The Brunardot Ellipse

[Epsilon=One]

The Brunardot Ellipse (BE)

The Brunardot Ellipse (BE) is a Conceptual Ellipse (www.101123.com/CE) (CE) when the Pulse (www.101123.com/Pul), P, is a Natural integer (www.101123.com/NI) greater than One.

Therefore, for every value of the Pulse, P, that is a Natural integer (www.101123.com/NI):the apogee, "o," hypotenuse, "h," hypotenuse radius, "Hr,"
key, "K," major diameter, "M," major radial, "m," perigee, "p,"
radius, "r," soliton, "s," vector, "v," wave, "w," and the square of
the diameter chord, "c," are, also, Natural integers (www.101123.com/NI).(If no image appears below, "Click" your browser "Refresh" icon.)
http://b.g2d.us/ce2-3-710.gif

Legend for Brunardot Acute and Obtuse Ellipses (CE)

Points B and D are the ellipses' foci.
Angles ADG and EBJ are right angles.
Angle BAD is the radius angle.
................(The angle between the hypotenuse and radius.)
Angle BHD is the elliptical angle.
................(The angle between two vectors.)
Angles CAD and CJB are the diagonal angles.
................(The angle between the radial diagonal and the radius.)
Angles CBH and CDH are the vector angles.
................(The angle between the vector and amplitude.)

Acute ellipse = elliptical angle BHD less than 60 degrees.
Equilateral ellipse = elliptical angle BHD = 60 degrees. (vector = wave)
Obtuse ellipse = elliptical angle BHD greater than 60 degrees.

Line AC = Line CJ = diagonal radial = d
Line AD = BJ = radius = r
Line AJ = diagonal = D
Line BA = hypotenuse = h
Line BC = Line CD = soliton = s
Line BD = wave = w
Line BE = Line DG = perigee = p
Line BG = Line DE = apogee = o
Line CE = Line CG = major radial = m = vector v
Line CF = Line CH = amplitude = a
Line BH = Line DH = vector = v = major radial = m
Line EF = diameter chord = c
Line EG = major diameter = M
Line FH = minor diameter = L
Line oK = Key = K = Hr = radius of hypotenuse circle inscribed in triangle ABD
Triangle ABD = hypotenuse right triangle
The Pulse, P, is either the perigee or soliton; and, the Pulse, P, is greater than One, 1.
The ellipse is acute if:
................the Pulse, P, is less than Two, 2, and is the perigee.
The ellipse is acute if:
................the Pulse, P, is greater than Two, 2, and is the soliton.
The ellipse is obtuse if:
................the Pulse, P, is less than Two, 2, and is the soliton.
The ellipse is obtuse if:
................the Pulse, P, is greater than Two, 2, and is the perigee.
The ellipse is equilateral if the Pulse, P, is Two, 2.
vector = Pulse squared. (v = P²)
Elliptical Constant (EC) = K – P = epsilon = e = One
For every ellipse: the vector equals the major radial.

v = m;

and,

For every ellipse: the perigee, soliton, vector, and the apogee are the first terms of a Brunardot Series sequence.
p + s = v; s + v = o;

and,

For every ellipse: the diameter chord squared equals twice the vector squared minus the soliton squared, which is the The Brunardot Theorem (www.101123.com/BT):

c² = 2v² – s²;

and, a corollary:

c = P(Square root(r + v)).



The hypotenuse radius, referred to as the key, "K," equals the Pulse minus one; or, the radius minus the perigee . . . or, the apogee minus the hypotenuse.

Hr = P – 1; or,
r – p; or,
o – h

........................One must ask:
Why ?
And, again,
Why ?
And, again,
Why ?
Over and over,
until . . . Infinity.
Primary Pythagorean triangles (www.CQthus.com/PTr) can be easily mapped to every Natural integer (www.101123.com/NI) by Brunardot Ellipses (BE):

........If p = any Natural integer (www.101123.com/NI) = perigee EB
Pythagorean triangle (www.CQthus.com/PTr) is:
............altitude = 2p – 1
............base = 2(p² – p)
............hypotenuse = 2(p² – p) + 1

There are many simple corollaries such as:

........If r = any odd Natural integer (www.101123.com/NI) = radius AD
Pythagorean triangle (www.CQthus.com/PTr) is:
............altitude = r
............base = (r² – 1)/2
............hypotenuse = (r² + 1)/2

The Brunardot Ellipse (BE) is, heuristically, associated with the Congeneric Realm of Coalescence (www.101123.com/CRR).

