Following to article of “The Change
Depends on the Direction of the Motion: The Symmetric Group Action (1)” posted
on link: https://emfps.blogspot.com/2017/08/the-change-depends-on-direction-of.html, the purpose of this article is to introduce some properties of
operators and transformations which are formed by moving three points on circle
and sphere.
But, before starting of this article, let me tell you more
explanations about theorems mentioned in previous article as follows:
1. All theorems in previous article
denote to get the maximum and minimum for vectors in all directions but if we need
to have the maximum and minimum of each point on surface or space exchanged by
operators and transformations, all equations should be changed as follows:
Theorem (1): │V│max
= (2^0.5).r1.r2 and │V│min = 0
Theorem (4): │V│max
= r2.max (z, r1)*(2^0.5) and
│V│min = r2.min (z, r1)*(2^0.5)
│V│min = r2.min (z, r1)*(2^0.5)
Where:
r1 = radius in operator or
transformation matrix
r2 = radius of
each point on surface or space in accordance with its polar coordinates
2. In Theorem (4),
if z = r1 then we can say this transformation matrix maps a random point on
surface to a random point on a sphere with radius equal to:
R = (2^0.5).r1.r2
R = (2^0.5).r1.r2
R = radius of
the sphere
An operator or transformation matrix formed by three points on
circle
Suppose three points on a circle are
rotating in which the distance among all three points are the same and equal
just like below figure:
For reaching to above conditions,
below polar coordinates for each point should be established:
A:
x = r cos θ
y = r sin θ
B:
x = - r sin (θ + 30)
y = r cos (θ +30)
C:
x = - r sin (30 – θ)
y = - r cos (30 – θ)
By considering any random number for
“r” and “θ”, you can see not only all distances are equal but also all three
points are on a circle.
Example:
r = 23 and θ
= 41 degree
AO = BO = CO = 23
AB = BC = CA = 39. 83716857
Above polar coordinates give us a
transformation matrix 3*2 as follows:
The properties
of transformation matrix 3*2 for R^3 to R^2
By multiplying matrix M by any 3D
vectors in the space, we can extract the properties of this transformation
matrix as follows:
Theorem (6):
The maximum magnitude among 2D vectors produced by three point’s transformation
matrix M is calculated by using below equation:
│V│max = 0.5. (6^0.5).r1.r2
Where:
r1 = radius in
transformation matrix M
r2 = radius of
each point in the space (3D) in accordance with its polar coordinates
The minimum
magnitude is obtained by using below equation:
│V│min = r1.r2 / Ф
Ф = the
constant coefficient equal to 176.943266509085
Here is a very interesting property:
Theorem (7): Always
there are six points or six 2D vectors produced by three point’s transformation
matrix M which give us the maximum magnitude while there is only one point or one
2d vector which gives us the minimum magnitude in which the direction of all
points or 2D vectors is between 0 degree to 180 degree.
The property of
transformation matrix 2*3 for R^2 to R^3
It is transpose of above matrix in
which we will have below matrix:
Theorem (8): This
transformation matrix maps a random point on surface to a random point on a sphere
with radius equal to: R = 0.5. (6^0.5).r1.r2
Where:
R = radius of
the sphere
r1 = radius in
transformation matrix M
r2 = radius of
each point on the surface (2D) in accordance with its polar coordinates
The properties
of an operator 3*3
If we want to study these three
points in 3D space rotating on circle or sphere, we will have an operator 3*3.
In this case, there are several statements where I have started three forms as
follows:
1. I added a constant coordinate (z) for each point and matrix will
be:
Theorem (9): Maximum
and minimum magnitudes among 3D vectors produced by operator M are calculated
by using below equations and conditions:
If r1 /z >
2^0.5 Then │V│max = 0.5. (6^0.5).r1.r2 and
│V│min = (3^0.5).r2 .min (z, r1)
If r1 /z <
2^0.5 Then │V│max = (3^0.5).r2 .max (z, r1) and
│V│min = 0.5. (6^0.5).r1.r2
Theorem (10):
If r1 /z = 2^0.5 then this operator maps a random point in the space to a
random point on a sphere with radius equal to:
R = 0.5. (6^0.5).r1.r2 or
R = (3^0.5).r2 .min (z, r1)
Where:
R = radius of
the sphere
r1 = radius in operator M
r2 = radius of
each point in the space (3D) in accordance with its polar coordinates
Theorem (11): If r1 /z >
2^0.5 Then, Always there are six points in the space or
six 3D vectors produced by three point’s operator M which give us the maximum
magnitude while there is only one point or one 3D vector which gives us the
minimum magnitude.
If r1 /z <
2^0.5 Then, Always there are six points in the space or
six 3D vectors produced by three point’s operator M which give us the minimum
magnitude while there is only one point or one 3D vector which gives us the
maximum magnitude.
2. I replaced the constant coordinate (y) instead of (z):
The properties of this operator are
similar to theorems (9), (10) and (11).
3. Suppose that three points in the space which have the same distance
are rotating on a sphere. In this case, we will have below polar coordinates
for all three points as follows:
Point A:
x = r * cos β * cos θ
y = r * cos β * sin θ
z = r * sin β
Point B:
x = -r * cos (60 – β) * cos θ
y = -r * cos (60 – β) * sin θ
z = r * sin (60 – β)
Point C:
x = -r * cos (60 + β) * cos θ
y = -r * cos (60 + β) * sin θ
z = -r * sin (60 + β)
Example:
Assume θ = 56, β =
17 and r = 31
According to above
coordinates, we can calculate the distance among points and also the distance
between points and center of sphere which answers are:
AO = BO = CO = 31
AB = BC = CA =
53.69358
The properties
of operator 3*3
Above coordinates of points A and B
and C make an operator 3*3 as follows:
Note: θ and β
have been introduced in my previous article of “The Change Depends on the Direction of the Motion: Generating All
Directions in 3D Space” posted on link: https://emfps.blogspot.com/2017/05/the-change-depends-on-direction-of.html
The property of this transformation
matrix is similar to theorems (6) and (7).
An operator or transformation matrix formed by four points on
circle
To be
continued….
