n-curve

Introduction

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 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 FTA , 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 of a Product of n-Curves

Products of n-curves often yield beautiful new curves. Let us take u, the unit circle centered at the origin and α, the astroid. Then,

$u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)$

and

$\alpha (t)=\cos ^{3}(2\pi t)+i\sin ^{3}(2\pi t)$

The parametric equations of $\alpha \cdot u_{n}$ 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-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 of n-Curving

Let ρ denote the Rhodonea Curve $r=\cos(2\theta )$, which is a loop at 1. Its parametric equations are

$x=\cos(4\pi t)\cos(2\pi t),y=\cos(4\pi t)\sin(2\pi t)$.

With the loop ρ we shall n-Curve the cosine curve $c(t)=2\pi t+i\cos(2\pi t),0\leq t\leq 1$. The curve $\phi _{\rho _{n},e}(c)$ has the parametric equations

$x=2\pi [t-1+\cos(4\pi nt)\cos(2\pi nt)],y=\cos(2\pi t)+2\pi \cos(4\pi nt)\sin(2\pi nt)$

References

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