We take the functional theoretic algebraC[0, 1] of curves. For each loop γ at 1, and each positive integer n, we define a curve 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.
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 .
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,
and the astroid is
The parametric equations of their product are
See the figure.
Since both are loops at 1, so is the product.
N-curve with .
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
and its n-curve is
The parametric equations of their product
are
See the figure.
Example 3: n-Curve of the Rhodonea minus the Rhodonea Curve
Let us take the Rhodonea Curve
If denotes the curve,
The parametric equations of are
n-Curving
If , then, as mentioned above, 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
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.
Generalized n-Curving
In the FTA C[0, 1] of curves, instead of e we shall take an arbitrary curve , a loop at 1.
This is justified since
Then, for a curve γ in C[0, 1],
and
If , the mapping
given by
is the n-curving.
We get the formula
Thus given any two loops and at 1, we get a transformation of curve
given by the above formula.
This we shall call generalized n-curving.
Example 1
Let us take and as the unit circle ``u.’’ and as the cosine curve
Note that
The transformed curve has the parametric equations