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| {{DISPLAYTITLE:''n''-Curve}}
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| We take the [[functional theoretic algebra]] ''C''[0, 1] of curves. For each loop ''γ'' at 1, and each positive integer ''n'', we define a curve <math>\gamma_n</math> called ''n''-curve. The ''n''-curves are interesting in two ways.
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| #Their f-products, sums and differences give rise to many beautiful curves.
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| #Using the ''n''-curves, we can define a transformation of curves, called ''n''-curving.
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|
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| == Multiplicative inverse of a curve ==
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| A curve ''γ'' in the [[functional theoretic algebra]] ''C''[0, 1], is invertible, i.e.
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| : <math>\gamma^{-1} \, </math>
| | By investing in a premium Word - Press theme, you're investing in the future of your website. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Should you go with simple HTML or use a platform like Wordpress. They found out all the possible information about bringing up your baby and save money at the same time. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br>Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. There are number of web services that offer Word press development across the world. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. This can be done by using a popular layout format and your unique Word - Press design can be achieved in other elements of the blog. <br><br>Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. The only problem with most is that they only offer a monthly plan, you never own the software and you can’t even install the software on your site, you must go to another website to manage your list and edit your autoresponder. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. User friendly features and flexibility that Word - Press has to offer is second to none. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog. <br><br>If you have any queries with regards to where by and how to use [http://www.nt-protect.host.sk/?wordpress_backup_77912 wordpress backup plugin], you can get hold of us at the page. A built-in widget which allows you to embed quickly video from popular websites. I didn't straight consider near it solon than one distance, I got the Popup Ascendancy plugin and it's up and lengthways, likely you make seen it today when you visited our blog, and I yet customize it to fit our Thesis Wound which gives it a rattling uncomparable visage and search than any different popup you know seen before on any added journal, I hump arrogated asset of one of it's quatern themes to make our own. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. The most important plugins you will need are All-in-One SEO Pack, some social bookmarking plugin, a Feedburner plugin and an RSS sign up button. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. Automated deal feed integration option to populate your blog with relevant deals. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. Thus, Word - Press is a good alternative if you are looking for free blogging software. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order. |
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| exists if
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| : <math>\gamma(0)\gamma(1) \neq 0. \, </math>
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| If <math>\gamma^{*}=(\gamma(0)+\gamma(1))e - \gamma </math>, where <math>e(t)=1, \forall t \in [0, 1]</math>, then
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| : <math>\gamma^{-1}= \frac{\gamma^{*}}{\gamma(0)\gamma(1)}. </math>
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| The set ''G'' of invertible curves is a non-commutative group under multiplication. Also the set ''H'' of loops at 1 is an Abelian subgroup of ''G.'' If <math>\gamma \in H</math>, then the mapping <math>\alpha \to \gamma^{-1}\cdot \alpha\cdot\gamma</math> is an inner automorphism of the group ''G.''
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| We use these concepts to define ''n''-curves and ''n''-curving.
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| == ''n''-curves and their products ==
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| If ''x'' is a real number and [''x''] denotes the greatest integer not greater than ''x'', then <math> x-[x] \in [0, 1].</math>
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| If <math>\gamma \in H</math> and ''n'' is a positive integer, then define a curve <math>\gamma_{n}</math> by
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| : <math>\gamma_n (t)=\gamma(nt - [nt]). \, </math>
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| <math>\gamma_{n}</math> is also a loop at ''1'' and we call it an ''n''-curve. | |
| Note that every curve in ''H'' is a 1-curve.
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| Suppose <math>\alpha, \beta \in H.</math>
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| Then, since <math>\alpha(0)=\beta(1)=1, \mbox{ the f-product } \alpha \cdot \beta = \beta + \alpha -e</math>.
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| === Example 1: Product of the astroid with the ''n''-curve of the unit circle ===
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| Let us take ''u'', the unit circle centered at the origin and α, the [[astroid]].
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| The ''n''-curve of ''u'' is given by, | |
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| : <math>u_n(t) = \cos(2\pi nt)+ i \sin(2\pi nt) \, </math>
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| and the astroid is | |
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| : <math>\alpha(t)=\cos^{3}(2\pi t)+ i \sin^{3}(2\pi t), 0\leq t \leq 1 </math>
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| The parametric equations of their product <math> \alpha \cdot u_{n} </math> are
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| :<math>x=\cos^3 (2\pi t)+ \cos(2\pi nt)-1,</math>
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| :<math>y=\sin^{3}(2\pi t)+ \sin(2\pi nt)</math>
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| See the figure.
