# The collinearity equations Lets use the following abbreviations by sdaferv

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```									The collinearity equations:
r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )
x = xp −c
r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )

r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )
y = yp −c
r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )

Let’s use the following abbreviations:
N x = r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )

N y = r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )

D = r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )

Partial derivatives with respect to interior orientation parameters; (xp, yp, c):
The x-coordinate equation
∂x                        ∂x                        ∂x     r ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )    N
=1                        =0                       = − 11                                                     =− x
∂x p                      ∂y p                       ∂c    r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )    D

The y-coordinate equation

∂y                        ∂y                        ∂y    r ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )       Ny
=0                        =1                       = − 12                                                      =−
∂x p                      ∂y p                       ∂c     r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )    D

Partial derivatives with respect to exterior orientation parameters; ground coordinates (Xo, Yo, Zo):
The x-coordinate equation
∂x      r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r13 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )]
= c 11 13
∂X o                                  [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2
∂x    r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r23 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )]
= c 21 13
∂Yo                                 [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
∂x     r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r33 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )]
= c 31 13
∂Z o                                 [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2
The y-coordinate equation
∂y      r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r13 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )]
= c 12 13
∂X o                                  [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
∂y    r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r23 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )]
= c 22 13
∂Yo                                 [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
∂y     r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r33 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )]
= c 32 13
∂Z o                                 [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
In abbreviated form
∂x      r D − r13 N x                        ∂x    r D − r23 N x                      ∂x     r D − r33 N x
= c 11                                      = c 21                                    = c 31
∂X o         D2                              ∂Yo        D2                            ∂Z o        D2

∂y     r12 D − r13 N y                       ∂y    r22 D − r23 N y                    ∂y     r32 D − r33 N y
=c                                          =c                                        =c
∂X o          D2                             ∂Yo          D2                          ∂Z o          D2
Partial derivatives with respect to exterior orientation parameters; attitude of image plane (ω, φ, κ):
The rotation matrix R is:

r11 = cos φ cos κ                               r12 = -cos φ sin κ                         r13 = sin φ

r21 = cos ω sin κ + sin ω sin φ cos κ           r22 = cos ω cos κ - sin ω sin φ sin κ      r23 = -sin ω cos φ

r31 = sin ω sin κ - cos ω sin φ cos κ           r32 = sin ω cos κ + cos ω sin φ sin κ      r33 = cos ω cos φ

∂ (Element )                                ∂ (Element )                             ∂ (Element )
∂ω                                           ∂φ                                       ∂κ
∂r11                                                    ∂r11                           ∂r11
=0                                                      = − sin φ cos κ                = − cos φ sin κ = r12
∂ω                                                      ∂φ                             ∂κ
∂r21                                                    ∂r21                           ∂r21
= − sin ω sin κ + cos ω sin φ cos κ = − r31             = sin ω cos φ cos κ            = cos ω cos κ − sin ω sin φ sin κ = r22
∂ω                                                      ∂φ                             ∂κ
∂r31                                                    ∂r31                           ∂r31
= cos ω sin κ + sin ω sin φ cos κ = r21                 = − cos ω cos φ cos κ          = sin ω cos κ + cos ω sin φ sin κ = r32
∂ω                                                      ∂φ                             ∂κ
∂r12                                                    ∂r12                           ∂r12
=0                                                    = sin φ sin κ                  = − cos φ cos κ = −r11
∂ω                                                      ∂φ                             ∂κ
∂r22                                                    ∂r22                           ∂r22
= − sin ω cos κ − cos ω sin φ sin κ = −r32            = − sin ω cos φ sin κ          = − cos ω sin κ − sin ω sin φ cos κ = − r21
∂ω                                                       ∂φ                            ∂κ
∂r32                                                    ∂r32                           ∂r32
= cos ω cos κ − sin ω sin φ sin κ = r22               = cos ω cos φ sin κ            = − sin ω sin κ + cos ω sin φ cos κ = − r31
∂ω                                                      ∂φ                             ∂κ
∂r13                                                    ∂r13                           ∂r13
=0                                                    = cos φ                        =0
∂ω                                                       ∂φ                            ∂κ
∂r23                                                    ∂r23                           ∂r23
= − cos ω cos φ = −r33                                  = sin ω sin φ                  =0
∂ω                                                      ∂φ                             ∂κ
∂r33                                                    ∂r33                           ∂r33
= − sin ω cos φ = r23                                   = − cos ω sin φ                =0
∂ω                                                      ∂φ                             ∂κ
The x-coordinate equation
∂x     [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )][0 − r31 ⋅ (Y − Yo ) + r21 ⋅ (Z − Z o )] + [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ (Z − Z o )][0 + r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z −
= −c 13
∂ω                                                                         [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
∂x      D[− r31 ⋅ (Y − Yo ) + r21 ⋅ ( Z − Z o )] + N x [r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z − Z o )]
= −c
∂ω                                             D2

∂x     [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )][− sin φ cosκ ⋅ ( X − X o ) + sin ω cosφ cosκ ⋅ (Y − Yo ) − cosω cosφ cosκ ⋅ (Z − Z o )]
= −c 13
∂φ                                                    [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
−c
[r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )][cos φ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cos ω sin φ ⋅ ( Z − Z o )]
[r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2
∂x      D[− sin φ cosκ ⋅ ( X − X o ) + sin ω cosφ cosκ ⋅ (Y − Yo ) − cosω cosφ cosκ ⋅ ( Z − Z o )] + N x [cosφ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cosω sin φ ⋅ ( Z − Zo )]
= −c
∂φ                                                                                    D2

