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Introduction
There are many ways of transforming a mathematical curve. Here we introduce a method using the principles of functional-theoretic algebra (FTA).
A curve γ in the FTA C[0, 1] of curves, is invertible, i.e.
exists if
![{\displaystyle \gamma (0)\gamma (1)\neq 0.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4ae4056d8edf8144b7d48814d144575c50abb6)
If
, then
![{\displaystyle \gamma ^{-1}={\frac {\gamma ^{*}}{\gamma (0)\gamma (1)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfed4e94db4518783584d13ab7c3e9e7fb5c7be9)
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
, then the mapping
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
If
and n is a positive integer, then define a curve
by
.
is also a loop at 1 and we call it an n-curve.
Note that every curve in H is a 1-curve.
Suppose
Then, since
, where
Example of a Product of n-Curves
Products of n-curves often yield beautiful new curves. Let us take u, the unit circle centered at the origin and α, the astroid.
Then,
![{\displaystyle u_{n}(t)=\cos(2\pi nt)+i\sin(2\pi nt)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09cc8db92f9b5556605e3ca41baa733e91f65bba)
and
![{\displaystyle \alpha (t)=\cos ^{3}(2\pi t)+i\sin ^{3}(2\pi t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9705e02dc983046724ff4ca77820acea19a1d74c)
The parametric equations of
are
See the figure. Since both
are loops at 1, so is the product.
n-Curving
If
, then the n-curve
. Therefore the mapping
is an inner automorphism of the group G. We extend this map to the whole of C[0, 1], denote it by
and call it n-curving with γ.
It can be verified that
. This new curve has the same initial and end points as α.
Example of n-Curving
Let ρ denote the Rhodonea Curve
, which is a loop at 1. Its parametric equations are
.
With the loop ρ we shall n-Curve the cosine curve
.
The curve
has the parametric equations
References
- Sebastian Vattamattam, ``Transforming Curves by n-Curving, in ``Bulletin of Kerala Mathematics Association, Vol. 5, No. 1, December 2008