KULLANILAN PROGRAMLARA İLİŞKİN FORMÜLASYONLAR

I. Temel Ödev Çözümü

Programda kullanılan Vincenty yöntemine ait formülasyon ve sembollere ilişkin açıklamalar aşağıdaki gibidir :

TanU1 = (1-f) Tan f1                                                                                               

Tans1= TanU1 / Cosa12                                                                                           

Sina = CosU1 Sina12                                                                                              

u2 = cos2a(a2-b2) / b2                                                                                             

A = 1 + (u2/16384) {4096 + u2[-768+ u2 (320-175u2)]}                                       

B = (u2/1024) {256 + u2[-128+ u2 (74-47u2)]}                                                           

s = (s/bA)                                                                                                               

yaklaşımıyla;

aşağıdaki 3 eşitlik, s değerinin değişimindeki fark önemsiz bir hale gelene kadar iterasyona sokulur.

2sm = 2s1 + s Ds

= BSins {Cos2sm + (B/4) [Coss (-1+2Cos22sm)-(B/6) Cos2sm   (-3+4Sin2s) (-3+4Cos22sm)]}s

= (s/bA) + Ds          

Daha sonra;

Tanf2 = (Sin U1 Coss + Cos U1 Sins Cosa12) / {(1-f)[Sin2a+(SinU1Sins -CosU1Coss Cosa12)2]½}                                                                                        

Tanl = (Sins Sina12) / (CosU1Coss - SinU1SinsCosa12)                                   

C = (f/16)Cos2a [4 + f (4 -3Cos2a)]                                                                     

w = l - (1-C) f Sina{s+CSins [Cos2sm+CCoss (-1+2Cos22sm)]}                    

l2 = l1 + w                                                                                                            

Tana21 = (Sina) / (-SinU1Sins + CosU1CossCosa12)                                       

eşitlikleri ile sonuç elde edilir.

f1        : 1. Noktanın enlemi

l1        : 1. Noktanın boylamı

a12       : 1. Noktadan 2. noktaya azimut

s          : Noktalar arasındaki elipsoidal mesafe

f           : Elipsoidin basıklığı

a          : Elipsoidin büyük yarı ekseni

b          : Elipsoidin küçük yarı ekseni

f2        : 2. Noktanın enlemi

l2        : 2. Noktanın boylamı

a21       : 2. Noktadan 1. noktaya azimut

II. Temel Ödev Çözümü

Programda kullanılan Vincenty yöntemine ait formülasyon ve sembollere ilişkin açıklamalar aşağıdaki gibidir :

TanU1 = (1-f) Tanf1                                                                                              

TanU2 = (1-f) Tanf2                                                                                              

l = w = l2 - l1                                                                                                     

yaklaşımından hareketle; aşağıdaki eşitlikler l’nın değişimindeki fark önemsiz oluncaya kadar iterasyona sokulur.

Sin2s = (CosU2 Sinl)2 + (CosU1 SinU2 – SinU1CosU2Cosl)2                            

Coss = SinU1SinU2 + CosU1CosU2 Cosl                                                            

Tans = Sins / Coss                                                                                              

Sina = CosU1CosU2 Sinl / Sins                                                                          

Cos2sm = Coss - (2SinU1SinU2 / Cos2a)                                                             

C = (f/16) Cos2a[4+f (4-3Cos2a)]                                                                        

l = w + (1-C)f Sina{s + C Sins [Cos2sm + C Coss (-1 + 2Cos22sm)]}           

Daha sonra;

u2 = cos2a(a2-b2)/b2                                                                                               

A = 1 + (u2/16384) {4096 + u2[-768+ u2 (320-175u2)]}                                     

B= (u2/1024) {256 + u2[-128+ u2 (74-47u2)]}                                                    

Ds = BSins {Cos2sm + (B/4) [Coss (-1+2Cos22sm)- (B/6)Cos2sm (-3 + 4Sin2s) (-3+4Cos22sm)]}                                                                                                  

s = bA(s - Ds)                                                                                                       

Tana12 = (CosU2 Sinl) / (Cos U1 SinU2 – SinU1CosU2 Cosl)                           

Tana21 = (CosU1 Sinl) / (-Sin U1 CosU2 + Cos U1 SinU2 Cosl)                       

elde edilir.

 

f1        : 1. Noktanın enlemi

f2        : 2. Noktanın enlemi

l1        : 1. Noktanın boylamı

l2        : 2. Noktanın boylamı

f           : Elipsoidin basıklığı

a          : Elipsoidin büyük yarı ekseni

b          : Elipsoidin küçük yarı ekseni

a12       : 1. Noktadan 2. noktaya azimut

a21       : 2. Noktadan 1. noktaya azimut

s          : Noktalar arasındaki elipsoidal mesafe

 

 

 

 

 

Kartezyen-Coğrafik Koordinatlar Dönüşümü

 

Kartezyen koordinatlardan coğrafi koordinatların hesabı  ve h’nin iterasyonunu gerektirir.

