The Change Depends on the Direction of the Motion: The Angles in Shadow

In the reference with my article of “The Change Depends on the Direction of the Motion: Generating All Directionsin 3D Space”, before going to any analysis in physics and engineering subjects, you have to calibrate and filter your template by using the method mentioned in this article. In fact, I had to make the necessary amendments to above article because maybe you will take some mistakes in your analysis when you utilize the template stated in my previous article (Generating All Directions in 3D Space).

What is the case?

The case is to be produced many angles which show the same direction. I name these angles:

“The Angles in Shadow”

What is the meaning “The angles in Shadow”?

Let me tell you an example to illustrate this concept.

Assume you are fixing your telescope on four points in the space. First you turn your telescope on Horizontal angle 104 degree (θ = 104) and Vertical angle 253 (β = 253). Then you turn it on Horizontal angle 284 degree (θ = 284) and Vertical angle 287 (β = 287). Then you turn it on Horizontal angle 104 degree (θ = 104) and Vertical angle 287 (β = 287) and finally you turn it on Horizontal angle 284 degree (θ = 284) and Vertical angle 253 (β = 253). If you use the equations stated in my previous article, you can calculate all directions as follows:

 As you can see, you are really looking at two points instead of four points. In fact, state number 1 and 2 are in the same direction and state number 3 and 4 are in the same direction.

How can we find the angles in shadow?

Here I am willing to introduce to you two methods. The method (1), which uses some trigonometric equations while method (2) follows the same method mentioned in article of “Can We Solve a Nonlinear Equation with Many Variables?” posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with.html

At the first, we should bear in mind that below conditions should be established for both methods:

Condition (1): For Vertical angles,

 Sin β = Sin β'   and    Cos β = - Cos β'

Condition (2):  For Horizontal angles, 

Sin θ = - Sin θ'   and   Cos θ = - Cos θ'

 Method (1):

We as well as know Trigonometric reduction formulas which are as follows:

Sin (90 – α) = Cos α    
   
Cos (90 – α) = Sin α   

Sin (90 + α) = Cos α   

Cos (90 + α) = - Sin α

To obtain condition (1), I use below tricks:

Sin (90 – α) = Cos α = Sin β         

Cos (90 – α) = Sin α = Cos β

Sin (90 + α) = Cos α = Sin β'        

 Cos (90 + α) = - Sin α = - Cos β'

And so, we have below formulas in Trigonometric:

Cos (180 + α) = - Cos α             

Sin (180 + α) = - Sin α

To get condition (2), I also use below formulas:

Cos (180 + α) = - Cos α = - Cos θ'       

 Sin (180 + α) = - Sin α = - Sin θ'

θ = α    and   θ' = 180 + α

According to above relationships, I can write a simple algorithm to generate all direction including symmetry direction and others as follows:

As you can see, I have fixed angle of (α) and have copied and pasted all angles on green, red, blue and yellow colors that if you change only angle of (α), you can easily get all the same directions just like below examples:

You can find the results for all 360 degrees by using this algorithm and a sensitivity analysis between α and θ, θ', β, and β'.

Method (2):

If we apply all 360 degrees for algorithm method (1), we will have 180 states for θ and θ' and 181 states for β and β' where we can not find the angles mentioned in above example (θ = 104 and β = 253). It means that method (1) gives us incomplete results. But, by using the method mentioned in article of “Can We Solve a Nonlinear Equation with Many Variables?” posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with.html, we can generate the complete results as follows:

For condition (1), we have:  tg β = - tg β'

For condition (2), we have:  tg θ = tg θ'

For establishing condition (1), we have to solve below equation:

tg β + tg β' = 0

This is an equation with two independent variables which can be solved with the method mentioned in article of “Can We Solve a Nonlinear Equation with Many Variables?” posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with.html,.

For establishing condition (2), we have to solve below equation:

tg β - tg β' = 0

This is an equation with two independent variables which can be also solved with above method.

By using method (2), we will have 360 states for θ and θ' and 358 states for β and β' in which the angles mentioned in above example (θ = 104 and β = 253) are also included in our results just like below cited:

Finally, method (2) says to us that there are (360 * 357) / 2 = 64260 repetitions which should be deducted from the total 130322 states where we should take our analysis in accordance with 66062 states.

