# N-curve

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 $\gamma _{n}$ called n-curve. The n-curves are interesting in two ways.
* Their f-products, sums and differences 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.
    $\gamma ^{-1}\,$
exists if
    $\gamma (0)\gamma (1)\neq 0.\,$
If $\gamma ^{*}=(\gamma (0)+\gamma (1))e-\gamma$, where $e(t)=1,\forall t\in [0,1]$, then
    $\gamma ^{-1}={\frac {\gamma ^{*}}{\gamma (0)\gamma (1)}}.$
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 $\gamma \in H$, then the mapping $\alpha \to \gamma ^{-1}\cdot \alpha \cdot \gamma$ 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 $x-[x]\in [0,1].$
If $\gamma \in H$ and n is a positive integer, then define a curve $\gamma _{n}$ by
    $\gamma _{n}(t)=\gamma (nt-[nt]).\,$
$\gamma _{n}$ is also a loop at 1 and we call it an n-curve. Note that every curve in H is a 1-curve.
Suppose $\alpha ,\beta \in H.$ Then, since $\alpha (0)=\beta (1)=1,{\mbox{ the f-product }}\alpha \cdot \beta =\beta +\alpha -e$.

### 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,
    $u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)\,$
and the astroid is
    $\alpha (t)=\cos ^{3}(2\pi t)+i\sin ^{3}(2\pi t),0\leq t\leq 1$
The parametric equations of their product $\alpha \cdot u_{n}$ are
    $x=\cos ^{3}(2\pi t)+\cos(2\pi nt)-1,$
    $y=\sin ^{3}(2\pi t)+\sin(2\pi nt)$
See the figure.
Since both $\alpha {\mbox{ and }}u_{n}$ are loops at 1, so is the product.
| n-curve with $N=53$ | Animation of n-curve for n values from 0 to 50 |

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

The unit circle is
    $u(t)=\cos(2\pi t)+i\sin(2\pi t)\,$
and its n-curve is
    $u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)\,$
The parametric equations of their product
    $u\cdot u_{n}$
are
    $x=\cos(2\pi nt)+\cos(2\pi t)-1,$
    $y=\sin(2\pi nt)+\sin(2\pi t)$
See the figure.

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

Let us take the Rhodonea Curve
    $r=\cos(3\theta )$
If $\rho$ denotes the curve,
    $\rho (t)=\cos(6\pi t)[\cos(2\pi t)+i\sin(2\pi t)],0\leq t\leq 1$
The parametric equations of $\rho _{n}-\rho$ are
    $x=\cos(6\pi nt)\cos(2\pi nt)-\cos(6\pi t)\cos(2\pi t),$
    $y=\cos(6\pi nt)\sin(2\pi nt)-\cos(6\pi t)\sin(2\pi t),0\leq t\leq 1$

### n-Curving

If $\gamma \in H$, then, as mentioned above, the n-curve $\gamma _{n}{\mbox{ also }}\in H$. Therefore, the mapping $\alpha \to \gamma _{n}^{-1}\cdot \alpha \cdot \gamma _{n}$ is an inner automorphism of the group G. We extend this map to the whole of C[0, 1], denote it by $\phi _{\gamma _{n},e}$ and call it n-curving with γ. It can be verified that
    $\phi _{\gamma _{n},e}(\alpha )=\alpha +[\alpha (1)-\alpha (0)](\gamma _{n}-1)e.\$
This new curve has the same initial and end points as α.

### Example 1 of n-curving

Let ρ denote the Rhodonea curve $r=\cos(2\theta )$, which is a loop at 1. Its parametric equations are
    $x=\cos(4\pi t)\cos(2\pi t),$
    $y=\cos(4\pi t)\sin(2\pi t),0\leq t\leq 1$
With the loop ρ we shall n-curve the cosine curve
    $c(t)=2\pi t+i\cos(2\pi t),\quad 0\leq t\leq 1.\,$
The curve $\phi _{\rho _{n},e}(c)$ has the parametric equations
    $x=2\pi [t-1+\cos(4\pi nt)\cos(2\pi nt)],\quad y=\cos(2\pi t)+2\pi \cos(4\pi nt)\sin(2\pi nt)$
See the figure.
It is a curve that starts at the point (0, 1) and ends at (2π, 1).

### Example 2 of n-curving

Let χ denote the Cosine Curve
    $\chi (t)=2\pi t+i\cos(2\pi t),0\leq t\leq 1$
With another Rhodonea Curve
    $\rho =\cos(3\theta )$
we shall n-curve the cosine curve.
The rhodonea curve can also be given as
    $\rho (t)=\cos(6\pi t)[\cos(2\pi t)+i\sin(2\pi t)],0\leq t\leq 1$
The curve $\phi _{\rho _{n},e}(\chi )$ has the parametric equations
    $x=2\pi t+2\pi [\cos(6\pi nt)\cos(2\pi nt)-1],$
    $y=\cos(2\pi t)+2\pi \cos(6\pi nt)\sin(2\pi nt),0\leq t\leq 1$
See the figure for $n=15$.

### Generalized n-curving

In the FTA C[0, 1] of curves, instead of e we shall take an arbitrary curve $\beta$, a loop at 1. This is justified since
    $L_{1}(\beta )=L_{2}(\beta )=1$
Then, for a curve γ in C[0, 1],
    $\gamma ^{*}=(\gamma (0)+\gamma (1))\beta -\gamma$
and
    $\gamma ^{-1}={\frac {\gamma ^{*}}{\gamma (0)\gamma (1)}}.$
If $\alpha \in H$, the mapping
    $\phi _{\alpha _{n},\beta }$
given by
    $\phi _{\alpha _{n},\beta }(\gamma )=\alpha _{n}^{-1}\cdot \gamma \cdot \alpha _{n}$
is the n-curving. We get the formula
    $\phi _{\alpha _{n},\beta }(\gamma )=\gamma +[\gamma (1)-\gamma (0)](\alpha _{n}-\beta ).$
Thus given any two loops $\alpha$ and $\beta$ at 1, we get a transformation of curve
    $\gamma$ given by the above formula.
This we shall call generalized n-curving.

### Example 1

Let us take $\alpha$ and $\beta$ as the unit circle ``u.’’ and $\gamma$ as the cosine curve
    $\gamma (t)=4\pi t+i\cos(4\pi t)0\leq t\leq 1$
Note that $\gamma (1)-\gamma (0)=4\pi$
For the transformed curve for $n=40$, see the figure.
The transformed curve $\phi _{u_{n},u}(\gamma )$ has the parametric equations

### Example 2

Denote the curve called Crooked Egg by $\eta$ whose polar equation is
    $r=\cos ^{3}\theta +\sin ^{3}\theta$
Its parametric equations are
    $x=\cos(2\pi t)(\cos ^{3}2\pi t+\sin ^{3}2\pi t),$
    $y=\sin(2\pi t)(\cos ^{3}2\pi t+\sin ^{3}2\pi t)$
Let us take $\alpha =\eta$ and $\beta =u,$
where $u$ is the unit circle.
The n-curved Archimedean spiral has the parametric equations
    $x=2\pi t\cos(2\pi t)+2\pi [(\cos ^{3}2\pi nt+\sin ^{3}2\pi nt)\cos(2\pi nt)-\cos(2\pi t)],$
    $y=2\pi t\sin(2\pi t)+2\pi [(\cos ^{3}2\pi nt)+\sin ^{3}2\pi nt)\sin(2\pi nt)-\sin(2\pi t)]$
See the figures, the Crooked Egg and the transformed Spiral for $n=20$.