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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 called n-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 , where , 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 , then the mapping 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
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 .
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.
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The unit circle is
and its n-curve is
The parametric equations of their product
are
See the figure.
Let us take the Rhodonea Curve
If denotes the curve,
The parametric equations of are
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.
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).
Let χ denote the Cosine Curve
With another Rhodonea Curve
we shall n-curve the cosine curve.
The rhodonea curve can also be given as
The curve has the parametric equations
See the figure for .
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
This we shall call generalized n-curving.
Let us take and as the unit circle ``u.’’ and as the cosine curve
Note that
For the transformed curve for , see the figure.
The transformed curve has the parametric equations
Denote the curve called Crooked Egg by whose polar equation is
Its parametric equations are
Let us take and
where is the unit circle.
The n-curved Archimedean spiral has the parametric equations
See the figures, the Crooked Egg and the transformed Spiral for .
![]() | This article has multiple issues. Please help
improve it or discuss these issues on the
talk page. (
Learn how and when to remove these template messages)
|
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 called n-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 , where , 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 , then the mapping 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
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 .
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.
![]() |
![]() |
The unit circle is
and its n-curve is
The parametric equations of their product
are
See the figure.
Let us take the Rhodonea Curve
If denotes the curve,
The parametric equations of are
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.
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).
Let χ denote the Cosine Curve
With another Rhodonea Curve
we shall n-curve the cosine curve.
The rhodonea curve can also be given as
The curve has the parametric equations
See the figure for .
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
This we shall call generalized n-curving.
Let us take and as the unit circle ``u.’’ and as the cosine curve
Note that
For the transformed curve for , see the figure.
The transformed curve has the parametric equations
Denote the curve called Crooked Egg by whose polar equation is
Its parametric equations are
Let us take and
where is the unit circle.
The n-curved Archimedean spiral has the parametric equations
See the figures, the Crooked Egg and the transformed Spiral for .