 By analysis of  the results extracted from the method (2), we can find the general formulas as follows:

θ' = θ +180    If     0 ≤ θ ≤ 180

   

β' = 180 – β   If     0 ≤ β ≤ 180

θ' = θ -180     If    180 ≤ θ ≤ 360   

β' = 540 – β   If    180 ≤  β ≤ 360

θ and β = degree

Example (1):

Suppose θ = 56 degree and β = 112 degree. According to above formulas, we should use below equations:

θ' = θ +180    If     0 ≤ θ ≤ 180

   

β' = 180 – β   If     0 ≤ β ≤ 180

Then we have:

θ' = 180 +56 = 236 degree

β' = 180- 112 = 68 degree

By using equations mentioned in 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, we will have below directions:

Direction V =(cos β * cos θ) i + (cos β * sin θ) j + (sin β) k

 For θ = 56   and    β = 112   , Direction V is:

Direction V = -0.20948 i -0.31056 j + 0.927184 k

For θ' = 236   and    β' = 68   , Direction V is:

Direction V = -0.20948 i -0.31056 j + 0.927184 k

You can see that the direction both of them are the same.

Example (2):

Suppose θ = 221 degree and β = 295 degree. According to above formulas, we should use below equations:

θ' = θ -180     If    180 ≤ θ ≤ 360   

β' = 540 – β   If    180 ≤  β ≤ 360

Then we have:

θ' = 221 - 180 = 41 degree

β' = 540 - 295 = 245 degree

Direction V =(cos β * cos θ) i + (cos β * sin θ) j + (sin β) k

 For θ = 221   and    β = 295   , Direction V is:

Direction V = -0.31895 i -0.27727 j – 0.90631 k

For θ' = 41   and    β' = 245   , Direction V is:

Direction V = -0.31895 i -0.27727 j – 0.90631 k

You can see that the direction both of them are the same.

In the reference with above formulas, we reach to a constant number equal to 64442 rows on spreadsheet of excel which shows us all directions in 3D space.

Conclusion

When we encounter the big data on our spreadsheet, the most crucial thing to bear in mind is to deduct the data in which there will not be any difference in our final results and analysis because it is possible that our laptop and computer will not be able to process the big data due to its technical characteristics.

The result of this article says to us that we can decrease 130321 rows on our spreadsheet to 64442 rows while the results and analysis will be finally the same.

Example:

Suppose you have a curve in 3D space as follows and the point P (x, y, z) wants to move from P0 (-1, 3, 2) on this curve toward all directions in amount of 4 unit (∆s = 4 unit). The question is:

What is the maximum change of below function when point P (x, y, z) moves in all directions? Which direction will the maximum change of the function occur?

Solution:

Suppose above function is a curve of heat equation to analyze thermal conduction in a room. Why do I say, assume this function is a curve of heat equation? Because this function follows the Laplace's equation.

Now, when I try above formula for “i” between 1 and 130321, the answer is the same.

The Images of The Universe

Here is the images of the Universe. Everything is the same

PS: “I think that it is not still the time to release all results around the world”

The Change Depends on the Direction of the Motion: Generating All Directions in 3D Space

In physics, science, and engineering, especially fluid dynamics and electromagnetism fields, we usually need to investigate the changes of a function in different directions. In this case, the best way for analyzing and designing is to have different directions all together in our hands, in which we will be able to compare all results to reach the new theorems and the new physical phenomena. The purpose of this article is to make a spreadsheet on an Excel file by using a new method where you will have all unit vectors in 3D space in different directions.

Let me give you an example:

Suppose you have a curve of f (x, y, z) = x^2 + y^2 - 2z^2 and the point P (x, y, z) wants to move from P0 (-1, 3, 2) on this curve toward all directions in amount of 4 unit (∆s = 4 unit). The question is:

What is the maximum change of the function when point P (x, y, z) moves in all directions? In which direction will the maximum change in the function occur?

Solution:

By making an analytic model in Excel, you will be able to answer the above questions. There is no need for any other software.

Unit Vectors in 2D and 3D

You as well as know, we can easily get all unit vectors in 2D surface by using bellow formula:

U = i cosθ + j sinθ

On the other hand, we can obtain the direction of a vector in 2D and 3D, by using below formula:

Direction of A = A / │A│

 But, how can we prove above formula?

A vector in 3D space can be modelled as the radius of a sphere in which we will have below function:

w = r (x, y, z) = (x^2 + y^2 + z^2) ^0.5

The gradient vector in any point is:

As you can see, the gradient vector proves above formula (formula:

Direction of A = A / │A│) where vector A = xi + yj + zk because we have:

Generating all directions in 3D space

 Consider a particle starts its circular motion on surface XY and simultaneously has a circular motion on surface XZ perpendicular to surface XY. In this case, this particle will produce a sphere where its circular angle on surface XY is “θ” and its circular angle on surface XZ is “β”.

