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 ${\displaystyle \gamma _{n}}$ called n-curve.^{[clarification needed]} 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.

${\displaystyle \gamma ^{-1}\,}$ 

exists if

${\displaystyle \gamma (0)\gamma (1)\neq 0.\,}$ 

If ${\displaystyle \gamma ^{*}=(\gamma (0)+\gamma (1))e-\gamma }$ , where ${\displaystyle e(t)=1,\forall t\in [0,1]}$ , then

${\displaystyle \gamma ^{-1}={\frac {\gamma ^{*}}{\gamma (0)\gamma (1)}}.}$ 

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 ${\displaystyle \gamma \in H}$ , then the mapping ${\displaystyle \alpha \to \gamma ^{-1}\cdot \alpha \cdot \gamma }$  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 ${\displaystyle x-[x]\in [0,1].}$ 

If ${\displaystyle \gamma \in H}$  and n is a positive integer, then define a curve ${\displaystyle \gamma _{n}}$  by

${\displaystyle \gamma _{n}(t)=\gamma (nt-[nt]).\,}$ 

${\displaystyle \gamma _{n}}$  is also a loop at 1 and we call it an n-curve. Note that every curve in H is a 1-curve.

Suppose ${\displaystyle \alpha ,\beta \in H.}$  Then, since ${\displaystyle \alpha (0)=\beta (1)=1,{\mbox{ the f-product }}\alpha \cdot \beta =\beta +\alpha -e}$ .

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,

${\displaystyle u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)\,}$ 

and the astroid is

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

The parametric equations of their product ${\displaystyle \alpha \cdot u_{n}}$  are

${\displaystyle x=\cos ^{3}(2\pi t)+\cos(2\pi nt)-1,}$ 

${\displaystyle y=\sin ^{3}(2\pi t)+\sin(2\pi nt)}$ 

See the figure.

Since both ${\displaystyle \alpha {\mbox{ and }}u_{n}}$  are loops at 1, so is the product.

n-curve with ${\displaystyle N=53}$

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

${\displaystyle u(t)=\cos(2\pi t)+i\sin(2\pi t)\,}$ 

and its n-curve is

${\displaystyle u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)\,}$ 

The parametric equations of their product

${\displaystyle u\cdot u_{n}}$ 

are

${\displaystyle x=\cos(2\pi nt)+\cos(2\pi t)-1,}$ 

${\displaystyle y=\sin(2\pi nt)+\sin(2\pi t)}$ 

See the figure.

 

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

Let us take the Rhodonea Curve

${\displaystyle r=\cos(3\theta )}$ 

If ${\displaystyle \rho }$  denotes the curve,

${\displaystyle \rho (t)=\cos(6\pi t)[\cos(2\pi t)+i\sin(2\pi t)],0\leq t\leq 1}$ 

The parametric equations of ${\displaystyle \rho _{n}-\rho }$  are

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

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

  

n-Curving

If ${\displaystyle \gamma \in H}$ , then, as mentioned above, the n-curve ${\displaystyle \gamma _{n}{\mbox{ also }}\in H}$ . Therefore, the mapping ${\displaystyle \alpha \to \gamma _{n}^{-1}\cdot \alpha \cdot \gamma _{n}}$  is an inner automorphism of the group G. We extend this map to the whole of C[0, 1], denote it by ${\displaystyle \phi _{\gamma _{n},e}}$  and call it n-curving with γ. It can be verified that

${\displaystyle \phi _{\gamma _{n},e}(\alpha )=\alpha +[\alpha (1)-\alpha (0)](\gamma _{n}-1)e.\ }$ 

This new curve has the same initial and end points as α.

