/Euler_s_Identity.html}}</ref> (Note that this formula, like all those involving exponentials of angles, assumes that the angle ''φ'' is expressed in [[radian]]s.) To convert between the rectangular and polar forms of a complex number, the conversion formulae given [[#Converting between polar and Cartesian coordinates|above]] can be used.
| |
| | |
| For the operations of [[multiplication]], [[division (mathematics)|division]], and [[exponentiation]] of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:
| |
| | |
| *Multiplication:
| |
| :: <math>r_0 e^{i\varphi_0} \cdot r_1 e^{i\varphi_1}=r_0 r_1 e^{i(\varphi_0 + \varphi_1)} \,</math>
| |
| *Division:
| |
| :: <math>\frac{r_0 e^{i\varphi_0}}{r_1 e^{i\varphi_1}}=\frac{r_0}{r_1}e^{i(\varphi_0 - \varphi_1)} \,</math>
| |
| *Exponentiation ([[De Moivre's formula]]):
| |
| :: <math>(re^{i\varphi})^n=r^ne^{in\varphi} \,</math>
| |
| | |
| ==Calculus==
| |
| [[Calculus]] can be applied to equations expressed in polar coordinates.<ref>{{Cite web|url=http://archives.math.utk.edu/visual.calculus/5/polar.1/index.html|title=Areas Bounded by Polar Curves|author=Husch, Lawrence S.|accessdate=2006-11-25}}</ref><ref>{{Cite web|url=http://archives.math.utk.edu/visual.calculus/3/polar.1/index.html|title=Tangent Lines to Polar Graphs|author=Lawrence S. Husch|accessdate=2006-11-25}}</ref>
| |
| | |
| The angular coordinate ''φ'' is expressed in radians throughout this section, which is the conventional choice when doing calculus.
| |
| | |
| ===Differential calculus===
| |
| Using {{nowrap|''x'' {{=}} ''r'' cos ''φ'' }} and {{nowrap|''y'' {{=}} ''r'' sin ''φ'' }}, one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function, ''u''(''x'',''y''), it follows that
| |
| :<math>r \frac{\partial u}{\partial r} = r \frac{\partial u}{\partial x}\frac{\partial x}{\partial r} + r \frac{\partial u}{\partial y}\frac{\partial y}{\partial r},</math>
| |
| :<math>\frac{\partial u}{\partial \varphi} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial \varphi} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial \varphi},</math>
| |
| or
| |
| :<math>r \frac{\partial u}{\partial r} = r \frac{\partial u}{\partial x} \cos \varphi + r \frac{\partial u}{\partial y} \sin \varphi = x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y},</math>
| |
| :<math>\frac{\partial u}{\partial \varphi} = - \frac{\partial u}{\partial x} r \sin \varphi + \frac{\partial u}{\partial y} r \cos \varphi = -y \frac{\partial u}{\partial x} + x \frac{\partial u}{\partial y}.</math>
| |
| | |
| Hence, we have the following formulae:
| |
| | |
| :<math>r \frac{\partial}{\partial r}= x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} \,</math>
| |
| :<math>\frac{\partial}{\partial \varphi} = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y} .</math>
| |
| | |
| Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function ''u''(''r'',''φ''), it follows that
| |
| :<math>\frac{\partial u}{\partial x} = \frac{\partial u}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial u}{\partial \varphi}\frac{\partial \varphi}{\partial x},</math>
| |
| :<math>\frac{\partial u}{\partial y} = \frac{\partial u}{\partial r}\frac{\partial r}{\partial y} + \frac{\partial u}{\partial \varphi}\frac{\partial \varphi}{\partial y},</math>
| |
| or
| |
| :<math>\frac{\partial u}{\partial x} = \frac{\partial u}{\partial r}\frac{x}{\sqrt{x^2+y^2}} - \frac{\partial u}{\partial \varphi}\frac{y}{x^2+y^2} = \cos \varphi \frac{\partial u}{\partial r} - \frac{1}{r} \sin \varphi \frac{\partial u}{\partial \varphi},</math>
| |
| :<math>\frac{\partial u}{\partial y} = \frac{\partial u}{\partial r}\frac{y}{\sqrt{x^2+y^2}} + \frac{\partial u}{\partial \varphi}\frac{x}{x^2+y^2} = \sin \varphi \frac{\partial u}{\partial r} + \frac{1}{r} \cos \varphi \frac{\partial u}{\partial \varphi}.</math>
| |
| | |
| Hence, we have the following formulae:
| |
| :<math>\frac{\partial}{\partial x} = \cos \varphi \frac{\partial}{\partial r} - \frac{1}{r} \sin \varphi \frac{\partial}{\partial \varphi} \,</math>
| |
| :<math>\frac{\partial}{\partial y} = \sin \varphi \frac{\partial}{\partial r} + \frac{1}{r} \cos \varphi \frac{\partial}{\partial \varphi}.</math>
| |
| | |
| To find the Cartesian slope of the tangent line to a polar curve ''r''(''φ'') at any given point, the curve is first expressed as a system of [[parametric equations]].
