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Views of a third degree hyperbola & its aperture circle.
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Mathematically, the circle and hyperbola can be modeled as expressions of the same formula,
(which may vary in the order and number of parameters used) .
For the figures in this post, i plotted x against the roots of (1-x^degree)^(1/degree).
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When degree equals 2, the formula's two roots produce a classic hyperbola, (round-nosed and symmetrical),
with a classic, (round), circle filling its aperture .
These are shown in the right-hand column, (in a blend from maroon to pale teal) .
For any given x value where the roots are distinct from each other, each root of a second degree hyperbola or circle
will be equivalent to the other if it were rotated by 2pi/2, (180°), around the X axis on the, (real, imaginary), complex plane .
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Where degree equals three, there will be three root curves .
These are shown in all figures in a blend from purple to pale blue .
For any given x value of a third degree circle and hyperbola, a point on one curve will be equivalent
to a point on another if that point is rotated by ±2pi/3, (120°), around the X axis on the complex plane .
The following principal applies to all hyperbolas and circles having an integral degree value that i have checked :
There will be degree root curves, and these will be separated from each other by (integer multiples of 2pi) / degree .
In part because of this, the aperture 'circle' of an evenly integral infinite degree hyperbola will appear
as a solid three dimensional object ; round in a cross section taken perpendicular to X .
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As far as i can tell, only even integer degree hyperbolas will have fully developed apertures …
apertures in which a closed form like a circle, an ellipse, square or rectangle can exist .
Odd degree hyperbolas, (of a degree higher than 1), develop their aperture on one side,
but leave the other open, producing a nipple-like form ; as can be seen in these figures .
However, the curves through the nipple appear to be symmetrical from their respective centers, and planar .
If so, i consider it an example of the 'innate poetry' of mathematics :
that where it appears an even degree hyperbola will fill a gap in its symmetrical and (often multi)planar form
with one or more symmetrical forms of a different nature, on one or more interleaved planes ;
an odd degree hyperbola bridges that space, making it the center of its own, continuous, symmetrical and planar form(s) .
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The straight green lines in the center-right column are what i call the "tending lines" of these hyperbolic limbs .
I believe that were these limbs to run to infinity, they would meet their lines ;
though for the purpose of observation, they are already close at ±8 .
I believe that the lines also select logical pairs among the hyperbolic branches,
(from the set where x is above the aperture zone, { >1 in these figures}, and that where x is below it) .
This (to me) reinforces the pairing implied by odd-degree hyperbolic branches' apparent symmetry from center-of-curvature .
I believe it is instructive that the odd-degree hyperbola curve pairs thus chosen cross from positive to negative,
(or vise-versa), on the complex plane as they pass through the zone of aperture .
If this principle is also applied to even-degree hyperbolae and circles,
i think a fuller understanding of any natural phenomena which may be linked to them can be had ;
(please see posts of 3, 6 & 14 December, 2010) .
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For the pi/2, (90°), arc between x=0 and x=1 where the second and third degree circles share the same plane,
and are both turning from a real value of 1 to a real value of 0, (please see right hand column),
the third degree circle is more squarish than the second .
This principal also holds as far as i have checked it : the higher the degree, the squarer the corner its 'circle' turns,
and the squarer the departure of the hyperbolic branch that meets it .
Mathematically, a square might be graphed as the aperture circle of two pi-rotationally-offset subsets of the root curves
of an evenly integral, infinite-degree hyperbola .
I am, as often without expertise, curious where the n-degree circles & hyperbolae might express themselves in nature and mathematics .
And whether, perhaps at their infinite or near-infinite degrees, these might provide a model for the electron,
for other sub-atomic particles, or be linked to the phenomenon of mass .
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Please note: In common usage, the terms, "cubic hyperbola", and to a much lesser extent, "third degree hyperbola", seem to be generally associated with a different function than i am presenting : y = constant / (x squared) . That association seems to rest on the observation that the function y = constant / x describes a second degree hyperbola, (which it does, though with limitations, and in what i consider a non-standard orientation) . I find hyperbolae as i am accustomed to deriving them : complex root = (constant ^ degree - ellipticalcompression * x ^ degree) ^ (1 / degree), more complete mathematically, as they exist in complex space, with one or more 'circles' bridging their 'aperture' . In part because of this, in the second degree, (that for which the two methods agree), i find them more beautiful in form . Further, i believe "n-degree hyperbolas", (my usage), by this method to be a broader, more flexible class of functions ; though one to answer different, (natural and mathematical), questions . Still, my use of the terms, "third degree hyperbola", (and "n-degree hyperbola", where 'n' is any number), is non-standard . Please do not confuse this form with the function y = 1 / (x squared) ; (or with the function y = a*x^3 + b*x^2 + c*x + d, where i have also seen the term applied) ; they are not the same . |
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