????????????????????????.. Subject : CLASSIFICATION OF CIRCLES : A MAIDEN CONCEPT geometry.research.independent
To the Mathematicians,
We know till date that, Euclidean circle, which, in the rectangular coordinate system (X, Y), (where, X and Y are certain functions of the parameter x), is defined as the path of a moving point P(X, Y), by maintaining a constant distance (radius) from a fixed point O(0,0), called centre. The starting point at angle x = 0, after encircling just one complete turn of x = 2*PI, returns back to the same point.
Proposed classification of circle : There exist a group of circles satisfying X^2 + Y^2 = 1, where, unlike the Euclidean circle, starting point A (at x = 0) and end point B (at x = 2*PI), are not the same point. This shows that the circular arc to trace less than or, in excess of one complete turn. These types of circles, including the Euclidean one, have been given a general name: ?Nbic circles?.
The angle measured in Nbic circles are in Nbic angle N (which is also a function of x). When geometrically measured, in terms of ?teN? angle (i.e., tan equivalent of the Nbic angle), it becomes: tan N = Y/X. For the point P(X, Y), the difference between the teN angle N and x, is the sweep angle = (arctan N ? x). The angular difference between B and A, measured in terms of the angle N<B-A>, or alternatively, the sweep angle sA, play the pivotal role for the classification of the Nbic circles.
Thus all the Nbic circles when categorized group-wise, are: Type1. Incomplete /Inferior /Negative circle: N<B-A> less than 2*PI, sweep angle = negative, identified in Hyperbolic circle, C<H>. Type2. Complete /Full / Zero circle: N<B-A> equal to 2*PI, sweep angle = 0, identified in Euclidean circle, C<0>. Type3. Over Complete / Superior /Positive circle: N<B-A> greater than 2*PI, sweep angle = positive, identified in Nbic circle, C<N>.
I have presented here only the proposed categorization of circles. For the detailed discussion and derivation, please refer to Chapter 6, pp159 ? 181, of my Book : ?Advanced Trigonometric Relations through Nbic functions? ? by Nisith K Bairagi, New Age International Publishes, New Delhi, (2012). [ISBN: 978-81-224-3023-3] Request : (1) please inform me whether any reference regarding this topic is available or not, also (2) please post this topic in your column for the readers? comments and criticism.
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