Awange J.,Grafarend E., Palancz B., Zaletnyik P. Algebraic Geodesy and Geoinformatics

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344 Appendix

Appendix A-4: Determined system of the 9-parameter

transformation N point problem

(a, b, c, σ

, σ

, X

, Y

, Z

) =

∂∆

∂a

− bNX

+ aNY

− cNX

+ aNZ

+ bY

i=1

+ cZ

i=1

+ bY

i=1

+ cZ

i=1

+ aY

i=1

(−2y

)

+ bX

i=1

+ bσ

i=1

(−x

) + bσ

i=1

(−X

) + aσ

i=1

+ aY

i=1

(−2Y

) + bX

i=1

+ Z

i=1

+ bσ

i=1

(−x

)

+ b

i=1

(−X

) + aσ

i=1

+ aZ

i=1

(−2z

) + cX

i=1

+ cσ

i=1

(−x

) + cσ

i=1

(−X

) + σ

i=1

(−2Y

) + aσ

i=1

+ Y

i=1

(−2Z

) + aZ

i=1

(−2Z

) + cX

i=1

+ cσ

i=1

(−x

)

+ c

i=1

(−X

) + σ

i=1

+ aσ

i=1

+ a

i=1



+ Z



= 0.

(a, b, c, σ

, σ

, X

, Y

, Z

) =

∂∆

∂b

bnX

− anX

− cnY

+ bnZ

+ bX

i=1

−2x

i=1

+ bσ

i=1

+ bX

i=1

−2X

+ Z

i=1

−2X

i=1

+ bσ

i=1

+ aX

i=1

+ cZ

i=1

aσ

i=1

−x

+ aσ

i=1

−X

+ aX

i=1

+ cZ

i=1

aσ

i=1

−x

+ a

i=1

−X

+ bZ

i=1

−2z

+ cY

i=1

+ cσ

i=1

−y

+ cσ

i=1

−Y

+ bσ

i=1

−2Z

+ cY

i=1

+ X

i=1

+ σ

i=1

−2x

cσ

i=1

−y

+ c

i=1

−Y

+ bσ

i=1

+ b

i=1



+ Z



= 0

App endix 345

(a, b, c, σ

, σ

, X

, Y

, Z

) =

∂∆

∂c

cnX

+ cnY

− anX

− bnY

+ cX

i=1

−2x

i=1

+ cσ

i=1

+ cX

i=1

−2X

+ aZ

i=1

+ Y

i=1

cσ

i=1

+ cY

i=1

−2y

+ bZ

i=1

+ σ

i=1

−2X

cσ

i=1

+ X

i=1

−2Y

+ cY

i=1

−2Y

+ bZ

i=1

+ σ

i=1

cσ

i=1

+ c

i=1



+ Y



+ aX

i=1

+ bY

i=1

aσ

i=1

−x

+ aσ

i=1

−X

+ bσ

i=1

−y

+ bσ

i=1

−Y

i=1

+ bY

i=1

+ aσ

i=1

−x

+ a

i=1

−X

+ bσ

i=1

−y

i=1

−Y

= 0

(a, b, c, σ

, σ

, X

, Y

, Z

) =

∂∆

∂X

+ b

+ c

− abnY

− acnZ

i=1

−x

+ b

i=1

−x

+ c

i=1

−x

+ b

i=1

−X

+ c

i=1

−X

i=1

+ abσ

i=1

+ cσ

i=1

−2Y

+ abσ

i=1

+ acσ

i=1

acσ

i=1

+ bσ

i=1

= 0

(a, b, c, σ

, σ

, X

, Y

, Z

) =

∂∆

∂Y

− abnX

+ nY

+ a

+ c

− bcnZ

abσ

i=1

+ abσ

i=1

+ cσ

i=1

+ σ

i=1

−y

+ a

i=1

−y

i=1

−y

+ a

i=1

−Y

+ c

i=1

−Y

+ σ

i=1

+ bcσ

i=1

aσ

i=1

−2Z

+ bcσ

i=1

= 0

346 Appendix

(a, b, c, σ

, σ

, X

, Y

, Z

) =

∂∆

∂Z

− acnX

− bcnY

+ nZ

+ a

+ b

acσ

i=1

+ bσ

i=1

−2X

+ acσ

i=1

+ bcσ

i=1

+ bcσ

i=1

aσ

i=1

+ σ

i=1

−z

+ a

i=1

−z

+ b

i=1

−z

+ a

i=1

−Z

i=1

−Z

+ σ

i=1

= 0

(a, b, c, σ

, σ

, X

, Y

, Z

) =

∂∆

∂σ

+ b

+ c

− abnX

− acnX

i=1

−2x

+ b

i=1

−2x

+ c

i=1

−2x

+ abY

i=1

acZ

i=1

+ σ

i=1

+ b

i=1

+ c

i=1

+ b

i=1

−X

i=1

−X

+ X

i=1

−x

+ b

i=1

+ c

i=1

abX

i=1

+ abσ

i=1

−x

+ cX

i=1

−2Y

+ abX

i=1

− 2Y

+ abX

i=1

−x

+ c

i=1

+ acX

i=1

+ acσ

i=1

−x

+ acX

i=1

+ b

i=1

−2x

+ ac

i=1

−x

= 0

(a, b, c, σ

, σ

, X

, Y

, Z

) =

∂∆

∂σ

− abnX

+ nY

+ a

+ c

− bcnY

abY

i=1

+ abY

i=1

+ cY

i=1

+ Y

i=1

−2y

i=1

−2y

+ c

i=1

−2y

+ abX

i=1

+ bcZ

i=1

abσ

i=1

−x

+ c

i=1

−2X

+ ab

i=1

−X

+ σ

i=1

+ a

i=1

+ a

i=1

−Y

+ c

i=1

−Y

+ Y

i=1

−y

i=1

+ c

i=1

+ bcY

i=1

+ bcσ

i=1

−y

+ aY

i=1

−2Z

bcY

i=1

+ bc

i=1

−y

+ a

i=1

= 0

App endix 347

(a, b, c, σ

, σ

, X

, Y

, Z

) =

∂∆

∂σ

− acnX

− bcnY

+ nZ

+ a

+ b

acZ

i=1

+ bZ

i=1

−2X

+ acZ

i=1

+ bcZ

i=1

+ bcZ

i=1

+ Z

i=1

−2z

+ a

i=1

−2z

+ b

i=1

−2z

acX

i=1

+ bcY

i=1

+ acσ

i=1

−x

+ ac

i=1

−X

− x

+ ac

i=1

−X

i=1

+ bcσ

i=1

−y

+ a

i=1

−2Y

+ bc

i=1

−Y

+ σ

i=1

+ b

i=1

+ a

i=1

−Z

+ b

i=1

−Z

+ Z

i=1

−z

+ a

i=1

+ b

i=1

= 0

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