n-curve

In the functional theoretic algebra C[0, 1] of continuous curves on the unit interval, an n-curve is a specific transformation of a given base curve. For a loop γ: [0, 1] → $\mathbb{ℂ}$ (or another suitable space) such that γ(0) = γ(1) = 1 (the base point), and for any positive integer n, the associated n-curve, denoted γ n, is defined by the pointwise functional power.

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

$\gamma (0)\gamma (1)\ne 0.$

If ${\gamma }^{*}=(\gamma (0)+\gamma (1))e-\gamma$, where $e(t)=1,\mathrm{\forall }t\in [0,1]$, 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 $\alpha (0)=\beta (1)=1,\text{ the f-product }\alpha \cdot \beta =\beta +\alpha -e$.

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

$\alpha (t)={\cos }^3(2\pi t)+i{\sin }^3(2\pi t),0\le t\le 1$

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 $\alpha \text{ and }{u}_n$ 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.

File:Unit Circle with n-Circle.jpg

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,

$\rho (t)=\cos (6\pi t)[\cos (2\pi t)+i\sin (2\pi t)],0\le t\le 1$

The parametric equations of ${\rho }_n-\rho$ are

$x=\cos (6\pi nt)\cos (2\pi nt)-\cos (6\pi t)\cos (2\pi t),$

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

File:Rhodonea Curve.jpgFile:Rhodonea-nRhodonea Curve.jpg

n-Curving

If $\gamma \in H$, then, as mentioned above, the n-curve ${\gamma }_n\text{ also }\in H$. 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 ${\varphi }_{{\gamma }_n,e}$ and call it n-curving with γ. It can be verified that

${\varphi }_{{\gamma }_n,e}(\alpha )=\alpha +[\alpha (1)-\alpha (0)]({\gamma }_n-1)e.$

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

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

$y=\cos (4\pi t)\sin (2\pi t),0\le t\le 1$

With the loop ρ we shall n-curve the cosine curve

$c(t)=2\pi t+i\cos (2\pi t),0\le t\le 1.$

The curve ${\varphi }_{{\rho }_n,e}(c)$ has the parametric equations

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

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

$\chi (t)=2\pi t+i\cos (2\pi t),0\le t\le 1$

With another Rhodonea Curve

$\rho =\cos (3\theta )$

we shall n-curve the cosine curve.

The rhodonea curve can also be given as

$\rho (t)=\cos (6\pi t)[\cos (2\pi t)+i\sin (2\pi t)],0\le t\le 1$

The curve ${\varphi }_{{\rho }_n,e}(\chi )$ has the parametric equations

$x=2\pi t+2\pi [\cos (6\pi nt)\cos (2\pi nt)-1],$

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

See the figure for $n=15$.

File:CosineRhodonea.jpg

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

${\varphi }_{{\alpha }_n,\beta }$

given by

${\varphi }_{{\alpha }_n,\beta }(\gamma )={\alpha }_n^{-1}\cdot \gamma \cdot {\alpha }_n$

is the n-curving. We get the formula

${\varphi }_{{\alpha }_n,\beta }(\gamma )=\gamma +[\gamma (1)-\gamma (0)]({\alpha }_n-\beta ).$

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

$\gamma (t)=4\pi t+i\cos (4\pi t)0\le t\le 1$

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

For the transformed curve for $n=40$, see the figure.

The transformed curve ${\varphi }_{u_n,u}(\gamma )$ has the parametric equations

File:N-curved cosine.jpg

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 $n=20$.

File:Crooked Egg.jpg File:Crooked Spiral.jpg

References

• 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

External links

• The Siluroid Curve