angle subtending arc and chord

[Jon]

It looks like the angle subtending arc A and chord B on a circle is,

ANGLE = 2*pi*[ (1/0!)(B/A)^0 -(1/1!) (B/A)^1 + (1/2!)(B/A)^2 - (1/3!)(B/A)^3
+ ... ] = 2pi*e^(-B/A)

since 3 terms into the recursive Maclaurin is always +/- 2pi.

I don't have time to check it out. I have to go to work. Anyone interested
look into it reply to this post or get back to me, ***@bellaire.tv

Later I'll draw a graph. see if these two graphs are the same:

y = - ln|x/(2pi)|
y =(2/x)sin(x/2)

[Ken Pledger]

Post by Jon
It looks like the angle subtending arc A and chord B on a circle is,
ANGLE = 2*pi*[ (1/0!)(B/A)^0 -(1/1!) (B/A)^1 + (1/2!)(B/A)^2 - (1/3!)(B/A)^3
+ ... ] = 2pi*e^(-B/A)
....

You can see that your formula must be wrong, by looking at special
cases. For example, a semicircle has B/A = 2/pi, but the subtending
angle pi is not equal to 2pi*e^(-2/pi).

A bit of trigonometry shows that you need to solve for t the equation

(sin(t/2))/t = B/2A.

That's not simple. For particular values of A and B you'd need to
tackle it numerically.

Ken Pledger.

[Jon]

What is the angle x subtending arc A and chord B on a circle?

radius = A/x

(Bx/2A) = sin (x/2)

I surrender. This is my best answer. Sorry for all the complicated mess,
but that's a recursive Maclaurin Series for you. I carried out the
derivatives 4 times.

http://jons-math.bravehost.com/angsub.html

Jon

[Jon]

Reciprocal Bases in 4-Space:

http://jons-math.bravehost.com/reciprocal4.html

Where time is a row or column of the determinant.

[Jon]

The reciprocal basis is supposed to facilitate deducing the contravariant
and covariant components of a vector. Part of these calculations are to
find the volume of a parallelpiped formed by 3 basis vectors in 3-space.
Then what is the 4-space equivalent? 4 basis vectors and the hypervolume of
a hyperparallelpiped? At any rate, I added to my page to touch on these
issues.

http://jons-math.bravehost.com/reciprocal4.html

[William Elliot]

Post by Jon
What is the angle x subtending arc A and chord B on a circle?
radius = A/x
(Bx/2A) = sin (x/2)

A central angle of t radians in a circle with radius r
subtends an arc length
s = rt
with a cord length
c = 2r.sin t/2.

Accordingly
ct/2s = (2rt.sin t/2)/2rt = sin t/2

Solve for t either by numerical approximation
or use the first few terms of the Taylor series
for sin and solve the resulting quadratic or cubic
equation for an approximation.

[Jon]

1 US fluid gallon = 231 cubic inches.

Particularly on a hypercube, if P(x,y,z) points to all points on the
surfaces, edges and corners of a 3-dimensional cube, then inverting it about
a sphere of radius r/2 results in,

P'(x',y',z') = [ P(x,y,z)*r/|P(x,y,z)| - P(x,y,z) ] where r/2 =
sqrt((sqrt(s^2+s^2))^2 + s^2) , the greatest extremity of a cube from the
center. Then

r = 2*s*sqrt(3) where s = the length of one side of the cube in 3-space.
Then

P'(x',y',z') = [ P(x,y,z)*2*s*sqrt(3)/|P(x,y,z)| - P(x,y,z) ]

One side of the 3-dimensional cube intersects the x-axis at, (s*cos(pi/4),
0, 0) The equation of the plane for that side is x = s*sqrt(2)/2 so
P(x,y,z)
so the position vector of that surface is, (s*sqrt(2)/2, y, z) and its
inversion is,

P(x',y',z') = [ (s*sqrt(2)/2, y, z)[ *2*s*sqrt(3)/sqrt((1/2)*s^2 + y^2 +
z^2) - (s*sqrt(2)/2, y, z) ] since |(s*sqrt(2)/2, y, z)|=sqrt((1/2)*s^2 +
y^2 + z^2)

.... the transformation of coordinates for one surface of a cube to
hypercube. By the same token,

P(x,y,z) = [ P(x',y',z')*r/|P(x',y',z')| - P(x',y',z') ] the inverse
transformation for the hypercube back to the cube.

[Jon]

Look at the animated graph on the homepage to my site:

http://mypeoplepc.com/members/jon8338/math/

The orbitals invert into an oscillating hyper-ellipse
- or -
The oscillating ellipse inverts into hyper-orbitals

Can you find the electrons at the center?

No. Because they have hit a brick wall due to the impermeability of space.
The only thing it allows is an expression with waves. So an electron is a
small mass forced to be a wave. All s,p,d,f and other orbitals transform
through the reverse transformation into spherical harmonic "hyperdynamics"
because it is understood. While several waves can occupy the same node at
the same time, it is not so for several masses.

[Jon]

Conflict occurs when

*two masses are forced to occupy the same space at the same time (E=mc^2 is
dissipated)
*a wave is forced to accommodate an unorthodox number of nodes. (an odd
number of nodes for circular harmonics)

See a simple analysis of some examples of circular harmonics at my web page,

http://mypeoplepc.com/members/jon8338/math/id39.html

When a deep sea earthquake generates a tsunami, the dynamics initially have
random nodes, but average to the surface to create a destructive wavefront.
When circular harmonics are forced to oscillate through 7 nodes, they cancel
and amplify chaotically in noise. Mass impressed on mass expresses as
waves, and wave impressed on wave expresses as mass. The wave is denied
cancellation or amplification and is in a perpetual state of random chaos
(noise), like the random probabilistic location of an electron in an
s-orbital.

The energy of a wave is dissipated when it decreases the number of nodes or
when it goes from an unorthodox number of nodes to an orthodox number of
nodes (such as going from 7 to 6 nodes in circular harmonics) like a busted
banjo with random spacing of frets.