 Ncurve

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 γ_{n} called ncurve. The ncurves are interesting in two ways.
 Their fproducts, sums and differences give rise to many beautiful curves.
 Using the ncurves, we can define a transformation of curves, called ncurving.
Multiplicative inverse of a curve
A curve γ in the functional theoretic algebra C[0, 1], is invertible, i.e.
exists if
If γ ^{*} = (γ(0) + γ(1))e − γ, where , then
The set G of invertible curves is a noncommutative 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 ncurves and ncurving.
nCurves 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 γ_{n} by
γ_{n} is also a loop at 1 and we call it an ncurve. Note that every curve in H is a 1curve.
Suppose Then, since .
Example 1: Product of the astroid with the ncurve of the unit circle
Let us take u, the unit circle centered at the origin and α, the astroid. The ncurve of “u” is given by,
and the astroid is
The parametric equations of their product are
 x = cos ^{3}(2πt) + cos(2πnt) − 1,
 y = sin ^{3}(2πt) + sin(2πnt)
See the figure.
Since both α and u_{n} are loops at 1, so is the product.
Example 2: Product of the unit circle and its ncurve
The unit circle is
and its ncurve is
The parametric equations of their product
are
 x = cos(2πnt) + cos(2πt) − 1,
 y = sin(2πnt) + sin(2πt)
See the figure.
Example 3: nCurve of the Rhodonea minus the Rhodonea curve
Let us take the Rhodonea Curve
 r = cos(3θ)
If ρ denotes the curve,
The parametric equations of ρ_{n} − ρ are
 x = cos(6πnt)cos(2πnt) − cos(6πt)cos(2πt),
nCurving
If , then, as mentioned above, the ncurve . 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 ncurving with γ. It can be verified that
This new curve has the same initial and end points as α.
Example 1 of ncurving
Let ρ denote the Rhodonea curve r = cos(2θ), which is a loop at 1. Its parametric equations are
 x = cos(4πt)cos(2πt),
With the loop ρ we shall ncurve 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).
Example 2 of ncurving
Let χ denote the Cosine Curve
With another Rhodonea Curve
 ρ = cos(3θ)
we shall ncurve the cosine curve.
The rhodonea curve can also be given as
The curve has the parametric equations
 x = 2πt + 2π[cos(6πnt)cos(2πnt) − 1],
See the figure for n = 15.
Generalized nCurving
In the FTA C[0, 1] of curves, instead of e we shall take an arbitrary curve β, a loop at 1. This is justified since
 L_{1}(β) = L_{2}(β) = 1
Then, for a curve γ in C[0, 1], γ ^{*} = (γ(0) + γ(1))β − γ
and
If , the mapping
given by
is the ncurving.
We get the formula
Thus given any two loops α and β at 1, we get a transformation of curve
 γ given by the above formula.
This we shall call generalized ncurving.
Example 1
Let us take α and β as the unit circle ``u.’’ and γ as the cosine curve
Note that γ(1) − γ(0) = 4π
For the transformed curve for n = 40, see the figure.
The transformed curve has the parametric equations
Example 2
Denote the curve called Crooked Egg by η whose polar equation is
 r = cos ^{3}θ + sin ^{3}θ
Its parametric equations are
 x = cos(2πt)(cos ^{3}2πt + sin ^{3}2πt),
 y = sin(2πt)(cos ^{3}2πt + sin ^{3}2πt)
Let us take α = η and β = u,
where u is the unit circle.
The ncurved Archimedean spiral has the parametric equations
 x = 2πtcos(2πt) + 2π[(cos ^{3}2πnt + sin ^{3}2πnt)cos(2πnt) − cos(2πt)],
 y = 2πtsin(2πt) + 2π[(cos ^{3}2πnt) + sin ^{3}2πnt)sin(2πnt) − sin(2πt)]
See the figures, the Crooked Egg and the transformed Spiral for n = 20.
See also
References
 Sebastian Vattamattam, "Transforming Curves by nCurving", in Bulletin of Kerala Mathematics Association, Vol. 5, No. 1, December 2008
Categories: Curves
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