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).r*

*1*

*.r*

*2*

*and │V│min = 0*

*Theorem (4): │V│max = r*

*2*

*.max (z, r*

*1*

*)*(2^0.5) and*

*│V│min = r*

*2*

*.min (z, r*

*1*

*)*(2^0.5)*

*Where:*

*r*

*1*

*= radius*

*in operator or transformation matrix*

*r*

*2*

*= 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).r**1**.r**2*

*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).r*

*1*

*.r*

*2*

*Where:*

*r*

*1*

*= radius*

*in transformation matrix M*

*r*

*2*

*= 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 = r*

*1*

*.r*

*2*

*/ 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).r*

*1*

*.r*

*2*

*Where:*

*R = radius of the sphere*

*r*

*1*

*= radius*

*in transformation matrix M*

*r*

*2*

*= 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 r*

*1*

*/z > 2^0.5 Then │V│max = 0.5. (6^0.5).r*

*1*

*.r*

*2*

*and*

*│V│min = (3^0.5).r*

*2*

*.min (z, r*

*1*

*)*

*If r*

*1*

*/z < 2^0.5 Then │V│max = (3^0.5).r*

*2*

*.max (z, r*

*1*

*) and*

*│V│min = 0.5. (6^0.5).r*

*1*

*.r*

*2*

*Theorem (10): If r*

*1*

*/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).r*

*1*

*.r*

*2*

*or*

*R = (3^0.5).r*

*2*

*.min (z, r*

*1*

*)*

*Where:*

*R = radius of the sphere*

*r*

*1*

*= radius*

*in operator M*

*r*

*2*

*= radius of each point in the space (3D) in accordance with its polar coordinates*

*Theorem (11): If r*

*1*

*/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 r*

*1*

*/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….**