The geometry of the earth - Physical Geodesy - page 202

Transformation of the ellipsoidal coordinates
Several important formulas for the transformation of coordinates may be
derived from Eqs. (5-61). First, let the position of P in space remain un-
changed; that is, let

δX = δY = δZ =0 . (5-62)

Determine the change of the ellipsoidal coordinates ϕ, λ, h if the dimensions
of the reference ellipsoid and its position are varied. Geometrically, this is
illustrated by Fig. 5.9. The problem is, thus, to solve equations (5-61) for
δϕ, δλ, δh , the left-hand sides being set equal to zero. To get δϕ , multiply the
first equation of (5-61) by

−

sin ϕ cos λ , the second equation of (5-61) by

− sin ϕ sin λ , and the third equation of (5-61) by cos ϕ and add all equations

obtained in this way. For δλ ,thefactorsare sin λ ,cos λ ,and0;for δh ,they

are cos ϕ cos λ ,cos ϕ sin λ ,andsin ϕ . The result is

−

aδϕ =sin ϕ cos λδx 0 +sin ϕ sin λδy 0 − cos ϕδz 0 +2 a sin ϕ cos ϕδf,

a cos ϕδλ =sin λδx 0 − cos λδy 0 ,

δa + a sin 2 ϕδf.
(5-63)
These formulas express the variations δϕ, δλ, δh at an arbitrary point in
terms of the variations δx 0 ,δy 0 ,δz 0 at a given point and the changes δa and
δf of the parameters of the reference ellipsoid. Thus, they relate two different
systems of ellipsoidal coordinates, provided these systems are so close to each
other that their differences may be considered as linear. Mathematically,
Eqs. (5-63) are infinitesimal coordinate transformations (essentially but not
exclusively orthogonal transformations); to the geodesist, they give the effect

δh = − cos ϕ cos λδx 0 − cos ϕ sin λδy 0 − sin ϕδz 0 −

P

E 2

Z

h+ ±h

h

E 1

'±' +

a+ ±a

± x 0

'

X

a

Y

Fig. 5.9. A small change of the reference ellipsoid together with a small
parallel shift