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» Online Classroom   » Celestial Navigation   » Public Discussion of Cel Nav   » Great Circle Computaion Casio Calc

   
Author Topic: Great Circle Computaion Casio Calc
splitfix


 - posted December 20, 2006 10:35 AM      Profile for splitfix           Edit/Delete Post 
L1=24.8667
L2=47.3167
D=59.8844

Formulae
C= Cos 1- (Sin L2-(Sin L1 X Cos D) divided by Cos L1 X Sin D)

OR this way

Cos C = Sin L2-(Sin L1 X Cos D)Divided by
Cos L1 X Sin D

My calc always results in 57.7 as C an correct ans = C-48.1 and Initial course is also 48.1 0r 48-06.0

This formulae also appears for Zn computations after Hc has been derived. So I have no problems w/ calc Distance or Hc. I like formulas opposed to tables. Taking a 2/M exam soon and thought I'd see if you could help.

Thanks D. Somers 3/M AMO

From: Tukwila
David Burch


 - posted December 20, 2006 11:37 AM      Profile for David Burch           Edit/Delete Post 
please clarify what you want to find. it seems like you have two latitudes and a distance... and are asking for a course, which must not be the right question.

is D the difference in longitude in degrees?

please define the constants and put units with them. thanks.

From: Starpath, Seattle, WA
David Burch


 - posted December 20, 2006 02:17 PM      Profile for David Burch           Edit/Delete Post 
i think i answered my own question with the answer you gave.

for those reading, the latitudes are decimal degrees and the longitude interval is decimal decrees as well, and we are calculating the initial heading of a great circle course.

the NIMA calculators we provide online in the freeware gives -048.222°, so let me check the discrepancies.

back shortly after checking the formulas given in the question.
this answer assumes a round earth. If interested we can compute this also for various ellipsoids such as WGS84, etc.

When it comes to looking for an answer to the degree or tenth of a degree, keep in mind that when computed from a GPS they actually use the ellipsoid you selected, and not a round earth solution.

=========
THIS TURNS OUT TO BE WRONG BECAUSE I MIS-INTREPTED D

From: Starpath, Seattle, WA
splitfix


 - posted December 20, 2006 03:48 PM      Profile for splitfix           Edit/Delete Post 
L1= dep lat

L2= Destination lat

D= Distance along G.Circle

Looking to compute Initial Course first getting course angle.

No problem w/ cos D =Sin L1 X Sin L2 + Cos L1 X Cos L2 X Cos Dlo

Solution = D =59.8844 degrees

all values are degrees and decimal

This is for a USCG Exam and only certain calculators are allowed and all my training for C-Nav is focused on correct answers therefore my interest and understanding is low of C- Nav theory.

The formula path is being followed w/ apparent correct entries yet the results are flawed.

From: Tukwila
splitfix


 - posted December 20, 2006 03:52 PM      Profile for splitfix           Edit/Delete Post 
Dlo= 71.75
From: Tukwila
David Burch


 - posted December 20, 2006 05:03 PM      Profile for David Burch           Edit/Delete Post 
I am sorry but i am confused. is this a new problem? i thought we were looking for a course ie C in the first question? please write out in words, ie "given this, this and this, find this."

what is there for D is exactly same as zenith distance is cel nav, and i think that looks right. that is, zenith distance is distance along a great circle between observer and GP. L1 = aLat, L2=declination, DLo = LHA. the forumla shown above is a correct answer for zenith distance or for GC distance between two points, lat 1 and lat 2 separated by DLo longitude interval.

From: Starpath, Seattle, WA
splitfix


 - posted December 20, 2006 07:14 PM      Profile for splitfix           Edit/Delete Post 
OK

All of this is Find (Course angle) to establish Initial Course w/o a programmed calculator for a great circle departure and arrival lat/long using formula computations. I've ommitted longtitudes except the value for Dlo East

All is in degrees and decimals not degrees and minutes

L1= lat of departure= 24.8667 North

L2= lat of destination= 47.3167 North

Dlo= Diff of longtitude= 71.75 East

D= distance along the track of the great circle= 59.8844

C= Initial course angle and equals North 48.1 degrees East as the books solution.

Solution is from the book. My solution using these values for the C= formula is 57.1 or 57-06

This is the problem.

Cn= Intial course to steer and in this case C and Cn are the same 048.1 or 048-06 T

My problem is plugging the above values into the formula path w/ a regular non programmable calculator and getting the same answer as the book. On page 342 Pub 9 Bowditch youll see an identical D= formula and a variation of C= formula however niether C= formulas seem to work.