If the amplitude, a, is a Natural integer (www.101123.com/NI) , a Brunardot Ellipse (BE) is referred to as a Pulsoidal Ellipse (www.101123.com/PE)[/URL] (PE).

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[ste]

How did you derive these ellipses, and determine what each of the lines represented? How can you connect these values with what is observed in reality? Can natural phenomenon be shown to coincide with the values of these ellipses, and if so, how?

[Epsilon=One]

How did you derive these ellipses, and determine what each of the lines represented? How can you connect these values with what is observed in reality? Can natural phenomenon be shown to coincide with the values of these ellipses, and if so, how?All ellipsoids are natural phenomena, which is referred to as Pulsoids (www.CQthus.com/PT/P).

Conceptual Ellipses (www.CQthus.com/PT/CE) (CE) (Brunardot Ellipses (BE) are special CEs) are two dimensional, heuristic representations that are derived from ellipsoids. They represent the orthogonal phenomena that is created from Dynamic, Emergent Separation (www.CQthus.com/PT/DES) (DES).

The major structural lines of CEs are vectors that are referred to as of Dynamic, Hyper-relativistic Vectors (www.CQthus.com/PT/DHV) (DHV).

[ste]

I get the impression that "Pulse" can be either soliton, or perigree. What exactly is the Pulse, and what determines whether it is a soliton or perigree?

[Epsilon=One]

I get the impression that "Pulse" can be either soliton, or perigree. What exactly is the Pulse, and what determines whether it is a soliton or perigree?The Pulse, P, represents symbolically the motion from Infinity (a radius of One) of an Emergent Ellipsoid (www.EmergentEllipsoid.com).

In any ellipse (heuristically describing in two dimensions a quantum) the Pulse is the distance of motion along the major diameter from its center (the infinitesimal) AND its ends (the infinite) which symbolically represents the motion of the described quantum; thus, motion from three points, which actual, simultaneous motion (when pulsing) is described as Triquametric Motion (www.CQthus.com/PT/TM) (TM).

The portion of the major radial that is not the Pulse is referred to as the Natural function (www.CQthus.com/PT/NF) (NF). As to whether the Pulse is the perigee or soliton, of an Emergent Ellipse, depends upon whether the Pulse exceeds twice the radius of Infinity (www.CQthus.com/PT/ROI) (ROI).

[ste]

The portion of the major radial that is not the Pulse is referred to as the Natural function (NF). As to whether the Pulse is the perigee or soliton, of an Emergent Ellipse, depends upon whether the Pulse exceeds twice the radius of Infinity (ROI).
The above statements seem to conflict with themselves in that, up until now, I've understood the soliton to be a representation of the natural function. At least, in terms of perigree (p). If both were true, the perigree would need to be the natural function when the Pulse is a soliton. Are you are referring to the Natural Function using different terms?

Let me get one thing straight. Can emergent ellipsoids (and by extention, everything that exists) be considered the resultant of complex (triquametric) motion emanating from three points?

[Epsilon=One]

The above statements seem to conflict with themselves in that, up until now, I've understood the soliton to be a representation of the natural function. At least, in terms of perigree (p).The soliton and perigee transpose for the obtuse and acute ellipses of an Emergent Ellipse, and depend upon whether the Pulse is greater than or less than two. If the Pulse is exactly two, the soliton and perigee are equal.

If both were true, the perigree would need to be the natural function when the Pulse is a soliton. Are you are referring to the Natural Function using different terms?Yes. The soliton and perigee are dependent upon shape. The Natural Function and Pulse depend upon direction of motion; thus, the terms are not directly related.

Can emergent ellipsoids (and by extention, everything that exists) be considered the resultant of complex (triquametric) motion emanating from three points?Yes, heuristically, in two dimensions. More accurately, in three dimensions (there are more), emanating from a point to and from all points on the inner surface of a spheroid. When considering the pulse, changes in time and shape, three forms of oscillation, etc., Triquametric Motion (www.CQthus.com/PT/TM) is extremely complex. Rather than using a paragraph or more of decription, I've coined the term, Triquametric Motion.