| |
| :<math>x=r(\varphi)\cos\varphi \,</math>
| |
| :<math>y=r(\varphi)\sin\varphi \,</math>
| |
| | |
| [[Derivative|Differentiating]] both equations with respect to ''φ'' yields
| |
| :<math>\frac{dx}{d\varphi}=r'(\varphi)\cos\varphi-r(\varphi)\sin\varphi \,</math>
| |
| :<math>\frac{dy}{d\varphi}=r'(\varphi)\sin\varphi+r(\varphi)\cos\varphi. \,</math>
| |
| | |
| Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point {{nowrap|(''r''(''φ''), ''φ'')}}:
| |
| :<math>\frac{dy}{dx}=\frac{r'(\varphi)\sin\varphi+r(\varphi)\cos\varphi}{r'(\varphi)\cos\varphi-r(\varphi)\sin\varphi}.</math>
| |
| | |
| For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see [[curvilinear coordinates]].
| |
| | |
| ===Integral calculus (arc length)===
| |
| The arc length (length of a line segment) defined by a polar function is found by the integration over the curve ''r''(''φ''). Let ''L'' denote this length along the curve starting from points ''A'' through to point ''B'', where these points correspond to ''φ'' = ''a'' and ''φ'' = ''b'' such that {{nowrap|0 < ''b'' − ''a'' < 2π}}. The length of ''L'' is given by the following integral
| |
| | |
| :<math>L = \int_a^b \sqrt{ \left[r(\varphi)\right]^2 + \left[ {{dr(\varphi) } \over { d\varphi }} \right] ^2 } d\varphi</math>
| |
| | |
| ===Integral calculus (area)===
| |
| [[Image:Polar coordinates integration region.svg|thumb|The integration region ''R'' is bounded by the curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b''.]]
| |
| Let ''R'' denote the region enclosed by a curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b'', where {{nowrap|0 < ''b'' − ''a'' ≤ 2π}}. Then, the area of ''R'' is
| |
| | |
| :<math>\frac12\int_a^b \left[r(\varphi)\right]^2\, d\varphi.</math>
| |
| | |
| [[Image:Polar coordinates integration Riemann sum.svg|thumb|The region ''R'' is approximated by ''n'' sectors (here, ''n'' = 5).]]
| |
| [[File:Planimeter.jpg|thumb|A [[planimeter]], which mechanically computes polar integrals]]
| |
| This result can be found as follows. First, the interval {{nowrap|[''a'', ''b'']}} is divided into ''n'' subintervals, where ''n'' is an arbitrary positive integer. Thus Δ''φ'', the length of each subinterval, is equal to {{nowrap|''b'' − ''a''}} (the total length of the interval), divided by ''n'', the number of subintervals. For each subinterval ''i'' = 1, 2, …, ''n'', let ''φ''<sub>''i''</sub> be the midpoint of the subinterval, and construct a [[circular sector|sector]] with the center at the pole, radius ''r''(''φ''<sub>''i''</sub>), central angle Δ''φ'' and arc length ''r''(''φ''<sub>''i''</sub>)Δ''φ''. The area of each constructed sector is therefore equal to
| |
| :<math>\left[r(\varphi_i)\right]^2 \pi \cdot \frac{\Delta \varphi}{2\pi} = \frac{1}{2}\left[r(\varphi_i)\right]^2 \Delta \varphi.</math>
| |
| Hence, the total area of all of the sectors is
| |
| :<math>\sum_{i=1}^n \tfrac12r(\varphi_i)^2\,\Delta\varphi.</math>
| |
| | |
| As the number of subintervals ''n'' is increased, the approximation of the area continues to improve. In the limit as {{nowrap|''n'' → ∞}}, the sum becomes the [[Riemann sum]] for the above integral.
