N-curve: Difference between revisions
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The parametric equations of their product <math>u \cdot u_{n}</math> are <math> x= \cos(2\pi nt)+ \cos(2\pi t)-1, y =\sin(2\pi nt)+ \sin(2\pi t)</math> |
The parametric equations of their product <math>u \cdot u_{n}</math> are <math> x= \cos(2\pi nt)+ \cos(2\pi t)-1, y =\sin(2\pi nt)+ \sin(2\pi t)</math> |
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[[File:Product of unit circle with n-circle.jpg ]] |
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==''n''-Curving== |
==''n''-Curving== |
Revision as of 23:17, 30 December 2010
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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 called n-curve. The n-curves are interesting in two ways.
- Their f-products give rise to many beautiful curves.
- Using the n-curves, we can define a transformation of curves, called n-curving.
Multiplicative inverse of a curve
A curve γ in the functional theoretic algebra C[0, 1], is invertible, i.e.
exists if
If , where , 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 .
Example 1: Product of the astroid with the n-curve of the unit circle
Let us take u, the unit circle centered at the origin and α, the astroid. The n-curve of “u” is given by,
and the astroid is
The parametric equations of their product are
See the figure.
Since both are loops at 1, so is the product.
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Example 2: Product of the unit circle and its n-curve
The unit circle is
and its n-curve is
The parametric equations of their product are
n-Curving
If , then, as mentioned above, the n-curve . Therefore the mapping is an inner automorphism of the group G. We extend this map to the whole of C[0, 1], denote it by and call it n-curving with γ. It can be verified that
This new curve has the same initial and end points as α.
Example of n-curving
Let ρ denote the Rhodonea curve , which is a loop at 1. Its parametric equations are
With the loop ρ we shall n-curve the cosine curve
The curve has the parametric equations
See the figure.
It is a curve that starts at the point (0, 1) and ends at (2π, 1).
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/N-curving.gif/450px-N-curving.gif)
See also
References
- Sebastian Vattamattam, "Transforming Curves by n-Curving", in Bulletin of Kerala Mathematics Association, Vol. 5, No. 1, December 2008