Again the problem is plugging in the above values into the C= formula on a non-progrmmable calculator.

From: Tukwila
splitfix


 - posted December 20, 2006 07:23 PM      Profile for splitfix           Edit/Delete Post 
Sorry

59.8844 X 60 = D

From: Tukwila
splitfix


 - posted December 20, 2006 07:27 PM      Profile for splitfix           Edit/Delete Post 
Sorry again
59.8844 degrees X 60= D in miles 3,593.0 miles. However the formula clearly asks for miles in degrees and is D= 59.8844 degrees.

From: Tukwila
David Burch


 - posted December 20, 2006 08:59 PM      Profile for David Burch           Edit/Delete Post 
i think i understand the problem, sorry it took me so long.

ie. Knowing the lat and lon of two positions on earth, find GC distance between them and the initial CG heading, leaving position 1 on the way to pos 2 by GC route.

this common problem can be formulated several ways, either directly using lat and lon of each location, or using lat 1 and lat 2 and the longitude difference between them.

this problem can be broken up into two problems, find the initial heading (call it C, which i think is what you want) and find the distance D.

the distance D is the easiest: the formula you have is correct.
and the correct answer for your data is

D = 3593.06 nmi = 59.884 degrees along a great circle

the correct answer for C is

C=048.100333 if headed east and 311.899667 if headed west.

And your question is: your formula in the top post does not give you that answer.

there are several ways to write the C formula, let me check them and get right back.

-------- the C equations are right but first one means cos to -1 power, which is another notation for arccos or ACOS. i will try the compuation and get back...

From: Starpath, Seattle, WA
David Burch


 - posted December 20, 2006 09:54 PM      Profile for David Burch           Edit/Delete Post 
there must be some input issue with the calculator. here is your exact input expressed as you could in an excel program

code:
B2=pi()/180  .... Excel uses radians not degrees so this is the conversion needed.
---------
A1=(SIN(47.3167*B2) - (SIN(24.8667*B2)*COS(59.884*B2)))/COS(24.8667*B2)/SIN(59.884*B2)
A1=0.667827637 .... this number should show up somewhere in the calculator sequence
---------
C1=ACOS(A1)/B2 = 48.10037859
... in the last step the answer is in radians, so we have to convert back to degrees.

So it looks like the error you see is some sequencing of the inputs... and i must admit that after bungling it up several times with two different calculators, i switched to Excel...

thus the message is... you are doing exactly the right thing by sorting this out ahead of time, since even knowing the formulas is not enough if the input seqeunce is tricky... ie am i dividing the whole top of the equation by this next input or just the term in the bottom etc.

note, for example, that A1 could also be correctly sequenced this way... look at the difference at the end

A1=(SIN(47.3167*B2) - (SIN(24.8667*B2)*COS(59.884*B2)))/(COS(24.8667*B2)*SIN(59.884*B2))

From: Starpath, Seattle, WA
splitfix


 - posted December 21, 2006 07:46 PM      Profile for splitfix           Edit/Delete Post 
Correct 0.6678 is the solution before inverting the Cos.

However I dont understand the B2 formula

B2= Pi()/180 I'm not good at math or C.N.
Is B2 a constant value

From: Tukwila
David Burch


 - posted December 21, 2006 11:14 PM      Profile for David Burch           Edit/Delete Post 
This factor of Pi/180 is just the math conversion between radians and degrees. It is required when you do the computations in Excel program or in some other languages like Basic, but it is not required in any calculator solution--the fact that excel requies us to write "pi/180" as "pi()/180" is just some quirk of that program.

For calculator solution just follow the steps in the next line after that but forget the B2 factor.

I only put it there so anyone can actually test to see that the numbers are correct, in otherwords that your second equation is in fact exactly right, and if the answer does not come out right then there is an input error or some sort.

(The first version of the equation, however, could be misinterpreted because of the notation.)

From: Starpath, Seattle, WA
David Burch


 - posted December 22, 2006 11:05 AM      Profile for David Burch           Edit/Delete Post 
As a curiosity, if you type the following (or cut and paste) this into the Google search engine

(180/pi)*acos((sin(47.3167*pi/180) - (sin(24.8667*pi/180)*cos(59.884*pi/180)))/cos(24.8667*pi/180)/sin(59.884*pi/180))

you will get this:

(180 / pi) * acos(((sin((47.3167 * pi) / 180) - (sin((24.8667 * pi) / 180) * cos((59.88400 * pi) / 180))) / cos((24.8667 * pi) / 180)) / sin((59.88400 * pi) / 180)) = 48.1003786

which is the answer we want.

From: Starpath, Seattle, WA


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