Boylam değeri ;

λ =  arctan ( Y / X)                                                                                               

eşitliğinden kolayca bulunur. h << N  olup , ilk aşama için h = 0 alınırsa;

                                                                                      

                                                           

                                                                                            

 ve h’nin iterasyonu ile coğrafi koordinatlar elde edilir.

Coğrafi koordinatlardan kartezyen koordinatların hesabı :

X = N cos cosλ                                                                                                

Y = N cos sinλ                                                                                                 

Z = (1 – f)2 N sin                                                                                            

Burada N değeri yine yukardaki gibi ;

                                                                                             

ile hesaplanabilir. Sonuç formülasyon :

X = (N+h) cos cosλ                                                                                           

Y = (N+h) cos sinλ                                                                                            

Z = ((1 – f)2 N + h) sin                                                                                       

 

a          : Elipsoidin büyük yarı ekseni

f           : Elipsoidin basıklığı

N         : Meridyen normalinin eğrilik yarıçapı

X, Y, Z: Noktanın kartezyen koordinat bileşenleri

f          : Noktanın enlemi

l          : Noktanın boylamı

h          : Noktanın elipsoit yüksekliği

 

Coğrafik – UTM Koordinatlar Dönüşümü ve Tersi

Coğrafi ve grid dönüşümlerini yapmak için gerekli formülasyonlar :

Öncelikli hesaplamalar:

Meridyen mesafesi hesabı

m = a(1-e2)[1-(e2sin2f)]-3/2df                                                                           

formülüyle yapılabileceği gibi, seri açılım kullanmak daha etkili olacaktır.

m = a{A0f -A2Sin2f+A4Sin4f -A6Sin6f}                                                           

A0 = 1- (e2/4)-(3e4/64)-(5e6/256)                                                                           

A2 = (3/8)(e2+e4/4+15e6/128)                                                                               

A4 = (15/256)(e4+3e6/4)                                                                                        

A6 = 35e6/3072                                                                                                     

 

Ayak noktası enlemi  (f' ):

n = (a-b)/(a+b) = f/(2-f)                                                                                        

G = a (1-n)(1-n2)(1+(9/4)n2+(225/64)n4)(p/180)                                                 

s = (mp)/(180G)                                                                                                    

f'=s+((3n/2)-(27n3/32))Sin2s +((21n2/16)-(55n4/32))Sin4s+(151n3/96) Sin6s+(1097n4/512)Sin8s                                                                                              

Eğrilik yarıçapı :

r =  a(1-e 2) / (1-e2Sin2f)3/2                                                                                  

n =  a / (1-e 2Sin2f)1/2                                                                                            

y = n / r                                                                                                                

Coğrafi koordinatlardan düzlem koordinatlarının hesabı :

t = Tan f                                                                                                                

w= l-l0                                               

E' = (K0nwCosf ){1 + Term1 + Term2 + Term3 }                                              

Term1 = (w2/6)Cos2f (y-t2)                                                                                  

Term2 =(w4/120)Cos4f[4y3(1-6t2)+y2(1+8t2)-y2t2+t4]                                      

Term3 = (w6/5040)Cos6f(61-479t2+179t4-t6)                                                       

E = E ' + 500 000                                                                                                  

N' = K0{m + Term1 + Term2 + Term3 + Term4 }                                              

Term1 = (w2/2 )nSinf Cosf                                                                                  

Term2 = (w4/24)nSinf Cos3f(4y2+y-t2)                                                            

Term3 = (w6/720)nSinf Cos5f[8y4(11-24t2)-28y3(1-6t2) +y2 (1-32t2)-y(2t2)+t4]                                                                                          

Term4 = (w8/40320)nSinf Cos7f (1385-3111t2+543t4-t6)                                   

N = N' + 0                                                                                                   

Meridyen yakınsama açısı :

g = Term1 + Term2 + Term3 + Term4                                                          

Term1 = -wSinf                                                                                                    

Term2 = -(w3/3)SinfCos2f (2 y2-y )                                                                   

Term3 = -(w5/15)SinfCos4f[y4(11-24t2)-y3(11-36t2)+2y2(1-7t2)+yt2]                 

Term4 = -(w7/315)SinfCos6f(17-26t2+2t4)                                                           

Haritalama ölçeği :

k = K0 + K0 Term1 + K0 Term2 + K0 Term3                                                       

Term1 = (w2/2) y Cos2 f                                                                                      

Term2 = (w4/24) Cos4f[4y3(1-6t2) + y2(1+24t 2) - 4y t2]                                   

Term3 = (w6/720) Cos6f (61-148t2+16t4)                                                            

Düzlem koordinatlardan coğrafi koordinatların hesabı :

E' =  E – 500 000                                                                                                   

x  =  E' / (K0n')                                                                                                                                                               

f = f' - Term1 + Term2 - Term3 + Term4                         

Term1 = (t' / (K0r')) (xE' / 2 )                                                                             

