Thursday, June 22, 2017

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 Directions in 3D Space” posted on link: https://emfps.blogspot.com/2017/05/the-change-depends-on-direction-of.html , 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.

Finally, by analyzing the results extracted from the method (2), we can find the general formulas as follows:

θ' = θ +180    If     0 ≤ θ ≤ 180   
θ' = θ -180     If    180 ≤ θ ≤ 360   
β' = 180 – β   If     0 ≤ β ≤ 180

β' = 540 – β   If    180 ≤  β ≤ 360