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.

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[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 $n=20$ .

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

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n-curve

Multiplicative inverse of a curve

n-curves and their products

Example 1: Product of the astroid with the n-curve of the unit circle

Example 2: Product of the unit circle and its n-curve

Example 3: n-Curve of the Rhodonea minus the Rhodonea curve

n-Curving

Example 1 of n-curving

Example 2 of n-curving

Generalized n-curving

Example 1

Example 2

See also

References