C10(3): The Ten Parameter Conformal Group as a Datum Transformation in Three-Dimensional Euclidean Space - Map Projections: Cartographic Information Systems - page 665

( 24.8 )
→

x μ / x 2 + c μ
(1 / x 2 +2 c v x v /x 2 )+ c 2 =

x μ + c μ x 2
1+2 c v x v + c 2 x 2

x μ =

(24.12)

Corollary 24.2 ( RT c R ).

x μ + c μ x 2
1+2 c v x v + c 2 x 2

RT c Rx μ = x μ =

(24.13)

End of Corollary.

Appendix 2

The proof that linearized observational equations for angles Ψ and distance ratios Ω of type
observed minus computed are equivariant with respect to C (3), formula ( 24.5 ), in

3 depart from
the Euclidean inner product/the Euclidean norm representation illustrated in Fig. 24.1 .

E

Definition 24.1 (angle, distance ratio).

2

2

cos Ψ := X 1 − X 0 | X 2 − X 0 , Ω = X 1 − X 0

(24.14)

X 1 − X 0 X 2 − X 0 X 2 − X 0

End of Definition.
If X 1 = X 0 + dX 1 ,X 2 = X 0 + dX 2 are infinitesimally related, then Ψ and Ω can be represented by

d X 1 d X 2

d X 1 | d X 2

d X 1 d X 1 d X 2 d X 2

cos Ψ := = (24.15)

d X 1 d X 2

2

2 = d X 1 d X 1

Ω =

d X 1

(24.16)

d X 2 d X 2 d X 2

The diffeomorphism (Jacobi map) d X → d Y = J dX
J = ∂x μ

∂x v = I + δ J

(24.17)

δ J := δ A + I δc δ A =[ δα μv ] ,δα μv := 2 ( δc μ x v

x μ δc v ) −

(24.18)

δc := 2 x λ δc λ

Corollary 24.3 (Jacobi map).

∀δ A = −δ A T

J = δ A + I δ c

(anti-symmetric) (24.19)

End of Corollary.