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.

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 $$n = 20$$.