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| | You will discover lawn or garden landscaping is easier than you think. You only need some key information with relation to its landscaping. That exactly what this article is all about. Use what you learn here to get your yard looking its best quickly.<br><br> |
| {{for|the topological usage|Rose (topology)}}
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| [[File:7_Petal_rose.svg|thumb|200px|right|Rose with ''k'' = 7 petals]]
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| [[File:8-Petal_rose.svg|thumb|200px|right|Rose with 8 petals (''k''=4).]]
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| [[File:Rose-rhodonea-curve-7x9-chart-improved.svg|thumb|200px|right|Rose curves defined by
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| <math>r=\cos k \theta</math>, for various values of ''k''=n/d.]]
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| <!-- [[Image:7-leafed rose.gif|thumb|200px|right|Rose with ''k'' = 7 petals]] --> | |
| <!-- [[Image:8-leafed rose.gif|thumb|200px|right|Rose with 8 petals (''k''=4).]] -->
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| In [[mathematics]], a '''rose''' or '''rhodonea curve''' is a [[sine wave|sinusoid]] plotted in [[polar coordinates]].
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| == General overview ==
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| Up to [[Similarity (geometry)|similar]]ity, these curves can all be expressed by a polar equation of the form
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| :<math>\!\,r=\cos(k\theta)</math>
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| or, alternatively, as a pair of Cartesian parametric equations of the form
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| :<math>\!\,x=\cos(kt)\sin(t)</math>
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| :<math>\!\,y=\cos(kt)\cos(t)</math>
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| If ''k'' is an integer, the curve will be rose shaped with | |
| *2''k'' petals if ''k'' is even, and
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| *''k'' petals if ''k'' is odd.
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| When ''k'' is even, the entire graph of the rose will be traced out exactly once when the value of θ changes from 0 to 2π. When ''k'' is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length 2π for ''k'' even, and π for ''k'' odd.)
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| If ''k'' ends in 1/2 (ex: 0.5, 2.5), the curve will be rose shaped with 4''k'' petals.
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| If ''k'' ends in 1/6 or 5/6 and is greater than 1 (ex: 1.16666667, 2.8333333), the curve will be rose shaped with 12''k'' petals.
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| If ''k'' ends in 1/3 and is greater than 1 (ex: 1.333333, 2.333333), the curve will be rose shaped and will have:
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| *3''k'' petals if ''k'' is even, and
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| *6''k'' petals if ''k'' is odd.
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| If ''k'' ends in 2/3 and is greater than 1 (ex: 1.666667, 2.666667), the curve will be rose shaped and will have:
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| *6''k'' petals if ''k'' is even, and
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| *3''k'' petals if ''k'' is odd.
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| If ''k'' is [[Rational number|rational]], then the curve is [[Closed (mathematics)|closed]] and has finite length. If ''k'' is [[Irrational number|irrational]], then it is not closed and has infinite length. Furthermore, the graph of the rose in this case forms a [[dense set]] (i.e., it comes arbitrarily close to every point in the unit disk).
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| Since
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| :<math>\sin(k \theta) = \cos\left( k \theta - \frac{\pi}{2} \right) = \cos\left( k \left( \theta-\frac{\pi}{2k} \right) \right)</math>
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| for all <math>\theta</math>, the curves given by the polar equations
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| :<math>\,r=\sin(k\theta)</math> and <math>\,r = \cos(k\theta)</math>
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| are identical except for a rotation of π/2''k'' radians.
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| Rhodonea curves were named by the Italian mathematician [[Guido Grandi]] between the year 1723 and 1728.<ref>{{MacTutor Biography|class=Curves|id=Rhodonea|title=Rhodonea}}</ref>
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| ==Area==
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| A rose whose polar equation is of the form
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| :<math>r=a \cos (k\theta)\,</math>
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| where ''k'' is a positive integer, has [[Polar coordinates#Integral calculus|area]]
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| :<math>
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| \frac{1}{2}\int_{0}^{2\pi}(a\cos (k\theta))^2\,d\theta = \frac {a^2}{2} \left(\pi + \frac{\sin(4k\pi)}{4k}\right) = \frac{\pi a^2}{2}
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| </math>
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| if ''k'' is even, and
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| :<math>
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| \frac{1}{2}\int_{0}^{\pi}(a\cos (k\theta))^2\,d\theta = \frac {a^2}{2} \left(\frac{\pi}{2} + \frac{\sin(2k\pi)}{4k}\right) = \frac{\pi a^2}{4}
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| </math>
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| if ''k'' is odd.
