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

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

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

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).

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

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

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

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 $n = 20$.

See also

• Functional-theoretic algebra
• The Siluroid Curve

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