Diseño de experimentación
4.1. Programación Genética aplicada a pronósticos
Units: We set R=1 throughout this section.
Consider an arbitrary conformal transformation described by the analytic function z(ζ ).
The scale factor m is the ratio of the lengths of small elements dz and dζ : m= |dz|
|dζ | =|z0(ζ )|, 1
m =|dζ |
|dz| =|ζ0(z)|. (4.70) The complex displacements may be written as
dζ =|dζ |expiα, dz=|dz|expiβ , (4.71)
so that the angle of rotation between the elements is β−α = arg z0(ζ ). This result apples to any small element but in particular it applies to an element along the ψ-axis in the ζ -plane.
If that element coincides with a meridian, as it does for NMS, then the corresponding angle in the projection z-plane defines the angle between the projected meridian and the y-axis.
This is just the convergence as defined in Section3.6. Therefore
γ= arg z0(ζ ) =−argζ0(z). (4.72) The derivative z0(ζ ) determines both the scale factor and the convergence.
The freedom to choose the derivatives of an analytic function in any direction allows us to take them along the real axis. Therefore
z0(ζ ) = xλ+ iyλ, ζ0(z) = λx+ iψx. (4.73) Since x and y are functions of λ and ψ, and vice-versa, we have
m(λ , ψ) = q
x2
λ+ y2λ, 1 m(x, y)=
q
λx2+ ψx2, (4.74) and
tan γ(λ , ψ) =yλ
xλ, tan γ(x, y) =−ψx
λx
. (4.75)
Note that
m(λ , ψ) = xλsec γ(λ , ψ), 1
m(x, y)= λxsec γ(x, y). (4.76)
Exact formulae for NMS to TMS scale factor and convergence
The real and imaginary parts of z(ζ ) = gd−1ζ are given in (4.20). Setting R=1 for clarity, tanh x= sin λ sech ψ, tan y= sec λ sinh ψ, (4.77) (sech2x) xλ = cos λ sech ψ, (sec2y) yλ = sin λ sec2λ sinh ψ, (4.78) tan λ = sinh x sec y tanh ψ= sech x sin y (4.79) (sec2λ)λx= cosh x sec y (sech2ψ)ψx=−sinhxsech2xsin y. (4.80) Simplify using
sech2x= 1− sin2λ sech2ψ= sech2ψ cosh2ψ− sin2λ, (4.81) sec2y= 1 + sec2λ sinh2ψ= sec2λ cosh2ψ− sin2λ, (4.82) sec2λ = 1 + sinh2xsec2y= sec2y[cosh2x− sin2y], (4.83) sech2ψ= 1− sech2xsin2y= sech2x[cosh2x− sin2y]. (4.84) The scale factors and convergence for the transformation z(ζ ) are therefore.
m(λ , ψ) =q
x2λ+ y2λ= 1 q
cosh2ψ− sin2λ
(4.85)
m(x, y) = 1 p
λx2+ ψx2 = q
cosh2x− sin2y, (4.86) tan γ(λ , ψ) = yλ
xλ = tan λ tanh ψ, (4.87)
tan γ(x, y) =−ψx
λx
= tanh x tan y. (4.88)
Exact formulae for sphere to TMS scale factor and convergence
The scale factor from the sphere to TMS is the product of the scale factor from the sphere to NMS and that for transformation z(ζ ) from NMS to TMS. Using the NMS scale fac-tors (2.61), (2.62) and equations (2.55), (4.79)b to express cosh ψ in terms of φ and(x, y) respectively
k(λ , φ ) = sec φ m(λ , ψ) = sec φ q
cosh2ψ− sin2λ
= 1
q
1− sin2λ cos2φ
(4.89)
k(x, y) = cosh ψ(x, y) m(x, y) =
pcosh2x− sin2y p1− sech2xsin2y
= cosh x, (4.90)
Convergence is additive but from sphere to NMS it is zero. Therefore equations4.87,4.88 give the sphere to TMS convergence. Setting tanh ψ= sin φ in4.87
γ(λ , φ ) = tan−1[tan λ sin φ ], (4.91) γ(x, y) = tan−1[tanh x tan y]. (4.92) These results are in agreement with Equations3.72–3.75(after restoring R).