| |
| | |
| A mechanical device that computes area integrals is the [[planimeter]], which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element [[Linkage (mechanical)|linkage]] effects [[Green's theorem]], converting the quadratic polar integral to a linear integral.
| |
| | |
| ====Generalization====
| |
| Using [[Cartesian coordinates]], an infinitesimal area element can be calculated as ''dA'' = ''dx'' ''dy''. The [[integration by substitution#Substitution_for_multiple_variables|substitution rule for multiple integrals]] states that, when using other coordinates, the [[Jacobian matrix and determinant|Jacobian]] determinant of the coordinate conversion formula has to be considered:
| |
| : <math>J = \det\frac{\partial(x,y)}{\partial(r,\varphi)}
| |
| =\begin{vmatrix}
| |
| \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \varphi} \\[8pt]
| |
| \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \varphi}
| |
| \end{vmatrix}
| |
| =\begin{vmatrix}
| |
| \cos\varphi & -r\sin\varphi \\
| |
| \sin\varphi & r\cos\varphi
| |
| \end{vmatrix}
| |
| =r\cos^2\varphi + r\sin^2\varphi = r.</math>
| |
| | |
| Hence, an area element in polar coordinates can be written as
| |
| :<math>dA = dx\,dy\ = J\,dr\,d\varphi = r\,dr\,d\varphi.</math>
| |
| | |
| Now, a function that is given in polar coordinates can be integrated as follows:
| |
| :<math>\iint_R f(x,y) \, dA = \int_a^b \int_0^{r(\varphi)} f(r,\varphi)\,r\,dr\,d\varphi.</math>
| |
| Here, ''R'' is the same region as above, namely, the region enclosed by a curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b''.
| |
| | |
| The formula for the area of ''R'' mentioned above is retrieved by taking ''f'' identically equal to 1. A more surprising application of this result yields the [[Gaussian integral]]
| |
| :<math> \int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt\pi.</math>
| |
| | |
| ===Vector calculus===
| |
| [[Vector calculus]] can also be applied to polar coordinates. For a planar motion, let <math>\mathbf{r}</math> be the position vector {{nowrap|(''r''cos(''φ''), ''r''sin(''φ''))}}, with ''r'' and ''φ'' depending on time ''t''.
| |
| | |
| We define the unit vectors
| |
| :<math>\hat{\mathbf{r}}=(\cos(\varphi),\sin(\varphi))</math>
| |
| in the direction of '''r''' and
| |
| :<math>\hat{\boldsymbol\varphi}=(-\sin(\varphi),\cos(\varphi)) = \hat {\mathbf{k}} \times \hat {\mathbf{r}} \ , </math>
| |
| in the plane of the motion perpendicular to the radial direction, where <math>\hat{\mathbf {k}}</math> is a unit vector normal to the plane of the motion.
| |
| | |
| Then
| |
| | |
| :<math> \mathbf{r} = (x, \ y ) = r (\cos \varphi ,\ \sin \varphi) = r \hat{\mathbf{r}}\ , </math>
| |
| | |
| :<math> \dot {\mathbf r} = (\dot x, \ \dot y ) = \dot r (\cos \varphi ,\ \sin \varphi) + r \dot \varphi (-\sin \varphi ,\ \cos \varphi) = \dot r \hat {\mathbf r} + r \dot \varphi \hat {\boldsymbol{\varphi}} \ , </math>
| |
| | |
| :<math> \ddot {\mathbf r } = (\ddot x, \ \ddot y ) = \ddot r (\cos \varphi ,\ \sin \varphi) + 2\dot r \dot \varphi (-\sin \varphi ,\ \cos \varphi) + r\ddot \varphi (-\sin \varphi ,\ \cos \varphi) - r {\dot \varphi }^2 (\cos \varphi ,\ \sin \varphi)\ = </math>
| |
| ::<math> \left( \ddot r - r\dot\varphi^2 \right) \hat{\mathbf r} + \left( r\ddot\varphi + 2\dot r \dot\varphi \right) \hat{\boldsymbol{\varphi}} \ = (\ddot r - r\dot\varphi^2)\hat{\mathbf{r}} + \frac{1}{r}\; \frac{d}{dt} \left(r^2\dot\varphi\right) \hat{\boldsymbol{\varphi}}</math>
| |
| | |
| ====Centrifugal and Coriolis terms====
| |
| {{See also|Mechanics of planar particle motion|Centrifugal force (rotating reference frame)}}
| |
| The term <math>r\dot\varphi^2</math> is sometimes referred to as the ''centrifugal term'', and the term <math>2\dot r \dot\varphi</math> as the ''Coriolis term''. For example, see Shankar.<ref name=Shankar>{{Cite book|title=Principles of Quantum Mechanics|author=Ramamurti Shankar|edition=2nd|page=81|url=http://books.google.com/?id=2zypV5EbKuIC&pg=PA81&dq=Coriolis+%22polar+coordinates%22|year=1994|isbn=0-306-44790-8|publisher=Springer}}</ref> Although these equations bear some resemblance in form to the [[centrifugal force|centrifugal]] and [[Coriolis effect]]s found in rotating reference frames, nonetheless these are not the same things.<ref name=angular>In particular, the angular rate appearing in the polar coordinate expressions is that of the particle under observation, <math>\dot{\varphi}</math>, while that in classical Newtonian mechanics is the angular rate Ω of a rotating frame of reference.</ref> For example, the physical centrifugal and Coriolis forces appear only in [[non-inertial frame]]s of reference. In contrast, these terms that appear when acceleration is expressed in polar coordinates are a mathematical consequence of differentiation; these terms appear wherever polar coordinates are used. In particular, these terms appear even when polar coordinates are used in [[inertial frame]]s of reference, where the physical centrifugal and Coriolis forces never appear.
| |
| | |
| [[Image:Co-rotating frame.PNG|thumb|300px|Inertial frame of reference ''S'' and instantaneous non-inertial co-rotating frame of reference ''S′''. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of ''S′'' at the particular moment ''t''. Particle is located at vector position '''r'''(''t'') and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle ''θ'' normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance ''r'' need not be related to the radius of curvature of the path.]]
| |
| | |
| =====Co-rotating frame=====
| |
| For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous ''co-rotating frame of reference''.<ref name=Taylor>For the following discussion, see {{Cite book|author=John R Taylor|title=Classical Mechanics|page=§9.10, pp. 358–359|isbn=1-891389-22-X|publisher=University Science Books|year=2005}}</ref> To define a co-rotating frame, first an origin is selected from which the distance ''r''(''t'') to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment ''t'', the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, ''dφ''/''dt''. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (''r(''t''), ''φ''(''t'')), and in the co-rotating frame be (''r(t), ''φ''′(t)''). Because the co-rotating frame rotates at the same rate as the particle, ''dφ''′/''dt'' = 0. The fictitious centrifugal force in the co-rotating frame is ''mrΩ<sup>2</sup>, radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because ''dφ''′/''dt'' = 0. The ''fictitious Coriolis force'' therefore has a value −2''m''(''dr''/''dt'')Ω, pointed in the direction of increasing ''φ'' only. Thus, using these forces in Newton's second law we find:
| |
| :<math>\boldsymbol{F} + \boldsymbol{F_{cf}} + \boldsymbol{F_{Cor}} = m \ddot{\boldsymbol{r}} \ , </math>
| |
| where over dots represent time differentiations, and '''F''' is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes:
| |
| :<math>F_r + mr\Omega^2 = m\ddot r</math>
| |
| :<math>F_{\varphi}-2m\dot r \Omega = mr \ddot {\varphi} \ , </math>
| |
| which can be compared to the equations for the inertial frame:
| |
| :<math>F_r = m \ddot r -mr \dot {\varphi}^2 \ </math>
| |
| :<math>F_{\varphi} = mr \ddot \varphi +2m \dot r \dot {\varphi} \ . </math>
| |
| This comparison, plus the recognition that by the definition of the co-rotating frame at time ''t'' it has a rate of rotation Ω = ''dφ''/''dt'', shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.
| |
| | |
| For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous [[osculating circle]] of its motion, not to a fixed center of polar coordinates. For more detail, see [[Centripetal force#Local coordinates|centripetal force]].
| |
| | |
| ==Connection to spherical and cylindrical coordinates==
| |
| The polar coordinate system is extended into three dimensions with two different coordinate systems, the [[cylindrical coordinate system|cylindrical]] and [[spherical coordinate system]].
| |
| | |
| ==Applications==
| |
| Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of [[circular motion|circular]] and [[orbital motion]].
| |
| | |
| ===Position and navigation===
| |
| Polar coordinates are used often in [[navigation]], as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, [[aircraft]] use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a [[clockwise]] direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to [[magnetic north]], while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.<ref>{{Cite web|url=http://www.thaitechnics.com/nav/adf.html|title=Aircraft Navigation System|accessdate=2006-11-26|first=Sumrit|last=Santhi}}</ref> Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read [[ICAO spelling alphabet|zero-niner-zero]] by [[air traffic control]]).<ref>{{Cite web|url=http://www.faa.gov/library/manuals/aircraft/airplane_handbook/media/faa-h-8083-3a-7of7.pdf|title=Emergency Procedures|format=PDF|accessdate=2007-01-15}}</ref>
| |
| | |
| ===Modeling===
| |
| Systems displaying [[radial symmetry]] provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the [[groundwater flow equation]] when applied to radially symmetric wells. Systems with a [[central force|radial force]] are also good candidates for the use of the polar coordinate system. These systems include [[gravitation|gravitational fields]], which obey the [[inverse-square law]], as well as systems with [[point source]]s, such as [[antenna (radio)|radio antennas]].
| |
| | |
| Radially asymmetric systems may also be modeled with polar coordinates. For example, a [[microphone]]'s [[Microphone#Microphone polar patterns|pickup pattern]] illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as {{nowrap|''r'' {{=}} 0.5 + 0.5sin(''φ'')}} at its target design frequency.<ref>{{Cite book|last=Eargle|first=John|authorlink=John M. Eargle|title=Handbook of Recording Engineering|year=2005|edition=Fourth|publisher=Springer|isbn = 0-387-28470-2}}</ref> The pattern shifts toward omnidirectionality at lower frequencies.
| |
| | |
| ==See also==
| |
| *[[Curvilinear coordinates]]
| |
| *[[List of canonical coordinate transformations]]
| |
| *[[Log-polar coordinates]]
| |
| | |
| ==References==
| |
| <div class="references-small">
| |
| ;General
| |
| *{{Cite book|last=Adams|first=Robert|coauthors=Christopher Essex|title=Calculus: a complete course|edition=Eighth|year=2013|publisher=Pearson Canada Inc.|isbn=978-0-321-78107-9}}
| |
| *{{Cite book|last=Anton|first=Howard|coauthors=Irl Bivens, Stephen Davis|title=Calculus|edition=Seventh|year=2002|publisher=Anton Textbooks, Inc.|isbn=0-471-38157-8}}
| |
| *{{Cite book|last=Finney|first=Ross|coauthors=George Thomas, Franklin Demana, Bert Waits|title=Calculus: Graphical, Numerical, Algebraic|edition=Single Variable Version|date=June 1994|publisher=Addison-Wesley Publishing Co.|isbn=0-201-55478-X}}
| |
| ;Specific
| |
| </div>
| |
| {{Reflist|2}}
| |
| | |
| ==External links==
| |
| {{wikibooks|Calculus|Polar Integration}}
| |
| * {{springer|title=Polar coordinates|id=p/p073410}}
| |
| *{{dmoz|Science/Math/Software/Graphing/|Graphing Software}}
| |
| *[http://www.random-science-tools.com/maths/coordinate-converter.htm Coordinate Converter — converts between polar, Cartesian and spherical coordinates]
| |
| *[http://scratch.mit.edu/projects/nevit/691690 Polar Coordinate System Dynamic Demo]
| |
| | |
| {{Orthogonal coordinate systems}}
| |
| | |
| {{DEFAULTSORT:Polar Coordinate System}}
| |
| [[Category:Coordinate systems]]
| |
| | |
| {{Link GA|es}}
| |
| {{Link GA|fr}}
| |
| {{Link GA|uk}}
| |
| {{Link FA|af}}
| |
| {{Link FA|ca}}
| |
| {{Link FA|eo}}
| |
| {{Link FA|ko}}
| |
| {{Link GA|ro}}
| |