According to above circular angles and radius of sphere (r), we can calculate coordination of point P (x, y, z) on sphere by using below equations:

x = r * cos β * cos θ

y = r * cos β * sin θ

z = r * sin β

Example:

Solve equation:    x^2 + y^2 + z^2 = 225

If θ = 25 degree and β = 56 degree

Answer:

We have:  r = (225) ^0.5 = 15

x = 15 * cos 56 * cos 25 = 7.602013

y = 15 * cos 56 * sin 25 = 3.544877

z = 15 * sin 56 = 12.43556

(7.602013 ^2) + (3.544877 ^2) + (12.43556 ^2) = 225

In fact, we have below vector:

V = xi + yj + zk

V = (r * cos β * cos θ)i + (r * cos β * sin θ)j + (r * sin β)k

For r = 1, we have below unit vector in 3D space:

U = (cos β * cos θ)i + (cos β * sin θ)j + (sin β)k

It is clear; the range of changes for “θ” and “β” is between 0 and 360 degree.

Therefore, for making a spreadsheet included all directions, we should go below steps:

- Choose ∆θ and ∆β between 0 and 360 degree. For instance, I considered ∆θ = ∆β = 1

- By using the method stated in article of “Can We Solve a Nonlinear Equation with Many Variables?” posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with.html, we should find all combinations of “θ” and “β” for ∆θ and ∆β between 0 and 360 degree. For example, when I considered ∆θ = ∆β = 1, I will have 130321 combinations on my excel spreadsheet (361^2 = 130321).

If you choose ∆θ = ∆β = 0.5, you will have 519841 combinations (directions) on your excel spreadsheet (721^2 = 519841).

Anyway, I think that ∆θ = ∆β = 1 is enough.

- Using from above equations for r = 1 and each set of combinations.  In this case, you have 130321 rows that it show you all directions which you need to your analysis.

In the next articles, I will show you how we can utilize this spreadsheet as a template to investigate the changes of some physical functions.

The change depends on direction of the motion: Generating Eigenvalues from special matrices (CON)

In the reference with the article of “Can We Solve a Nonlinear Equation With Many Variables? (Con)” posted on link: : http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with_19.html and following to article of "The change depends on direction of the motion: Generating Eigenvalues from special matrices" posted on link: https://emfps.blogspot.com/2016/06/the-change-depends-on-direction-of.html, here is another new theorem as follows:

Theorem: “If and only if there are three points on a sphere which are members of natural number, they make a matrix 3*3 where the eigenvalue of this matrix is easily calculated by using below 

formulas:

A =

 a11    a12    a13

a21     a22    a23

a31     a32    a33

λ1 = a11 - a12  

 

λ2 = a11+ a12 + a13

Example (1):

The points P1 (1, 2, 2) and P2 (2, 1, 2) and P3 (2, 2, 1) are on a sphere with radius of 3 (r =3), these points make matrix A where we can calculate eigenvalue of this matrix as follows:

A = 

1	2	2
2	1	2
2	2	1

λ1 = 1 – 2 = -1  

λ2 = 1+ 2 +2 = 5

Example (2):

The points P1 (8, 16, 16) and P2 (16, 8, 16) and P3 (16, 16, 8) are on a sphere with radius of 24 (r =24), these points make matrix A where we can calculate eigenvalue of this matrix as follows:

A = 

8	16	16
16	8	16
16	16	8

λ1 = 8 – 16 = -8  

λ2 = 8+ 16 +16 = 40

Example (3):

The points P1 (4, 8, 8) and P2 (8, 4, 8) and P3 (8, 8, 4) are on a sphere with radius of 12 (r =12), these points make matrix A where we can calculate eigenvalue of this matrix as follows:

A = 

4	8	8
8	4	8
8	8	4

λ1 = 4 – 8 = -4 

λ2 = 4+ 8 +8 = 20

Energy Saving by Using the Solution of a Nonlinear Equation With Many Variables

Following to article of "Can We Solve a Nonlinear Equation with Many Variables? (Con)" posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with_19.html, the purpose of this article is to solve a nonlinear equation with five variables in limited domain and range. I have started it with a sample of thermal conduction in curtain wall system in buildings in which this example pursues the energy saving and cost management in shopfront glass in malls, shopping centers and mini-markets. This is a real example because the people can directly use from the data and results of this article to save energy.

Before starting of example, let me tell you about a problem and a restriction when we are applying 

this method to solve the equations as follows:

1. As you understood, this method gives us the same domain and range for all variables while we 

need to different domains and ranges to solve the equations in the fields of engineering. For instance, domains for variables can be between 1 - 100 or 1 – 1000 but sometimes we have to have real numbers for the variables including: x (0.0025, 0.48) or

y (-23.5, 0.78) or z (4, 563.75) and so on. I can say to you that we are able to solve easily this problem by using of some theorems in mathematics and change the domains and ranges of variables to ideal and desired ones. I have shown you this changes in below example.

