Curve transformation
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[edit] Curve Transformations
By a curve we mean a curve in the two-dimensional Cartesian plane, like a straight line, a circle, an ellipse, or a parabola. Given a curve, there are different methods of obtaining another curve, called an associative curve. By curve transformations, we mean such methods. Transformations that give the envelop, inversion etc. of curves are well known. Here is presented a new type of curve transformations, developed by this author.
[edit] L and R transformations of paths in C
We consider a family of curves called paths. A continuous function f:[0, 1] → C, the complex field, is called a path from f(0) to f(1). We shall call f(0) and f(1) the initial and terminal points of the path f. If f(0) = f(1), then f is called a loop.
The set of paths, denoted by C[0, 1], is a vector space over the field C. We present three linear transformations on this space, each of which transforms a path f into another path that has the same initial and terminal points as f.
Given any path f and a loop c with c(0) = c(1) = 1, we can have three other paths listed below, all of which are from f(0) to f(1).
1. Lf(t) = f(t) + f(1)[c(t) -1]
2. Rf(t) = f(t) + f(0)[c(t) -1]
3. LRf(t) = f(t) + (f(o) + f(1))[c(t) -1]
Let us call these three associated curves as the L, R, and LR transformations of f. One can easily verify that the mappings f → Lf, f → Rf, and f → LRf are linear transformations on C[0, 1].
[edit] Illustrative Example
c(t) = cos(2πt) + isin(2πt), t ε [0, 1]. c is the unit circle and it is a loop with c(0) = c(1) = 1. f(t) = t + 1, t ε [0, 1], the part of the x-axis form x = 1 to x = 2. f(0) = 1, f(1) = 2. The path is from the point (1, 0) to (2, 0).
On computation we get the transformations as follows: Lf(t) = (t – 1 + 2cos2πt) +i(2sin2πt) Rf(t) = (t + cos2πt) +i(2sin2πt) RLf(t) = (t – 2 + 3cos2πt) +i(3sin2πt)
These three new paths are shown in the following figure. 
[edit] Transformation of the Archimedean Spiral
The Archimedean Spiral r = t, t ε [0, 6п] can be treated as a path in C defined on the closed interval [0, 6п]. In the Cartesian Coordinate system, its parametric equations are x = tcost, y = tsint and so the path is given by f(t) = tcost + itsint, t ε [0, 6п]. f(0) = 0 and f(6п) = 6п. The path is from the point (0, 0) to (6п, 0) in the xy-plane.
Lf(t) = tcost + itsint + 6п(cost + isint – 1) = [(t + 6п)cost - 6п] +i[(t + 6п)sint]
Rf = f, since f(0) = 0 
Note: The function could be redefined on [0, 1] without affecting the results.
[edit] Beautiful Curve Patterns
Using the above transformations we can get families of curves, which form beautiful patterns. As the first example, five families of such curves are given below, obtained by transforming the curves y = x^n, n an even natural number. All the equations are given in parametric form.
Let n be an odd natural number and I = [-1, 1]. We take this interval instead of [0, 1]. Consider the family of paths: f(t) = t + it^n, t ε I. Applying the above transformations we get 5 families of curves, including the given:
Family 1: x = t, y = t^n, t ε I
Family 2: x = cosпt - sinпt – t + 1, y = cosпt + sinпt – t^n + 1
Family 3: x = -sinпt -cosпt – t -1, y = -sinпt + cosпt +1 – t^n
Family 4: x = cosпt + sinпt + t + 1, y = -cosпt + sinпt + t^n – 1
Family 5: x = sinпt -cosпt + t -1, y = -sinпt - cosпt -1 + t^n
All the members of F1, F4, and F5 have their end points at (-1, 1) and (1, 1) in the xy-plane. All the members of the other two families have their end points at (-1, -1) and (1, -1). Look at the figure to see how they form a beautiful pattern. 
Aesthetic application is the best and only application known so far for this type of transformations. If anyone can suggest any other application, this author will be happy to share with him or her the theory behind the whole thing.
