Matinga's Cyclical Function

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Matinga's Cyclical Function

The universal sine wave generator is a transposition of Pythagoras' Theorem, as shown below. Test the function of x, in (2) on computer plotter. There are free online plotters available:

Pythagoras' Theorem Transposed:

(1)     B    = ( A^2 - C^2 ) ^ 1/2 ,

The natural Sine function is

f(x), A = ( a ^ 2 - x ^ 2 ) ^ 1/2 ,

a * Sin ( x ) ,

f(x), A = ( a ^ 2 - (x - h ) ^ 2 ) ^ 1/2 ,

a * Sin ( θ + phi )

f(x), A = ( a ^ 2 - ( x - h) ^ 2 ) ^ 1/2,

a * Sin ( Pi * (x + h) ) ,

a * Sin ( θ + phi )

The natural sine squared function is

f(x), A = a ^ 2 - x ^ 2 ,

a * Sin ^ 2 ( Pi *( x + 1/2) ) ,

a * Sin ^ 2 ( Pi * x + Pi / 2 )

In the standard Cartesian plane, the plane is reduced to breadth of 2a and height of 2a, by

( a ^ 2 ) ^ 1/2. As the for the full expression ( a ^ 2 - x ^ 2 ) ^ 1/2 , the terms in the brackets are 'sequentially equivalent' to, yet not 'instantaneously equal' to a - x . That is because you cannot 'collapse' or simply an algorithm further than the opening expression is programming. The rules of programming also apply to algebra. The two disciplines have parity, and they are both forms of algebra. The Commutative Law of algebra applies to transposition about the equals sign as well as on any single given side of it. That proves the claim.

(2)     Circular functions are equivalent to parabolic functions. They are both cyclical. The conclusion is that the function is a Cyclical Function (Matinga's), in either case. The example immediately below produces the cycle commonly understood as the 1:1, x:y ratio of parabola, or better perceived as a circle.

y    =   ( r^2 - x^2 ) ^ 1/2 ,          

(3)     Sin(x) as a function also benefits from the same equation, recycled.

y    =   ( r^2 - x^2 ) ^ 1/2 ,    f (x)        

(4)     produces a cycle - Matinga's Cyclical Function & the fundamental Sine Wave as functions

The square root will produce a negative and positive set of y results every time. The function will produce both a parabola and a circle at the origin of coordinates ( 0, 0), with diameter, d. ( The same d or diameter corresponds to distance, in physics. )

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