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| The same applies to roses with polar equations of the form
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| :<math>r=a \sin (k\theta)\,</math>
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| since the graphs of these are just rigid rotations of the roses defined using the cosine.
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| ==How the parameter k affects shapes==
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| In the form ''k'' = ''n'', for integer n, the shape will appear similar to a flower. If ''n'' is odd half of these will overlap, forming a flower with ''n'' petals. However if it is even the petals will not overlap, forming a flower with 2''n'' petals.
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| When ''d'' is a prime number then ''n''/''d'' is a least common form and the petals will stretch around to overlap other petals. The number of petals each one overlaps is equal to the how far through the sequence of primes this prime is +1, i.e. 2 is 2, 3 is 3, 5 is 4, 7 is 5, etc.
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| In the form ''k'' = 1/''d'' when ''d'' is even then it will appear as a series of ''d''/2 loops that meet at 2 small loops at the center touching (0, 0) from the vertical and is symmetrical about the ''x''-axis.
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| If ''d'' is odd then it will have ''d'' div 2 loops that meet at a small loop at the center from ether the left (when in the form ''d'' = 4''n'' − 1) or the right (''d'' = 4''n'' + 1).
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| If ''d'' is not prime and n is not 1, then it will appear as a series of interlocking loops.
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| If ''k'' is an irrational number (e.g. <math>\pi</math>, <math>\sqrt{2}</math>, etc.) then the curve will have infinitely many petals, and it will be [[Dense set|dense]] in the unit disc.
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| ==See also==
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| *[[Lissajous curve]]
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| *[[quadrifolium]] - a rose curve with ''k'' = 2.
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| *[[Maurer rose]]
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| *[[Rose (topology)]]
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| ==Notes==
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| <references /> | |
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| ==External links==
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| {{commonscat|Rose curves}}
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| * {{MathWorld|Rose|Rose}}
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| * [http://www25.brinkster.com/denshade/PolarFlower.html Applet to create rose with k parameter]
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| * [http://xahlee.org/SpecialPlaneCurves_dir/Rose_dir/rose.html Visual Dictionary of Special Plane Curves] Xah Lee
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| * [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rose Interactive example with JSXGraph]
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| [[Category:Curves]]
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You will discover lawn or garden landscaping is easier than you think. You only need some key information with relation to its landscaping. That exactly what this article is all about. Use what you learn here to get your yard looking its best quickly.
If you're traditionalist, the nice looking picket fence is the style and design that need your name. The pickets come in three sizes and they are tall, small, and medium. You could choose 3 if somebody an eclectic look. This fence can be quite easy to install. Since the fence is pickets the open space, then investing some plastic sheeting shall keep the earth in place.
Another thing is your shrubbery. An individual going to wish to trim your trees, hedges and vines around your house to make sure everything is clean-cut. Therefore be efficient at have the debris all wheeled away in a wheel barrow and taken away the actual landscapers. All you need become done properly.
Buying your dream house also lets you have insight as a section of a permanent community. On the other guitar hand, in the rented apartment or home, one might feel temporary and less involved.
Last year's "other" cinderella, the Demon Deacons, travel to Boston College in a tricky ACC opener that should tell whether have the capacity to compete extra league title or not solely.
Almost many folks adore a campfire! If you have a fireplace, a lot more use it for its worth. Season permitting, light a fire every time the residence is being shown, and during open houses. A firelit room looks and smells stylish. You can also prove individuals that your fireplace does actually perform.
Personalized baby blankets are trendy right away. This will give your baby to exercise his individualism and creativity. Now, your baby will become in a snugly mood with a warm blanket wrapping him around during cold winter nights.
If you liked this report and you would like to receive extra info concerning hedgingplants kindly go to our own web-site.