Example 1 of n-curving

Let ρ denote the Rhodonea curve ${\displaystyle r=\cos(2\theta )}$ , which is a loop at 1. Its parametric equations are

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

${\displaystyle y=\cos(4\pi t)\sin(2\pi t),0\leq t\leq 1}$ 

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

${\displaystyle c(t)=2\pi t+i\cos(2\pi t),\quad 0\leq t\leq 1.\,}$ 

The curve ${\displaystyle \phi _{\rho _{n},e}(c)}$  has the parametric equations

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

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

${\displaystyle \chi (t)=2\pi t+i\cos(2\pi t),0\leq t\leq 1}$ 

With another Rhodonea Curve

${\displaystyle \rho =\cos(3\theta )}$ 

we shall n-curve the cosine curve.

The rhodonea curve can also be given as

${\displaystyle \rho (t)=\cos(6\pi t)[\cos(2\pi t)+i\sin(2\pi t)],0\leq t\leq 1}$ 

The curve ${\displaystyle \phi _{\rho _{n},e}(\chi )}$  has the parametric equations

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

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

See the figure for ${\displaystyle n=15}$ .

 

Generalized n-curving

In the FTA C[0, 1] of curves, instead of e we shall take an arbitrary curve ${\displaystyle \beta }$ , a loop at 1. This is justified since

${\displaystyle L_{1}(\beta )=L_{2}(\beta )=1}$ 

Then, for a curve γ in C[0, 1],

${\displaystyle \gamma ^{*}=(\gamma (0)+\gamma (1))\beta -\gamma }$ 

and

${\displaystyle \gamma ^{-1}={\frac {\gamma ^{*}}{\gamma (0)\gamma (1)}}.}$ 

If ${\displaystyle \alpha \in H}$ , the mapping

${\displaystyle \phi _{\alpha _{n},\beta }}$ 

given by

${\displaystyle \phi _{\alpha _{n},\beta }(\gamma )=\alpha _{n}^{-1}\cdot \gamma \cdot \alpha _{n}}$ 

is the n-curving. We get the formula

${\displaystyle \phi _{\alpha _{n},\beta }(\gamma )=\gamma +[\gamma (1)-\gamma (0)](\alpha _{n}-\beta ).}$ 

Thus given any two loops ${\displaystyle \alpha }$  and ${\displaystyle \beta }$  at 1, we get a transformation of curve

${\displaystyle \gamma }$  given by the above formula.

This we shall call generalized n-curving.

Example 1

Let us take ${\displaystyle \alpha }$  and ${\displaystyle \beta }$  as the unit circle ``u.’’ and ${\displaystyle \gamma }$  as the cosine curve

${\displaystyle \gamma (t)=4\pi t+i\cos(4\pi t)0\leq t\leq 1}$ 

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

For the transformed curve for ${\displaystyle n=40}$ , see the figure.

The transformed curve ${\displaystyle \phi _{u_{n},u}(\gamma )}$  has the parametric equations

 

Example 2

Denote the curve called Crooked Egg by ${\displaystyle \eta }$  whose polar equation is

${\displaystyle r=\cos ^{3}\theta +\sin ^{3}\theta }$ 

Its parametric equations are

${\displaystyle x=\cos(2\pi t)(\cos ^{3}2\pi t+\sin ^{3}2\pi t),}$ 

${\displaystyle y=\sin(2\pi t)(\cos ^{3}2\pi t+\sin ^{3}2\pi t)}$ 

Let us take ${\displaystyle \alpha =\eta }$  and ${\displaystyle \beta =u,}$ 

where ${\displaystyle u}$  is the unit circle.

The n-curved Archimedean spiral has the parametric equations

${\displaystyle x=2\pi t\cos(2\pi t)+2\pi [(\cos ^{3}2\pi nt+\sin ^{3}2\pi nt)\cos(2\pi nt)-\cos(2\pi t)],}$ 

${\displaystyle y=2\pi t\sin(2\pi t)+2\pi [(\cos ^{3}2\pi nt)+\sin ^{3}2\pi nt)\sin(2\pi nt)-\sin(2\pi t)]}$ 

See the figures, the Crooked Egg and the transformed Spiral for ${\displaystyle n=20}$ .

   

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

External links

• The Siluroid Curve