N-curve: Difference between revisions
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==Introduction== |
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There are many ways of transforming a mathematical curve. Here we introduce a method using the principles of [[functional-theoretic algebra]] (FTA). |
There are many ways of transforming a mathematical curve. Here we introduce a method using the principles of [[functional-theoretic algebra]] (FTA). |
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== ''n''-curves and their products== |
== ''n''-curves and their products== |
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If ''x'' is a real number |
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 |
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>. <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. |
<math>\gamma_{n}(t)=\gamma(nt - [nt])</math>. <math>\gamma_{n}</math> is also a loop at ''1'' and we call it an n-curve. |
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Note that every curve in ''H'' is a 1-curve. |
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Suppose <math>\alpha, \beta \in H.</math> Then, since <math>\alpha(0)=\beta(1)=1, \alpha \cdot \beta = \beta + \alpha -e</math>, where <math>e(t)=1, \forall t \in [0, 1].</math> |
Suppose <math>\alpha, \beta \in H.</math> |
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Then, since <math>\alpha(0)=\beta(1)=1, \alpha \cdot \beta = \beta + \alpha -e</math>, where <math>e(t)=1, \forall t \in [0, 1].</math> |
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== Example of a |
== Example of a Product of ''n''-Curves == |
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Let us take ''u'', the unit circle centered at the origin and |
Products of ''n''-curves often yield beautiful new curves. Let us take ''u'', the unit circle centered at the origin and α, the [[astroid]]. |
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Then, |
Then, |
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: <math>u_{n}(t)=\cos(2\pi nt)+ i \sin(2\pi nt)</math> |
: <math>u_{n}(t)=\cos(2\pi nt)+ i \sin(2\pi nt)</math> |
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and |
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: <math>\alpha(t)=\cos^{3}(2\pi t)+ i \sin^{3}(2\pi t)</math> |
: <math>\alpha(t)=\cos^{3}(2\pi t)+ i \sin^{3}(2\pi t)</math> |
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The parametric equations of <math>\alpha\cdot u_{n}</math> are <math>x=\cos^3 (2\pi t)+ \cos(2\pi nt)-1, y=\sin^{3}(2\pi t)+ \sin(2\pi nt)</math> |
The parametric equations of <math>\alpha\cdot u_{n}</math> are <math>x=\cos^3 (2\pi t)+ \cos(2\pi nt)-1, y=\sin^{3}(2\pi t)+ \sin(2\pi nt)</math> |
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See the figure. Since both <math>\alpha \mbox{ and } u_{n}</math> are loops at 1, so is the product. |
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See the figure. |
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[[File:curve3.jpg]] |
[[File:curve3.jpg]] |
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==References== |
==References== |
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Introduction
There are many ways of transforming a mathematical curve. Here we introduce a method using the principles of functional-theoretic algebra (FTA).
A curve γ in the FTA C[0, 1] of curves, is invertible, i.e.
exists if
If , then
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 , then the mapping is an inner automorphism of the group G.
We use these concepts to define n-curves and n-curving.
n-curves and their products
If x is a real number and [x] denotes the greatest integer not greater than x, then
If and n is a positive integer, then define a curve by
. is also a loop at 1 and we call it an n-curve. Note that every curve in H is a 1-curve.
Suppose Then, since , where
Example of a Product of n-Curves
Products of n-curves often yield beautiful new curves. Let us take u, the unit circle centered at the origin and α, the astroid. Then,
and
The parametric equations of are
See the figure. Since both are loops at 1, so is the product.
References
- Sebastian Vattamattam, ``Transforming Curves by n-Curving, in ``Bulletin of Kerala Mathematics Association, Vol. 5, No. 1, December 2008