∂x      − D cosκ [− sin φ ⋅ ( X − X o ) + sin ω cosφ ⋅ (Y − Yo ) − cosω cosφ ⋅ ( Z − Zo )] + N x [cosφ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cosω sin φ ⋅ ( Z − Z o )]
= −c
∂φ                                                                                  D2

∂x      − D 2 cosκ + N x [cosφ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cosω sin φ ⋅ ( Z − Z o )]
= −c
∂φ                                               D2

∂x
= −c
[
− D 2 cosκ + N x − N x ⋅ cosκ + N y sin κ     ]
∂φ                        D2

∂x     [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )][r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )]
= −c 13
∂κ                               [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2
∂x      Ny
= −c
∂κ      D
In Summary:

∂x
= −c
D[− r31 ⋅ (Y − Yo ) + r21 ⋅ ( Z − Z o )] + N x [r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z − Z o )]                   ∂x
= −c
[
− D 2 cosκ + N x − N x ⋅ cosκ + N y sin κ    ]           ∂x
= −c
Ny
∂ω                                             D2                                                              ∂φ                        D2                                     ∂κ      D
The y-coordinate equation

∂y     [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )][0 − r32 ⋅ (Y − Yo ) + r22 ⋅ (Z − Z o )] + [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ (Z − Z o )][0 + r33 ⋅ (Y − Yo ) − r23 ⋅
= −c 13
∂ω                                                                        [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
∂y      D[− r32 ⋅ (Y − Yo ) + r22 ⋅ ( Z − Z o )] + N y [r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z − Z o )]
= −c
∂ω                                             D2

∂y     [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Zo )][sin φ sin κ ⋅ ( X − X o ) − sin ω cosφ sin κ ⋅ (Y − Yo ) + cosω cosφ sin κ ⋅ (Z − Zo )]
= −c 13
∂φ                                                   [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Zo )]2

−c
[r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )][cos φ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cos ω sin φ ⋅ ( Z − Z o )]
[r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2
∂y      D[sin φ sin κ ⋅ ( X − X o ) − sin ω cosφ sin κ ⋅ (Y − Yo ) + cosω cosφ sin κ ⋅ ( Z − Z o )] + N y [cosφ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cosω sin φ ⋅ ( Z − Z o )]
= −c
∂φ                                                                                       D2
∂y      D sin κ [sin φ ⋅ ( X − X o ) − sin ω cos φ ⋅ (Y − Yo ) + cos ω cos φ ⋅ ( Z − Z o )] + N y [cos φ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cos ω sin φ ⋅ ( Z − Z o )]
= −c
∂φ                                                                                     D2
∂y      D 2 sin κ + N y [cos φ ⋅ ( X − X o ) + sin ω sin φ ⋅ (Y − Yo ) − cos ω sin φ ⋅ ( Z − Z o )]
= −c
∂φ                                                 D2
∂y
= −c
[
D 2 sin κ + N y N x cos κ + N y sin κ           ]
∂φ                       D2

∂y     [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )][−r11 ⋅ ( X − X o ) − r21 ⋅ (Y − Yo ) − r31 ⋅ ( Z − Z o )]
= −c 13
∂κ                               [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
∂y   N
=c x
∂κ    D
In Summary:

∂y      D[− r32 ⋅ (Y − Yo ) + r22 ⋅ ( Z − Z o )] + N y [r33 ⋅ (Y − Yo ) − r23 ⋅ ( Z − Z o )] ∂y      D 2 sin κ + N y [N x cos κ + N y sin κ ]                                 ∂y   N
= −c                                                                                         = −c                                                                             =c x
∂ω                                             D2                                            ∂φ                        D2                                                     ∂κ    D
Partial derivatives with respect to ground coordinates of tie points(X, Y, Z)
The x-coordinate equation
∂x     ∂x      r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r13 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )]
=−      = −c 11 13
∂X    ∂X o                                  [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2
∂x     ∂x     r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r23 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )]
=−     = −c 21 13
∂Y    ∂Yo                                  [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2
∂x     ∂x      r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r33 [r11 ⋅ ( X − X o ) + r21 ⋅ (Y − Yo ) + r31 ⋅ ( Z − Z o )]
=−      = −c 31 13
∂Z    ∂Z o                                  [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )]2
The y-coordinate equation
∂y     ∂y      r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r13 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )]
=−      = −c 12 13
∂X    ∂X o                                  [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
∂y     ∂y     r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r23 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )]
=−     = −c 22 13
∂Y    ∂Yo                                  [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2
∂y     ∂y      r [r ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ ( Z − Z o )] − r33 [r12 ⋅ ( X − X o ) + r22 ⋅ (Y − Yo ) + r32 ⋅ ( Z − Z o )]
=−      = −c 32 13
∂Z    ∂Z o                                  [r13 ⋅ ( X − X o ) + r23 ⋅ (Y − Yo ) + r33 ⋅ (Z − Z o )]2

In abbreviated form
∂x      r D − r13 N x                            ∂x     r D − r23 N x                         ∂x     r D − r33 N x
= −c 11                                          = −c 21                                      = −c 31
∂X          D2                                   ∂Y         D2                                ∂Z         D2

∂y      r12 D − r13 N y                          ∂y      r22 D − r23 N y                      ∂y      r32 D − r33 N y
= −c                                             = −c                                         = −c
∂X            D2                                 ∂Y            D2                             ∂Z            D2

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