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We take the functional theoretic algebra C of curves. For each loop γ at 1, and each positive integer n, we define a curve ${\displaystyle \gamma _{n}}$ 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, 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 ${\displaystyle e(t)=1,\forall t\in }$, 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 denotes the greatest integer not greater than x, then ${\displaystyle x-\in .}$

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

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

${\displaystyle \gamma _{n}}$ 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.

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,

${\displaystyle \rho (t)=\cos(6\pi t),0\leq t\leq 1}$

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, denote it by ${\displaystyle \phi _{\gamma _{n},e}}$ and call it n-curving with γ. It can be verified that

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

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

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

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

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

With another Rhodonea Curve

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

we shall n-curve the cosine curve.

The rhodonea curve can also be given as

${\displaystyle \rho (t)=\cos(6\pi t),0\leq t\leq 1}$

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

${\displaystyle x=2\pi t+2\pi ,}$

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

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

Generalized n-curving

In the FTA C 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,

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

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

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

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

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

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

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

• Curves