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| Since both <math>\alpha \mbox{ and } u_{n}</math> are loops at 1, so is the product.
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| {|
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| | [[File:N-curve.svg|thumb|''n''-curve with <math>N=53</math>]]
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| | [[File:N-curve.gif|thumb|350px|Animation of ''n''-curve for ''n'' values from 0 to 50]]
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| |}
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| === Example 2: Product of the unit circle and its ''n''-curve ===
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| The unit circle is
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| : <math> u(t) = \cos(2\pi t)+ i \sin(2\pi t) \, </math>
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| and its ''n''-curve is | |
| : <math> u_n(t) = \cos(2\pi nt)+ i \sin(2\pi nt) \, </math>
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| The parametric equations of their product
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| :<math>u \cdot u_{n}</math>
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| are
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| :<math> x= \cos(2\pi nt)+ \cos(2\pi t)-1,</math>
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| :<math> y =\sin(2\pi nt)+ \sin(2\pi t)</math>
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| See the figure.
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| [[File:Unit_Circle_with_n-Circle.jpg ]]
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| === Example 3: ''n''-Curve of the Rhodonea minus the [[Rhodonea curve]] ===
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| Let us take the Rhodonea Curve
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| : <math> r = \cos(3\theta)</math>
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| If <math> \rho </math> denotes the curve, | |
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| : <math> \rho(t) = \cos(6\pi t)[\cos(2\pi t) + i\sin(2\pi t)], 0 \leq t \leq 1 </math> | |
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| The parametric equations of <math> \rho_{n}- \rho </math> are
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| : <math> x = \cos(6\pi nt)\cos(2\pi nt) - \cos(6\pi t)\cos(2\pi t), </math>
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| :<math> y = \cos(6\pi nt)\sin(2\pi nt)-\cos(6\pi t)\sin(2\pi t), 0 \leq t \leq 1 </math>
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| [[File:Rhodonea Curve.jpg]][[File:Rhodonea-nRhodonea_Curve.jpg]]
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| === ''n''-Curving ===
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| If <math>\gamma \in H</math>, then, as mentioned above, the ''n''-curve <math>\gamma_{n} \mbox{ also } \in H</math>. Therefore the mapping <math>\alpha \to \gamma_n^{-1}\cdot \alpha\cdot\gamma_n</math> is an inner automorphism of the group ''G.'' We extend this map to the whole of ''C''[0, 1], denote it by <math>\phi_{\gamma_n,e}</math> and call it ''n''-curving with γ.
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| It can be verified that
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| : <math>\phi_{\gamma_n ,e}(\alpha)=\alpha + [\alpha(1)-\alpha(0)](\gamma_{n}-1)e. \ </math>
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| This new curve has the same initial and end points as α.
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| === Example 1 of ''n''-curving ===
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| Let ρ denote the [[Rhodonea curve]] <math> r = \cos(2\theta)</math>, which is a loop at 1. Its parametric equations are
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| : <math> x = \cos(4\pi t)\cos(2\pi t), </math>
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| :<math> y = \cos(4\pi t)\sin(2\pi t), 0\leq t \leq 1 </math>
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| With the loop ρ we shall ''n''-curve the cosine curve
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| : <math>c(t)=2\pi t + i \cos(2\pi t),\quad 0 \leq t \leq 1. \,</math>
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| The curve <math>\phi_{\rho_{n},e}(c)</math> has the parametric equations
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| : <math>x=2\pi[t-1+\cos(4\pi nt)\cos(2\pi nt)], \quad y=\cos(2\pi t)+ 2\pi \cos(4\pi nt)\sin(2\pi nt)</math>
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| See the figure.
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| It is a curve that starts at the point (0, 1) and ends at (2π, 1).
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| [[File:N-curving.gif|thumb|450px|center|Notice how the curve starts with a [[cosine]] curve at ''N''=0. Please note that the parametric equation was modified to center the curve at origin.]]
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| === Example 2 of ''n''-curving ===
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| Let χ denote the [[Cosine Curve]]
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| : <math> \chi(t) = 2\pi t +i\cos(2\pi t), 0\leq t \leq 1 </math>
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| With another [[Rhodonea Curve]]
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| :<math> \rho = \cos(3 \theta) </math>
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| we shall ''n''-curve the cosine curve.
