N-curve


We take the functional theoretic algebra C 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.
  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

A curve γ in the functional theoretic algebra C, 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 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.

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
See the figure.

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

Let us take the Rhodonea Curve
If denotes the curve,
The parametric equations of 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, 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

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 and ends at.
curve at N=0. Please note that the parametric equation was modified to center the curve at origin.

Example 2 of ''n''-curving

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

In the FTA C 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,
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
This we shall call generalized n-curving.

Example 1

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

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.