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

[clarification needed] The n-curves are interesting in two ways.

A curve γ in the functional theoretic algebra C[0, 1], is invertible, i.e. exists if If

, then 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

is an inner automorphism of the group G. We use these concepts to define n-curves and n-curving.

If x is a real number and [x] denotes the greatest integer not greater than x, then

and n is a positive integer, then define a curve

is also a loop at 1 and we call it an n-curve.

α ( 0 ) = β ( 1 ) = 1 ,

α ⋅ β = β + α − e

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

The unit circle is and its n-curve is The parametric equations of their product are See the figure.

denotes the curve, The parametric equations of

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.

Let ρ denote the Rhodonea curve

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).

Let χ denote the Cosine Curve With another Rhodonea Curve we shall n-curve the cosine curve.

has the parametric equations See the figure for

This is justified since Then, for a curve γ in C[0, 1], and If

at 1, we get a transformation of curve This we shall call generalized n-curving.

as the cosine curve Note that

Denote the curve called Crooked Egg by

The n-curved Archimedean spiral has the parametric equations See the figures, the Crooked Egg and the transformed Spiral for