N-curve

We take the functional theoretic algebra ''C'' , 1of 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.
#Their f-products, sums and differences give rise to many beautiful curves.
#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'' , 1 is invertible, i.e.

: $\gamma^ \,$

exists if

: $\gamma(0)\gamma(1) \neq 0. \,$

If $\gamma^=(\gamma(0)+\gamma(1))e - \gamma$, where $e(t)=1, \forall t \in$, 1/math>, then

: $\gamma^= \frac.$

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 $\alpha \to \gamma^\cdot \alpha\cdot\gamma$ 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- \in , 1

If $\gamma \in H$ and ''n'' is a positive integer, then define a curve $\gamma_$ by

: \gamma_n (t)=\gamma(nt - t. \,

$\gamma_$ 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 $\alpha(0)=\beta(1)=1, \mbox \alpha \cdot \beta = \beta + \alpha -e$.

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

: $\alpha(t)=\cos^(2\pi t)+ i \sin^(2\pi t), 0\leq t \leq 1$

The parametric equations of their product $\alpha \cdot u_$ are

:$x=\cos^3 (2\pi t)+ \cos(2\pi nt)-1,$
:$y=\sin^(2\pi t)+ \sin(2\pi nt)$

See the figure.

Since both $\alpha \mbox u_$ 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
:$u \cdot u_$
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 $\rho_- \rho$ 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 $\gamma_ \mbox \in H$. Therefore, the mapping $\alpha \to \gamma_n^\cdot \alpha\cdot\gamma_n$ is an inner automorphism of the group ''G.'' We extend this map to the whole of ''C'' , 1 denote it by $\phi_$ and call it ''n''-curving with γ.
It can be verified that

: \phi_(\alpha)=\alpha + alpha(1)-\alpha(0)\gamma_-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 $\phi_(c)$ has the parametric equations

: x=2\pi -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 $\phi_(\chi)$ 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'' , 1of 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'' , 1
:$\gamma^=(\gamma(0)+\gamma(1))\beta - \gamma$
and
: $\gamma^= \frac.$

If $\alpha \in H$, the mapping
:$\phi_$
given by
:$\phi_(\gamma) = \alpha_n^\cdot \gamma \cdot \alpha_n$

is the ''n''-curving. We get the formula

: \phi_(\gamma)=\gamma + gamma(1)-\gamma(0)\alpha_-\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 $n = 20$.

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 201
Book of Beautiful Curves

External links

The Siluroid Curve

Curves