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.

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[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

$\cos(6\pi nt)\cos(2\pi nt)-\cos(6\pi t)\cos(2\pi t),y=\cos(6\pi nt)\sin(2\pi nt)-\cos(6\pi t)\sin(2\pi t),0\leq t\leq 1$

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).

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

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

$x=2\pi t+2\pi [\cos(6\pi nt)\cos(2\pi nt)-1],\quad y=\cos(2\pi t)+2\pi \cos(6\pi nt)\sin(2\pi nt),0\leq t\leq 1$

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 $\gamma (1)-\gamma (0)=4\pi$

The transformed curve $\phi _{u_{n},u}(\gamma )$ has the parametric equations

$x=4\pi t+4\pi [\cos(2\pi nt)-\cos(2\pi t)],y=\cos(4\pi t)+4\pi [\sin(2\pi nt)-\sin(2\pi t)]$

For the transformed curves, see the figure.

See also

• Functional-theoretic algebra

References

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