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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