C10(3): The Ten Parameter Conformal Group as a Datum Transformation in Three-Dimensional Euclidean Space - Map Projections: Cartographic Information Systems - page 658

Finally Appendix 1 introduces in more detail the special conformal transformation composed by
R T C R . Appendix 2 presents the proofs that angles and distance ratios are locally equivariant with
respect to C (3) extending results by Baarda ( 1967a , b ), Grafarend and Schaffrin ( 1976 )aswellas
Grafarend et al. ( 1982 ).

24-1 The Ten Parameter Conformal Group C 10 (3)

in Three Dimensional Euclidean Space

3 := 3 ,δ μv

Dimensionless quantities like infinitesimal angles and distance ratios in E { R } equipped

with the canonical metric ds 2 =( dx 1 ) 2 +( dx 2 ) 2 +( dx 3 ) 2 = dx μ δ μv dx v , ( μ, v ) (summation

convention over repeated indices) are equivariant with respect to the ten parameter conformal
group C (3). The transformation group C (3) is build up by four irreducible constituents called
translation, rotation, dilatation and special conformal. They are represented either finite or close
to the identity in the sense of x μ = x μ + δx μ , in particular
1. translation (three parameters)

∈{ 1 , 2 , 3 }

T ( α ) x μ = x μ + α μ
δx μ = δα μ

(24.1)

SO (3) ,Λ T Λ = I 3 )

2. rotation (three parameters, orthonormal matrix Λ ∈

T ( ω ) x μ = Λ v x v

δx μ = δω v x v δΩ = δΩ T

(24.2)

−

3. dilatation (one parameter)

T ( λ ) x μ = λx μ
δx μ = x μ δλ

(24.3)

4. special conformal transformation (three parameters)
T ( λ ) x μ =

x μ + c μ x 2
1+2 c v x v + c 2 x 2

(24.4)

δx μ =( x 2 δ μv 2 x μ x v ) δc v

−

x 2 := ( x 1 ) 2 +( x 2 ) 2 +( x 3 ) 2 =

, c T x = c v x v := c 1 x 1 + c 2 x 2 + c 3 x 3 =

. The special
conformal transformation is a composite transformation R T C R where R is an inversion Rx μ / x 2
and T c a translation T c x μ = x μ + c μ .Since T 0 =1and R 2 = 1 we note this transformation goes
smoothly to the identity as the parameter c μ

x , x c , x

→ 0.
In total, the ten parameter conformal group C (3) is the composite transformation T ( c ) T ( λ )
T ( ω ) T ( α ), in particular
x μ = λΛ v 1+2 c λ x λ + c 2 x 2 − 1 ( x v + c v x 2 + α μ )
δx μ = δα μ + x v δω v + x μ δλ +( x 2 δ μv

(24.5)

− 2 x μ x v ) δc v

In terms of the ten parameters of the conformal group C (3) close to the identity ( 24.5 )canbe
written according to