Know thyself and awaken to the warrior god within you
Stand erect and move through the world with a gentle hand and ease of passage
For you are the Kings of a long forgotten era of Kings
Proud in form I move through this world with dignity and grace
For it is the divine birthright of my embodied being
and the Responsibility of my Godhood
In my mind I am quite
In my heart I am Alive
I am. I am Alive.
In my soul I am ablaze in the radiance of the God flame.
To be is to be. There is no question.
There is no escaping it.
There is no understanding it.
To do so is to mistake the map for the territory,
the finger for the moon.
You can’t eat a picture of delicious food,
all it does is make you more hungry.
Is is, from now within eternity.
In you and me are the rocks and the trees.
Last night I dies a man. This morning I am reborn as god.
In the experience of the I am there is nothing
temporal because there is never a beginning.
No search. No end.
There is only is.
Arise alive my soul rises over the earth
Arise alive wings on my back, light on foot
Whatever does an Angel’s footprint look like
A windswept kiss upon a child’s rose taut cheeks
Whispering forgiveness upon the earls of the condemned
Or a blessing of death to a wounded heart
Murmering eyes of tattered love
Angels be near
Whisper in my ear
So that I might hear
Your eternal voice
Calling me closer
Forever beckoning me
Into the arms of the Lord
Calling me home to the ground of my creation
The Salivation of my Salvation
The eternal expression of the I am presence
That is you. That is me.
That is your divine exhalation in manifest form.
May I know this today in the moment
And awaken unto you.
Amen. Amien. I am all beings.
Am I? I am.
Am I? I am.
Am I? I am.
Am I? I am.
Am I-I am.
Am I, I Am?
Am I, I Am.
Am I I Am.
AmIIAm.
Am I-I Am.
Am I, I Am.
Am I? I Am.
Am I? I Am.
Am I? I Am.
Am I? I Am.
Am I? I Am.
Am I? I Am.
Am I, I Am?
All One
Alone
We are ultimately alone
I feel alone
Alone, but infinitely connected
Alone. All One
One, why does it fucking have to be one
It is so goddam l-one-ly, lonely, being just one
One in the infinite space that is my being
Alone in the darkenss
Alone in the infinite space that presides in my being, that is my being
Never ending. Never really beginning
All one in the darkness.
A poet with his pen sets mind afire
Blazing the houses of forgotten slumber
In the brilliant allure of truth’s caress
And twilight’s release.
Geometry > Plane Geometry > Circles >
Number Theory > Constants > Transcendental Root Constants >
Interactive Entries > Interactive Demonstrations >
Circle-Circle Intersection
Two circles may intersect in two imaginary points, a single degenerate point, or two distinct points.
The intersections of two circles determine a line known as the radical line. If three circles mutually intersect in a single point, their point of intersection is the intersection of their pairwise radical lines, known as the radical center.
CircleCircleIntersection
Let two circles of radii R and r and centered at (0,0) and (d,0) intersect in a region shaped like an asymmetric lens. The equations of the two circles are
x^2+y^2 = R^2
(1)
(x-d)^2+y^2 = r^2.
(2)
Combining (1) and (2) gives
(x-d)^2+(R^2-x^2)=r^2.
(3)
Multiplying through and rearranging gives
x^2-2dx+d^2-x^2=r^2-R^2.
(4)
Solving for x results in
x=(d^2-r^2+R^2)/(2d).
(5)
The chord connecting the cusps of the lens therefore has half-length y given by plugging x back in to obtain
y^2 = R^2-x^2=R^2-((d^2-r^2+R^2)/(2d))^2
(6)
= (4d^2R^2-(d^2-r^2+R^2)^2)/(4d^2).
(7)
Solving for y and plugging back in to give the entire chord length a=2y then gives
a = 1/dsqrt(4d^2R^2-(d^2-r^2+R^2)^2)
(8)
= 1/dsqrt((-d+r-R)(-d-r+R)(-d+r+R)(d+r+R)).
(9)
This same formulation applies directly to the sphere-sphere intersection problem.
To find the area of the asymmetric “lens” in which the circles intersect, simply use the formula for the circular segment of radius R^’ and triangular height d^’
A(R^’,d^’)=R^(‘2)cos^(-1)((d^’)/(R^’))-d^’sqrt(R^(‘2)-d^(‘2))
(10)
twice, one for each half of the “lens.” Noting that the heights of the two segment triangles are
d_1 = x=(d^2-r^2+R^2)/(2d)
(11)
d_2 = d-x=(d^2+r^2-R^2)/(2d).
(12)
The result is
A = A(R,d_1)+A(r,d_2)
(13)
= r^2cos^(-1)((d^2+r^2-R^2)/(2dr))+R^2cos^(-1)((d^2+R^2-r^2)/(2dR))-1/2sqrt((-d+r+R)(d+r-R)(d-r+R)(d+r+R)).
(14)
The limiting cases of this expression can be checked to give 0 when d=R+r and
A = 2R^2cos^(-1)(d/(2R))-1/2dsqrt(4R^2-d^2)
(15)
= 2A(1/2d,R)
(16)
when r=R, as expected.
Circle-CircleIntersectionHalf
In order for half the area of two unit disks (R=1) to overlap, set A=piR^2/2=pi/2 in the above equation
1/2pi=2cos^(-1)(1/2d)-1/2dsqrt(4-d^2)
(17)
and solve numerically, yielding d=0.8079455… (OEIS A133741).
Circle3Intersection
If three symmetrically placed equal circles intersect in a single point, as illustrated above, the total area of the three lens-shaped regions formed by the pairwise intersection of circles is given by
A=pi-3/2sqrt(3).
(18)
Circle4Intersection
Similarly, the total area of the four lens-shaped regions formed by the pairwise intersection of circles is given by
A=2(pi-2).
(19)
SEE ALSO:
Borromean Rings, Brocard Triangles, Circle-Ellipse Intersection, Circle-Line Intersection, Circular Segment, Circular Triangle, Double Bubble, Goat Problem, Johnson’s Theorem, Lens, Lune, Mohammed Sign, Moss’s Egg, Radical Center, Radical Line, Reuleaux Triangle, Sphere-Sphere Intersection, Steiner Construction, Triangle Arcs, Triquetra, Venn Diagram, Vesica Piscis
REFERENCES:
Sloane, N. J. A. Sequence A133741 in “The On-Line Encyclopedia of Integer Sequences.”
This article is directly from the link below and will be referenced throughout this site as mathematical constants in the conversation between Circles. Weisstein, Eric W. “Circle-Circle Intersection.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Circle-CircleIntersection.html