2. But, there is a restrictive factor something like discrete functions instead of continuous functions so that we have to follow this limitation forever. It means that we should consider sub-intervals between domains and ranges for our variables. In fact, the numbers which belong to variables between a domain and rage are discrete not continuous. Of course, we can decrease and decrease sub-intervals step by step. The best way is, to apply Arithmetic Progression for sub-intervals.

"Please be informed that we can use many methods to reach our targets which are new data and results. Therefore, the most important thing is, to extract new data and results by any method. In fact, the analysis of these new data and results lead us to reach the gates of new worlds. The methods are only tools to reach our targets."

A real example of energy saving: Thermal conduction in shopfront glass

The law of thermal conduction (Fourier's Law) gives us the great opportunity to calculate the rate of energy transfer by heat for a slab of infinitesimal thickness (dx) and temperature difference (dT) in which the rate of│dT/dx│is named the temperature gradient. Simple conductive heat transfer (in watts) through a uniform body can be calculated from below equation: 

P = k. A.│∆T/ ∆x│      and       ∆T = Th – Tc   and     Th > Tc

Where:

- A is the area of the body (m2)

- k is the body's thermal conductivity (W/m°C)

- ∆T is the temperature difference across the body (°C)

- ∆x is the body's thickness (m)

- Th is one face of the slab with high temperature (°C)

- Tc is other face of the slab with low temperature (°C)

- P is the rate of heat transfer (Watt)

The heat transfer occurs only if there is a difference in temperature between two parts of a slab.

For this example, I used from manual of "CODE OF PRACTICE FOR USE OF GLASS IN BUILDINGS" an Indian Standard which was adopted by the Bureau of Indian Standards. You can find this paper posted on below link:

https://www.aisglass.com/sites/default/files/pdfs/technical%20papers/AIS-59.pdf

The page 27 of above Standard shows us the boundaries for maximum area of designed normal glass in compare with its thickness in shop fronts. (See Table 5.3)  

According to Table 5.3, I chose below domains for A (area) and ∆x (thickness):

A = (6, 15) and ∆x = (15, 25)

And so, I consider domains for the temperature of outside and inside as follows:

Th = (15, 32) and Tc = (-20, 4)

In this case, we will have a maximum of heat transfer equal to 41600 W and a minimum heat transfer equal to 2112 W. (kglass = 0.8 W/m°C)  

P max = 41600 W,       and Pmin = 2112 W

Therefore, the range for heat transfer is:  P = (2112, 41600)

Now, I apply previous method and solve above equation for five domains and range. The rate of heat transfer (P) to the Number of Results has been presented on below graph:

I only apply one sub-interval and get the total sum of results equal to 195 in which we will have 30 results for P = 12800 W. If you spend more time and apply many sub-intervals, it is possible, you will find much more results.

Analysis of findings and results

The most important part of this article is, to analyze the results to reach our target which is the energy saving in the same conditions accompanied by cost management.

According to the results, here is many outcomes which lead us to handle the cost management and the energy saving. Let me present only several states as follows:

1. For instance, if we compare maximum transfer rate with another results, we will have below analysis: (Please see below figure)

According to above table, if you decrease the temperature of your shop from 32 °C to 15°C and also increase the thickness of glass from 15 mm to 16.935 mm, then you will save about 40% of your energy consumption. What is your cost management?

Usually shop front glasses have around 8 years guarantee. If you open your shop 12 hour per day, total hours for energy consumption is: Total hours = 8*365*12 = 35040 hour

Then, you have around 40% (0.403846) energy saving which is equal to 16800 W. Therefore, you will be able to save the energy totally around 588772 KWh in the period of 8 years (16800*35040). The cost of electricity power is about 12 cent per KWh. In the result, you will save totally amount of USD70, 640 for 8 years and USD8, 830 per year. If we add future value of each annual saving with an average return rate (Cost of capital) equal to 11% per year, it means that you are saving amount of money around USD 104,718. What is your costs? You should purchase normal glass with thickness 17 mm instead of 15mm. Therefore, you should pay more USD 40 per square meter (Price of normal glass with 15mm thickness is around USD 130 and 17mm is around USD170). Consequently, your additional investment is USD 600. It means that you have still saved the amount of money around USD 104,000. 

2. If we compare some results together, sometimes we can see very exiting outcomes as follows:

According to above results, if we increase the thickness from 15mm to 25mm, we are saving around 42% energy consumption and if we increase the thickness from 15mm to 23 mm, we are saving around 35% energy consumption. For calculation of cost management, you can apply the same method mentioned in Item (1).

3. In the reference with below results, if we are able to change the design for area of 7.74 m2 to 6 m2, we have already saved around 23% energy consumption.

As I told you, we can obtain 30 results for P = 12800 W. You can find these results on below figure for your better analysis and design: