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| | Inside the CCTV wireless security camera globe, though, most cameras remain analog as well as their resolution is measured differently from that which you cctv dvr software suppliers are employed to. CCTV systems are successful, be it within the area of public surveillance or [http://Mahatourism.com/groups/what-you-dont-know-about-samsung-surveillance-cameras-could-be-costing-to-more-than-you-think/ private security]. Open source dvr software 16 channel video for cctv DVRs usually form part of the deals that you order, but you also have the choice to buy premium products from retailers.<br><br> |
| |[[Image:Logarithmic Spiral Pylab.svg|260px|thumb|Logarithmic spiral (pitch 10°)]]
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| |[[Image:NautilusCutawayLogarithmicSpiral.jpg|200px|thumb|Cutaway of a [[nautilus]] shell showing the chambers arranged in an approximately logarithmic spiral]]
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| |[[File:Fractal Broccoli.jpg|200px|thumb|[[Romanesco broccoli]], which grows in a logarithmic spiral]]
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| |[[Image:Mandel zoom 04 seehorse tail.jpg|thumb|200px|A section of the [[Mandelbrot set]] following a logarithmic spiral]]
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| |[[Image:Low pressure system over Iceland.jpg|thumb|200px|A [[low pressure area]] over [[Iceland]] shows an approximately logarithmic spiral pattern]]
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| |[[File:Messier51 sRGB.jpg|thumb|200px|The arms of [[Spiral galaxy|spiral galaxies]] often have the shape of a logarithmic spiral, here the [[Whirlpool Galaxy]]]]
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| |[[Image:Polygon spiral.svg|thumb|200px|Polygon spiral]]
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| A '''logarithmic spiral''', '''equiangular spiral''' or '''growth spiral''' is a [[self-similarity|self-similar]] [[spiral]] [[curve]] which often appears in nature. The logarithmic spiral was first described by [[René Descartes|Descartes]] and later extensively investigated by [[Jacob Bernoulli]], who called it ''Spira mirabilis'', "the marvelous spiral".
| | The following is info was swallowed from the Swann 2011 Catalog you will find additional cctv digital video recorder software DVRs which are available from Swann. The CCTV [http://big5.qikan.com/gate/big5/cctvdvrreviews.com build your own cctv dvr] that can perform its duty without connected to a ware is known as wireless CCTV.<br><br>[http://www.suzuki.nl/?set=cookiepermission&action=accepted&return=http://cctvdvrreviews.com cctv dvr viewer linux] So [http://www.seo-kensaku-engine.net/rank.cgi?mode=link&id=258&url=http://cctvdvrreviews.com Cctv Dvr With Hard Drive] too inside the academic world, you'll find rules to adhere to [http://c.yam.com/msnews/IRT/r.c?http://cctvdvrreviews.com 8 channel cctv dvr reviews] which demonstrate which you understand the expectations of people reading academic texts. Here's where we learn how to "cluster" or "map" our creativity process. [http://tsptalk.mobi/leaving.php?u=http%3A%2F%2Fcctvdvrreviews.com&error=DIFFERENT_DOMAIN&back=http%3A%2F%2Ftsptalk.mobi%2F&imz_s=1a7os3d995291ar05boma6td93 Cctv dvr ireland] genie cctv dvr Visit for additional info on how Carma can assist you to nurture a web based footprint that supports your goals being a writer. Feeling an increased-quality emotion with [http://get.hoop.la/t/trustubb?r=http%3A//cctvdvrreviews.com ordered] this issue can look your ideas inside artless technique. |
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| ==Definition==
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| In [[polar coordinates]] <math>(r, \theta)</math> the [[logarithm|logarithmic]] curve can be written as<ref>{{cite book | title = Divine Proportion: Φ Phi in Art, Nature, and Science | author = Priya Hemenway | isbn = 1-4027-3522-7 | publisher = Sterling Publishing Co | year = 2005}}</ref>
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| :<math>r = ae^{b\theta}\,</math>
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| or
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| :<math>\theta = \frac{1}{b} \ln(r/a),</math>
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| with [[e (mathematical constant)|<math>e</math>]] being the base of natural logarithms, and <math>a</math> and <math>b</math> being arbitrary positive real constants.
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| In parametric form, the curve is
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| :<math>x(t) = r(t) \cos(t) = ae^{bt} \cos(t)\,</math>
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| :<math>y(t) = r(t) \sin(t) = ae^{bt} \sin(t)\,</math>
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| with [[real number]]s <math>a</math> and <math>b</math>.
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| The spiral has the property that the angle ''φ'' between the [[tangent]] and [[radial line]] at the point <math>(r, \theta)</math> is constant. This property can be expressed in [[differential geometry of curves|differential geometric terms]] as | |
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| :<math>\arccos \frac{\langle \mathbf{r}(\theta), \mathbf{r}'(\theta) \rangle}{\|\mathbf{r}(\theta)\|\|\mathbf{r}'(\theta)\|} = \arctan \frac{1}{b} = \phi.</math> | |
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| The [[derivative]] of <math>\mathbf{r}(\theta)</math> is proportional to the parameter <math>b</math>. In other words, it controls how "tightly" and in which direction the spiral spirals. In the extreme case that <math>b = 0</math> (<math>\textstyle\phi = \frac{\pi}{2}</math>) the spiral becomes a [[circle]] of radius <math>a</math>. Conversely, in the [[Limit of a function|limit]] that <math>b</math> approaches [[Extended real number line|infinity]] (''φ'' → 0) the spiral tends toward a straight half-line. The [[Complementary angles|complement]] of ''φ'' is called the ''pitch''.
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| ==''Spira mirabilis'' and Jacob Bernoulli==
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| '''''Spira mirabilis''''', [[Latin]] for "miraculous spiral", is another name for the logarithmic spiral. Although this curve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve by [[Jacob Bernoulli]], because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as [[self-similarity]]. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as [[nautilus]] shells and [[sunflower]] heads. Jacob Bernoulli wanted such a spiral engraved on his [[headstone]] along with the phrase "[[Eadem mutata resurgo]]" ("Although changed, I shall arise the same."), but, by error, an [[Archimedean spiral]] was placed there instead.<ref name="livio">{{cite book|last=Livio|first=Mario|year=2002|title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number|publisher=Broadway Books|location=New York|isbn=0-7679-0815-5}}</ref><ref>Yates, R. C.: ''A Handbook on Curves and Their Properties'', J. W. Edwards (1952), "Evolutes." p. 206</ref>
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| ==Properties== | |
| The logarithmic spiral can be distinguished from the [[Archimedean spiral]] by the fact that the distances between the turnings of a logarithmic spiral increase in [[geometric progression]], while in an Archimedean spiral these distances are constant.
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| Logarithmic spirals are self-similar in that the result of applying any [[similarity (geometry)|similarity transformation]] to the spiral is [[congruence (geometry)|congruent]] to the original untransformed spiral. Scaling by a factor <math>e^{2 \pi b}</math>, for an integer ''b'', with the center of scaling at the origin, gives the same curve as the original; other scale factors give a curve that is rotated from the original position of the spiral. Logarithmic spirals are also congruent to their own [[involute]]s, [[evolute]]s, and the [[pedal curve]]s based on their centers.
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| Starting at a point <math>P</math> and moving inward along the spiral, one can circle the origin an unbounded number of times without reaching it; yet, the total distance covered on this path is finite; that is, the [[limit (mathematics)|limit]] as <math>\theta</math> goes toward <math>-\infty</math> is finite. This property was first realized by [[Evangelista Torricelli]] even before [[calculus]] had been invented.<ref>
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| {{cite book
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| | title = The history of the calculus and its conceptual development
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| | author = Carl Benjamin Boyer
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| | publisher = Courier Dover Publications
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| | year = 1949
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| | isbn = 978-0-486-60509-8
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| | page = 133
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| | url = http://books.google.com/books?id=KLQSHUW8FnUC&pg=PA133
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| }}</ref>
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| The total distance covered is <math>\textstyle\frac{r}{\cos(\phi)}</math>, where <math>r</math> is the straight-line distance from <math>P</math> to the origin.
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| The [[exponential function]] exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at 0. ([[Up to]] adding integer multiples of <math>2\pi i</math> to the lines, the mapping of all lines to all logarithmic spirals is [[onto]].) The pitch angle of the logarithmic spiral is the angle between the line and the imaginary axis.
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| The function <math>x \mapsto x^k</math>, where the constant <math>k</math> is a [[complex number]] with non-zero [[imaginary part]], maps the [[real line]] to a logarithmic spiral in the complex plane.
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| One can construct a [[golden spiral]], a logarithmic spiral that grows outward by a factor of the [[golden ratio]] for every 90 degrees of rotation (pitch about 17.03239 degrees), or approximate it using [[Fibonacci number]]s.