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| The rhodonea curve can also be given as
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| : <math> \rho(t) = \cos(6\pi t)[\cos (2\pi t)+ i\sin(2\pi t)], 0\leq t \leq 1 </math>
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| The curve <math>\phi_{\rho_{n},e}(\chi)</math> has the parametric equations
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| : <math> x=2\pi t + 2\pi [\cos( 6\pi nt)\cos(2\pi nt)- 1], </math>
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| :<math> y=\cos(2\pi t) + 2\pi \cos( 6\pi nt)\sin(2 \pi nt), 0\leq t \leq 1 </math>
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| See the figure for <math>n = 15 </math>.
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| [[File:CosineRhodonea.jpg]]
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| === Generalized ''n''-curving ===
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| In the FTA ''C''[0, 1] of curves, instead of ''e'' we shall take an arbitrary curve <math>\beta</math>, a loop at 1.
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| This is justified since | |
| :<math> L_1(\beta)=L_2(\beta) = 1 </math>
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| Then, for a curve ''γ'' in ''C''[0, 1],
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| :<math>\gamma^{*}=(\gamma(0)+\gamma(1))\beta - \gamma </math>
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| and | |
| : <math>\gamma^{-1}= \frac{\gamma^{*}}{\gamma(0)\gamma(1)}. </math>
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| If <math>\alpha \in H</math>, the mapping
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| :<math>\phi_{\alpha_n,\beta}</math>
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| given by
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| :<math>\phi_{\alpha_n,\beta}(\gamma) = \alpha_n^{-1}\cdot \gamma \cdot \alpha_n</math>
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| is the ''n''-curving. We get the formula
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| : <math>\phi_{\alpha_n ,\beta}(\gamma)=\gamma + [\gamma(1)-\gamma(0)](\alpha_{n}-\beta). </math>
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| Thus given any two loops <math>\alpha</math> and <math>\beta</math> at 1, we get a transformation of curve
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| :<math>\gamma</math> given by the above formula.
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| This we shall call generalized ''n''-curving.
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| === Example 1 ===
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| Let us take <math>\alpha</math> and <math> \beta </math> as the unit circle ``u.’’ and <math> \gamma </math> as the cosine curve
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| :<math> \gamma (t) = 4\pi t + i\cos(4\pi t) 0 \leq t \leq 1</math>
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| Note that <math> \gamma (1) - \gamma (0) = 4\pi</math>
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| For the transformed curve for <math>n = 40</math>, see the figure.
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| The transformed curve <math> \phi_{u_n, u}( \gamma )</math> has the parametric equations
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| [[File:N-curved_cosine.jpg]]
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| === Example 2 ===
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| Denote the curve called [[Crooked Egg]] by <math> \eta </math> whose polar equation is
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| : <math> r = \cos^3 \theta + \sin^3 \theta </math>
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| Its parametric equations are
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| : <math> x = \cos(2\pi t) (\cos^3 2\pi t + \sin^3 2\pi t), </math>
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| : <math> y = \sin(2\pi t) (\cos^3 2\pi t + \sin^3 2\pi t) </math>
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| Let us take <math> \alpha = \eta </math> and <math> \beta = u, </math>
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| where <math> u</math> is the unit circle.
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| The ''n''-curved [[Archimedean spiral]] has the parametric equations
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| : <math> x = 2\pi t \cos(2\pi t)+ 2\pi [(\cos^3 2\pi nt+\sin^3 2\pi nt) \cos(2\pi nt)- \cos(2\pi t)], </math>
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| :<math> y = 2\pi t \sin(2\pi t)+ 2\pi [(\cos^3 2\pi nt)+\sin^3 2\pi nt)\sin(2\pi nt)- \sin(2\pi t)] </math>
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| See the figures, the Crooked Egg and the transformed Spiral for <math>n = 20</math>.
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| [[File:Crooked_Egg.jpg]]
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| [[File:Crooked_Spiral.jpg]]
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| == See also ==
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| * [[Functional-theoretic algebra]]
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| * [http://siluroid.dejudicibus.i The Siluroid Curve]
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| == References ==
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| * Sebastian Vattamattam, "Transforming Curves by ''n''-Curving", in ''Bulletin of Kerala Mathematics Association'', Vol. 5, No. 1, December 2008
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| [[Category:Curves]]
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By investing in a premium Word - Press theme, you're investing in the future of your website. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Should you go with simple HTML or use a platform like Wordpress. They found out all the possible information about bringing up your baby and save money at the same time. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day.
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