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
.
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
, 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, then [x] denotes the greatest integer not greater than x and
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
Let us take u, the unit circle centered at the origin and
Products of n-curves often yield beautiful new curves.
α, the astroid.
Then,
The parametric equations of
are