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| ==Logarithmic spirals in nature==
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| In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons:
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| *The approach of a [[hawk]] to its prey. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.<ref>{{Citation |first=Gilbert J. |last=Chin |date=8 December 2000 |title=Organismal Biology: Flying Along a Logarithmic Spiral |journal=[[Science (journal)|Science]] |volume=290 |issue=5498 |page=1857 |url=http://www.sciencemag.org/content/290/5498/1857.3.short |doi=10.1126/science.290.5498.1857c}}</ref>
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| *The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will result in a practically straight line.<ref>
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| {{cite book
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| | title = Discovering Moths: Nighttime Jewels in Your Own Backyard
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| | author = John Himmelman
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| | publisher = Down East Enterprise Inc
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| | year = 2002
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| | isbn = 978-0-89272-528-1
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| | page = 63
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| | url = http://books.google.com/books?id=iGn6ohfKhbAC&pg=PA63
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| }}</ref>
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| *The arms of spiral [[galaxy|galaxies]].<ref>
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| {{cite book
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| | title = Spiral structure in galaxies: a density wave theory
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| | author = G. Bertin and C. C. Lin
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| | publisher = MIT Press
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| | year = 1996
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| | isbn = 978-0-262-02396-2
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| | page = 78
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| | url = http://books.google.com/books?id=06yfwrdpTk4C&pg=PA78
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| }}</ref> Our own galaxy, the [[Milky Way]], has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees.<ref>
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| {{cite book
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| | title = The universal book of mathematics: from Abracadabra to Zeno's paradoxes
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| | author = David J. Darling
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| | publisher = John Wiley and Sons
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| | year = 2004
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| | isbn = 978-0-471-27047-8
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| | page = 188
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| | url = http://books.google.com/books?id=nnpChqstvg0C&pg=PA188
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| }}</ref>
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| *The nerves of the [[cornea]] (this is, corneal nerves of the subepithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).<ref name="Yu">C. Q. Yu CQ and M. I. Rosenblatt, "Transgenic corneal neurofluorescence in mice: a new model for in vivo investigation of nerve structure and regeneration,"
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| Invest Ophthalmol Vis Sci. 2007 Apr;48(4):1535-42.</ref>
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| *The [[rainband|bands]] of [[tropical cyclone]]s, such as hurricanes.<ref>
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| {{cite book
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| | title = Treatise on physics, Volume 1
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| | author = Andrew Gray
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| | publisher = Churchill
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| | year = 1901
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| | page = 356–357
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| | url = http://books.google.com/books?id=ArELAAAAYAAJ&pg=PA357
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| }}</ref>
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| | |
| *Many [[Biology|biological]] structures including the shells of [[Mollusca|mollusk]]s.<ref>
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| {{cite book
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| | title = Spiral symmetry
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| | chapter = The form, function, and synthesis of the molluscan shell
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| | author = Michael Cortie
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| | editor = István Hargittai and Clifford A. Pickover
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| | publisher = World Scientific
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| | year = 1992
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| | isbn = 978-981-02-0615-4
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| | page = 370
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| | url = http://books.google.com/books?id=Ga8aoiIUx1gC&pg=PA370
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| }}</ref> In these cases, the reason may be construction from expanding similar shapes, as shown for [[polygon]]al figures in the accompanying graphic.
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| *[[Logarithmic spiral beaches]] can form as the result of wave refraction and diffraction by the coast. [[Half Moon Bay, California]] is an example of such a type of beach.<ref>
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| {{cite book
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| | title = Beach management: principles and practice
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| | author = Allan Thomas Williams and Anton Micallef
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| | publisher = Earthscan
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| | year = 2009
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| | isbn = 978-1-84407-435-8
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| | page = 14
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| | url = http://books.google.com/books?id=z_vKEMeJXKYC&pg=PA14
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| }}</ref>
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| ==See also==
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| *[[Archimedean spiral]]
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| *[[Epispiral]]
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| *[[Golden spiral]]
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| ==References==
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| {{reflist}}
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| * {{mathworld|urlname=LogarithmicSpiral|title=Logarithmic Spiral}}
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| * Jim Wilson, [http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/related%20curves/related%20curves.html Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves], University of Georgia (1999)
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| * Alexander Bogomolny, [http://www.cut-the-knot.org/Curriculum/Geometry/Mirabilis.shtml Spira Mirabilis - Wonderful Spiral], at [[cut-the-knot]]
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| *
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| ==External links== | |
| {{commonscat}}
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| * [http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/golden%20spiral/logspiral-history.html Spira mirabilis] history and math
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| * [http://apod.nasa.gov/apod/ap030925.html ''Astronomy Picture of the Day''], [[Hurricane Isabel]] vs. the [[Whirlpool Galaxy]]
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| * [http://apod.nasa.gov/apod/ap080517.html ''Astronomy Picture of the Day''], [[Typhoon Rammasun (2008)|Typhoon Rammasun]] vs. the [[Pinwheel Galaxy]]
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| * [http://SpiralZoom.com/ ''SpiralZoom.com''], an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
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| * [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Logarithmic_spiral Online exploration using JSXGraph (JavaScript)]
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| [[Category:Spirals]]
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| [[Category:Logarithms]]
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| [[Category:Curves]]
| |
| {{Link GA|ja}}
| |
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