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.[clarification needed] The n-curves are interesting in two ways.

  1. Their f-products, sums and differences give rise to many beautiful curves.
  2. Using the n-curves, we can define a transformation of curves, called n-curving.

Multiplicative inverse of a curve

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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

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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

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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.

 
n-curve with  
 
Animation of n-curve for n values from 0 to 50

Example 2: Product of the unit circle and its n-curve

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The unit circle is

 

and its n-curve is

 

The parametric equations of their product

 

are

 
 

See the figure.

 

Example 3: n-Curve of the Rhodonea minus the Rhodonea curve

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Let us take the Rhodonea Curve

 

If   denotes the curve,

 

The parametric equations of   are

 
 

  

n-Curving

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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 1 of n-curving

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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).

 
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.

Example 2 of n-curving

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Let χ denote the Cosine Curve

 

With another Rhodonea Curve

 

we shall n-curve the cosine curve.

The rhodonea curve can also be given as

 

The curve   has the parametric equations

 
 

See the figure for  .

 

Generalized n-curving

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In the FTA C[0, 1] of curves, instead of e we shall take an arbitrary curve  , a loop at 1. This is justified since

 

Then, for a curve γ in C[0, 1],

 

and

 

If  , the mapping

 

given by

 

is the n-curving. We get the formula

 

Thus given any two loops   and   at 1, we get a transformation of curve

  given by the above formula.

This we shall call generalized n-curving.

Example 1

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Let us take   and   as the unit circle ``u.’’ and   as the cosine curve

 

Note that  

For the transformed curve for  , see the figure.

The transformed curve   has the parametric equations

 

Example 2

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Denote the curve called Crooked Egg by   whose polar equation is

 

Its parametric equations are

 
 

Let us take   and  

where   is the unit circle.

The n-curved Archimedean spiral has the parametric equations

 
 

See the figures, the Crooked Egg and the transformed Spiral for  .

   

References

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  • Sebastian Vattamattam, "Transforming Curves by n-Curving", in Bulletin of Kerala Mathematics Association, Vol. 5, No. 1, December 2008
  • Sebastian Vattamattam, Book of Beautiful Curves, Expressions, Kottayam, January 2015 Book of Beautiful Curves
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