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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.[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[edit]

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[edit]

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[edit]

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[edit]

The unit circle is

and its n-curve is

The parametric equations of their product


See the figure.

Unit Circle with n-Circle.jpg

Example 3: n-Curve of the Rhodonea minus the Rhodonea curve[edit]

Let us take the Rhodonea Curve

If denotes the curve,

The parametric equations of are

Rhodonea Curve.jpgRhodonea-nRhodonea Curve.jpg


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[edit]

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[edit]

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[edit]

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],


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[edit]

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

N-curved cosine.jpg

Example 2[edit]

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 .

Crooked Egg.jpg Crooked Spiral.jpg

See also[edit]


  • 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