Epsilon=One
08-11-2005, 03:27 PM
The Conceptual Ellipse (CE)
The significance of Conceptual Ellipses (CE) is that
they establish the Elliptical Constant (www.101123.com/EC) (EC) (epsilon = One)
for every elliptical shape;
and,
all the salient structural parts of
each Conceptual Ellipse (CE) are
related by simple equations.
(If no image appears below, "Click" your browser "Refresh" icon.)
http://b.g2d.us/celeg660p.gif
Legend for Conceptual 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 hypotenus 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 (www101123.com/BT).
c² = 2v² – s²
A unique concept of Conceptual Ellipses
is referred to as the Pulse, "P."
It is the Pulse that generates Conceptual Ellipses
from three points on a straight line: the center and the endpoints.
The Pulse, "P," can be any value that is greater than epsilon, the Elliptical Constant (www.101123.com/EC), which equals One, "1."
If the Pulse, "P," is greater than Two, 2, and is the perigee, "p," or the Pulse, "P," is less than Two, 2, and is the soliton, "s," the CE is obtuse; and, the equation for the radius, r, is: r = 2P - e; and, the equation for the hypotenuse, h, is: h = w + e.
If the Pulse, "P," is less than Two, 2, and is the perigee, "p," or the Pulse, "P," is greater than Two, 2, and is the soliton, "s," the CE is acute; and, the equation for the radius, r', is: r' = 2P - e; and, the equation for the hypotenuse, h', is: h' = w + e.
For an equilateral CE the perigee, "p," equals the soliton, "s"; thus, both are equal to the Prime, "P"; and the wave, "w," = the vector, "v." Thus, the equations for the radius and hypotenuse of either the acute or obtuse CE are interchangeable.
A Conceptual Ellipse (CE) is any elliptical shape with the vector, "v," the square of the Pulse, "P." Then, the Elliptical Constant (www.101123.com/EC) (EC), which is epsilon equals One (e = 1) is the Pulse, "P," minus the Key, "K."
A Conceptual Ellipse (CE) has the major radial, "m" (which equals the vector, "v") minus the Pulse, "P," equal to the Natural function (NF) (www.101123.com/NF).
The Natural function (www.101123.com/NF) (NF) is the Pulse squared minus the Pulse. (P² - P)
The Natural function (www.101123.com/NF) (NF) of an acute CE equals the pergee (p); the Natural function (www.101123/NF) (NF) of an obtuse CE equals the soliton, "s". For an equilateral CE the perigee, "p," equals the soliton, "s"; therefore either equals the NF.
Every Conceptual Ellipse (CE) has the same algebraic ratio between respective internal parts of each elliptical shape. That is, for example, for every CE: the vector, "v," equals the Elliptical Constant (www.101123.com/EC) (EC) times the Pulse, "P," squared.
v = eP²
(A Conceptual Ellipse).
There are five types of Conceptual Ellipses (CE): two are regular, referred to as acute and obtuse; three are special, referred to as equilateral, a line, and a circle.
All five types of CEs can be categorized by the elliptical angle, BHD, that is the intersection of two vectors with the amplitude and the ellipse's locus. An acute CE has an elliptical angle, BHD, less than 60 degrees; an obtuse CE has an elliptical angle, BHD, greater than 60 degrees; and, an equilateral CE has an elliptical angle of exactly 60 degrees; a line has an elliptical angle, BHD of 180 degrees; and, a circle has no elliptical angle, BHD. (The length of the vector, "v," equals the amplitude, "a") A line is an ellipse with the foci infinitely distant from one another, while a circle is an ellipse with the foci congruent. An equilateral ellipse has the vectors, "v," equal to the wave, "w," such that an equilateral triangle is formed.
If the Pulse, P, is a Natural integer (www.101123.com/NI) greater than One, a Conceptual Ellipse (CE) is referred to as a Brunardot Ellipse (www.101123.com/BE) (BE).
(If no image appears below, "Click" your browser "Refresh" icon.)
http://b.g2d.us/ceformfull.gif
The Equilateral Ellipse can use either the Acute Ellipse or the Obtuse Ellipse Formulas.
To distinguish acute ellipse symbols from obtuse ellipse symbols, an accent mark (p' s' v' w' etc.) or a subscript lowercase "a" can be used with the acute ellipse symbols.©Copyright 2005-2008 by Brunardot. All rights reserved.
Terms: PhysicsMathForums.com, Brunardot, and Pulsoid Theory must be cited.
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with the requested information.Please ask questions. :)With questions it’s possible to know if
comments are logical and convincing;
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The significance of Conceptual Ellipses (CE) is that
they establish the Elliptical Constant (www.101123.com/EC) (EC) (epsilon = One)
for every elliptical shape;
and,
all the salient structural parts of
each Conceptual Ellipse (CE) are
related by simple equations.
