The Change Depends on the Direction of the Motion: The Symmetric Group Action (2)

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) 

 

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 = 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….