Term2 = ( t' / (K0r')) (E'x3 / 24 ) [-4y'2 + 9y' (1-t'2) +12t'2 ]                                

Term3=(t'/(K0r')) (E'x5 ) / 720) [ 8y'4 (11-24t'2 ) -12y'3 (21-71t'2) +15y'2 (15-98t'2 +15t'4 ) +180y' ( 5t'2-3t'4 ) +360t'4 ]                                                                     

Term4 = (t'/(K0r'))(E'x7/40320)(1385+3633t'2+4095t'4+1575t'6)                       

w = Term1 - Term2 + Term3 - Term4                                                                

Term1 = x Secf'                                                                                                   

Term2 = (x3/6) Secf' (y' +2t'2)                                                                             

Term3 = (x5/120) Secf' [-4y'31-6t'2 ) + y'2 ( 9-68t'2 ) +72y't'2 +24t'4 ]                 

Term4 = (x7/5040) Secf' (61+662t'2 +1320t'4 + 720t'6 )                                       

l = l0 + w                                                                                                              

Meridyen yakınsama açısı :

x = E'/K0n'                                                                                                             

t' = Tanf'                                                                                                               

g = Term1 + Term2 + Term3 + Term4                                                                  

Term1 =  -t' x                                                                                                         

Term2 =  (t' x3/3 ) (-2y'2+3y' +t'2 )                                                                       

Term3=(-t' x5 /15 )[y'4 (11-24t'2 )-3y'3 (8-23t'2 )+5y'2 (3-14t'2)+30y't'2 +3t'4 ]     

Term4 =  (t' x7 / 315 ) ( 17+77 t'2+ 105t'4 +45t'6 )                                                 

 

f          : Enlem

l          : Boylam

a          : Elipsoidin büyük yarı ekseni

b          : Elipsoidin küçük yarı ekseni

f           : Elipsoidin basıklığı

e          : Elipsoidin birinci dış merkezliliği

E          : Sağa değer

N         : Yukarı değer

g          : Meridyen yakınsama açısı

k          : Haritalama ölçeği

K0        : Küçültme faktörü

Datum Dönüşümü Hesabı

Datum dönüşümü ile ilgili temel formülasyon aşağıdaki gibidir :

                            

 

Yukardaki formülde, Harita Genel Komutanlığı tarafından belirlenen dönüşüm parametreleri aşağıdaki gibidir:

 

WGS-84’den ED-50’ye dönüşüm parametreleri :

 

X ekseni etrafındaki dönüklük

εX  =   0.0183"  

Y ekseni etrafındaki dönüklük

εY = -0.0003"

Z ekseni etrafındaki dönüklük

εZ =   0.4738"

X ekseni yönündeki öteleme

tX  =   84.003 m

Y ekseni yönündeki öteleme

tY = 102.315 m

Z ekseni yönündeki öteleme

tZ =  129.879 m

Ölçek

k  = -1.0347

 

 

ED-50’den WGS-84’e dönüşüm parametreleri :

 

X ekseni etrafındaki dönüklük

εX  =  - 0.0183"

Y ekseni etrafındaki dönüklük

εY =    0.0003"

Z ekseni etrafındaki dönüklük

εZ =  - 0.4738"

X ekseni yönündeki öteleme

tX  =   - 84.003 m

Y ekseni yönündeki öteleme

tY = - 102.315 m

Z ekseni yönündeki öteleme

tZ =  -129.879 m

Ölçek

k =   1.0347

 

Julian Günü, GPS Haftası Hesabı

Julian gününün (JD) hesabı için gerekli formülasyon  ;

yıl için Y, ay için M, gün için D ve saat için UT kullanılırsa

JD=INT [365.25*y] + INT [30.6001* (m+1)] + D + UT/24 + 1720981.5            

y = Y-1     ve     m = M+12             eğer M ≤ 2                                                     

y = Y        ve     m = M                    eğer M > 2                                                    

formülasyonu ile hesaplanabilir. Julian gününden tarihi hesaplamak için ise aşağıdaki adımlar kullanılabilir :

a = INT [JD + 0.5]                                                                                               

b = a + 1537                                                                                                        

c = INT [(b-122.1)/365.25 ]                                                                               

d = INT [365.25 * c ]                                                                                          

e = INT [(b – d) / 30.6001 ]                                                                                

D = b – d – INT [30.6001*e] + FRAC [JD + 0.5 ]                                              

M = e – 1 – 12*INT [e/14]                                                                                 

Y = c – 4715 – INT [(7 + M)/10 ]                                                                      

Haftanın gününü hesaplamak için mod 7’ ye göre aşağıdaki formül kullanılabilir :

N = modulo {INT[JD + 0.5 ] ,  7 }                                                                     

GPS haftası ise aşağıdaki bağıntı ile elde edilebilir :

HAFTA = INT [(JD – 2444244.5)/7]