(If no image appears below, "Click" your browser "Refresh" icon.)
http://b.g2d.us/celeg660p.gif
Legend for Conceptual 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 hypotenus 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 (www101123.com/BT).
c² = 2v² – s²
A unique concept of Conceptual Ellipses
is referred to as the Pulse, "P."
It is the Pulse that generates Conceptual Ellipses
from three points on a straight line: the center and the endpoints.
The Pulse, "P," can be any value that is greater than epsilon, the Elliptical Constant (www.101123.com/EC), which equals One, "1."
If the Pulse, "P," is greater than Two, 2, and is the perigee, "p," or the Pulse, "P," is less than Two, 2, and is the soliton, "s," the CE is obtuse; and, the equation for the radius, r, is: r = 2P - e; and, the equation for the hypotenuse, h, is: h = w + e.
If the Pulse, "P," is less than Two, 2, and is the perigee, "p," or the Pulse, "P," is greater than Two, 2, and is the soliton, "s," the CE is acute; and, the equation for the radius, r', is: r' = 2P - e; and, the equation for the hypotenuse, h', is: h' = w + e.
For an equilateral CE the perigee, "p," equals the soliton, "s"; thus, both are equal to the Prime, "P"; and the wave, "w," = the vector, "v." Thus, the equations for the radius and hypotenuse of either the acute or obtuse CE are interchangeable.
A Conceptual Ellipse (CE) is any elliptical shape with the vector, "v," the square of the Pulse, "P." Then, the Elliptical Constant (www.101123.com/EC) (EC), which is epsilon equals One (e = 1) is the Pulse, "P," minus the Key, "K."
A Conceptual Ellipse (CE) has the major radial, "m" (which equals the vector, "v") minus the Pulse, "P," equal to the Natural function (NF) (www.101123.com/NF).
The Natural function (www.101123.com/NF) (NF) is the Pulse squared minus the Pulse. (P² - P)
The Natural function (www.101123.com/NF) (NF) of an acute CE equals the pergee (p); the Natural function (www.101123/NF) (NF) of an obtuse CE equals the soliton, "s". For an equilateral CE the perigee, "p," equals the soliton, "s"; therefore either equals the NF.
Every Conceptual Ellipse (CE) has the same algebraic ratio between respective internal parts of each elliptical shape. That is, for example, for every CE: the vector, "v," equals the Elliptical Constant (www.101123.com/EC) (EC) times the Pulse, "P," squared.
v = eP²
(A Conceptual Ellipse).
There are five types of Conceptual Ellipses (CE): two are regular, referred to as acute and obtuse; three are special, referred to as equilateral, a line, and a circle.
All five types of CEs can be categorized by the elliptical angle, BHD, that is the intersection of two vectors with the amplitude and the ellipse's locus. An acute CE has an elliptical angle, BHD, less than 60 degrees; an obtuse CE has an elliptical angle, BHD, greater than 60 degrees; and, an equilateral CE has an elliptical angle of exactly 60 degrees; a line has an elliptical angle, BHD of 180 degrees; and, a circle has no elliptical angle, BHD. (The length of the vector, "v," equals the amplitude, "a") A line is an ellipse with the foci infinitely distant from one another, while a circle is an ellipse with the foci congruent. An equilateral ellipse has the vectors, "v," equal to the wave, "w," such that an equilateral triangle is formed.
If the Pulse, P, is a Natural integer (www.101123.com/NI) greater than One, a Conceptual Ellipse (CE) is referred to as a Brunardot Ellipse (www.101123.com/BE) (BE).
(If no image appears below, "Click" your browser "Refresh" icon.)
http://b.g2d.us/ceformfull.gif
The Equilateral Ellipse can use either the Acute Ellipse or the Obtuse Ellipse Formulas.
To distinguish acute ellipse symbols from obtuse ellipse symbols, an accent mark (p' s' v' w' etc.) or a subscript lowercase "a" can be used with the acute ellipse symbols.©Copyright 2005-2008 by Brunardot. All rights reserved.
Terms: PhysicsMathForums.com, Brunardot, and Pulsoid Theory must be cited.
Sorry! This Thread has not been completed.
Please Bookmark and return to this site often.
If there is an immediate need for information,
please e-mail directly at the below "Click" link.
Please note that any private correspondence
may be edited and anonymously posted unless
requested otherwise.
Every effort will be made to expedite a reply
with the requested information.Please ask questions. :)With questions it’s possible to know if
comments are logical and convincing;
or whether clarification is required.
http://1.g2d.us/e.gifhttp://2.g2d.us/e.gifhttp://3.g2d.us/e.gifhttp://4.g2d.us/e.gifhttp://5.g2d.us/e.gif
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If images don’t display, "click" the Refresh Icon.