Direct series for NMS to TMS scale factor and convergence
We work from the series solutions (4.37) and (4.38) in which we neglected terms of O(λ5).
Therefore we must neglect terms of O(λ4) in the expressions for the derivatives, (4.41) and (4.42), and any expressions derived from them. Inserting these derivatives into equa-tion (4.87) gives (R=1, s= sin φ etc. )
Using the series for arctan (equationE.20),
γ= tan−1(tan γ) = tan γ−1 We calculate the NMS to TMS scale factor by using (4.76) rather than (4.74). (The method is easier for the more involved series for TME.)
m(λ , φ ) = xλsec γ= c
Direct series for sphere to TMS scale factor and convergence
The sphere to TMS scale factor is given by multiplying this result by the sphere to NMS scale factor, sec φ= 1/c. in agreement with the leading terms of the expansion in equation3.74. The series (4.96) for the convergence is unchanged:
where s= sin φ etc. This result in agreement with equation3.72, neglecting O(λ4).
Inverse series for NMS to TMS scale factor and convergence
We neglected terms of O(x5) in the series for the inverse transformation, (4.51) and (4.52).
Therefore we must neglect terms of O(x4) in the expressions for the derivatives, (4.53) and (4.54) and any expressions derived from them. Inserting the derivatives into equa-tion (4.87) gives (R=1, s1= sin φ1etc.)
Using the series for arctan (equationE.20),
γ= tan−1(tan γ) = tan γ−1 Therefore the NMS to TMS scale factor expressed in projection coordinates is given by
1
Inverse series for sphere to TMS scale factor and convergence
For the TMS scale factor we must multiply the above result by the sphere to NMS scale factor, cosh ψ in (2.62). We can no longer simply use (4.79)b to express this factor in terms of x and y: we have only the series for ψ at (4.52) with R=1: derivatives of cosh are simply sinh, cosh, . . . alternating, we have
cosh ψ= cosh ψ1+ 1
1!sinh ψ1(ψ− ψ1) + 1
2!cosh ψ1(ψ− ψ1)2+··· . (4.109)
Substitute ψ−ψ1 from4.108and neglect terms of order O(x4) and, at the same time, set cosh ψ1= sec φ1and sinh ψ1= tan φ1from (2.55):
cosh ψ= 1 c1
1−1
2t12
x R
2
+···
(4.110) Therefore the sphere to TMS scale factor is
k(x, y) = cosh ψ m(x, y) =
1+1
2x2+···
, (4.111)
in agreement with the two leading terms of equation3.75 (after restoring R) which are just the leading terms of the expansion of cosh x. This result is of course trivial, but we it demonstrates the method we shall follow for the TME series.
The series (4.96) for the convergence is unchanged:
γ(x, y) = t1x
1−1
3(1 + t12)x2+···
. (4.112)
where t1= tan φ1and m(φ1) = y. This result is in agreement with equation3.73.
Chapter 5
The geometry of the ellipsoid
Abstract
Geodetic and geocentric latitude. Parameters of the ellipsoid. Relation of Cartesian and geographical coordinates. Reduced or parametric latitude. Cur-vature. Distances on the ellipsoid. Meridian distance and its inverse. Auxiliary latitudes: conformal, rectifying and authalic.
5.1 Coordinates on the ellipsoid
The Earth is more accurately modelled as an oblate ellipsoid of revolution. If the symmetry axis is taken as OZ the Cartesian equation with respect to its centre is
X2 a2 +Y2
a2 +Z2
b2 = 1, a> b. (5.1)
The definition of longitude λ is exactly the same as on the sphere. The geodetic latitude φ , which we will simply call ’latitude’, is the angle at which the normal at P intersects the equatorial plane (Z= 0). The new feature is that the normal does not pass through the
φ
λ
φ
φ
ν=
Figure 5.1
centre of the ellipsoid (except when P is on the equator and at the poles). The line joining Pto the centre defines the geocentric latitude φc. We introduce the notation p(φ ) for the distance PN of a point P from the central axis and we also set ν(φ ) for the length CP of the normal at P to its intersection with the symmetry axis. Therefore
p(φ ) = ν(φ ) cos φ =p
X2+Y2. (5.2)