I-2 The optimal polycylindric projection of conformal type (UPC) 631
Definition I.8 (Generalized polycylindric projection, mapping equations).
The conformal mapping of the ellipsoid-of-revolution (I.32) with semi-major axis A 1 , semi-minor axis
A 2 , and relative eccentricity squared E
2 := (A
2
1 − A
2
2 )/A
2
1 onto the developed circular cylinder C
2
R
of radius (I.31) is called a generalized polycylindric projection if the parallel circle-of-reference Φ 0 is
mapped equidistantly except for a dilatation factor ρ 0 such that the mapping equations (I.33) hold.
R = ρ 0 A 1 cos Φ 0 /
1 − E 2 sin
2 Φ 0 ,
(I.31)
X ∈ E
2
A 1 ,A 2
:=
X ∈ R
3
X
2 + Y
2
A 2
1
+
Z
2
A 2
2
= 1, A 1 ∈ R
+ , A 2 ∈ R
+ , A 1 > A 2
,
(I.32)
x = ρ 0 A 1
cos Φ 0 (Λ − Λ 0 )
1 − E 2 sin
2 Φ 0
,
y = ρ 0 A 1
cos Φ 0
1 − E 2 sin
2 Φ 0
ln
tan
π
4
+
Φ
2
1 − E sin Φ
1 + E sin Φ
E/2
.
(I.33)
The plane covered by the chart (x, y). Cartesian coordinates, with an Euclidean metric, namely
{R
2 , δ kl } (Kronecker delta, unit matrix) is generated by developing the circular cylinder C
2
R of radius
(I.31) with respect to the surface normal latitude Φ 0 of reference.
End of Definition.
Lemma I.9 (Generalized polycylindric projection, principal stretches).
With respect to the left Tissot distortion measure represented by the matrix C l G
−1
l
of the left Cauchy–
Green deformation tensor C l = J
T
l G r J l multiplied by the inverse of the left metric tensor G l , the matrix
of the metric tensor of E
2
A 1 ,A 2
, the left principal stretches of the generalized polycylindric projection
are given by
Λ 1 = Λ 2 = ρ 0
cos Φ 0
1 − E 2 sin
2 Φ 0
1 − E 2 sin
2 Φ
cos Φ
.
(I.34)
The eigenvalues {Λ 1 , Λ 2 } cover the eigenspace of the left Tissot matrix C l G
−1
l . Due to conformality,
they are identical, Λ 1 = Λ 2 = Λ S . J l denotes the left Jacobi map (dx, dy) → (dΛ, dΦ), G r the matrix of
the right metric tensor of the plane generated by developing the circular cylinder C
2
R of radius (I.31),
namely the unit matrix G r = I 2 .
End of Lemma.
Definition I.10 (Generalized polycylindric projection, Airy optimum).
The generalized polycylindric projection of conformal type of (I.33) and (I.34) of the ellipsoid-ofrevolution E
2
A 1 ,A 2
onto the developed circular cylinder C
2
R of radius (I.31) is called Airy optimal if the
deviation from an isometry (I.35) in terms of the left principal stretches {Λ 1 , Λ 2 }, in particular (I.34),
averaged over a mapping area of interest, namely the surface integral (I.36), is minimal with respect
to the unknown dilatation factor ρ 0 .
[(Λ 1 − 1)
2 + (Λ 2 − 1)
2 ]/2 ,
(I.35)
J lA :=
1
2S
area
[(Λ 1 − 1)
2 + (Λ 2 − 1)
2 ]dS = min
ρ 0
.
(I.36)
End of Definition.
Definition I.8 (Generalized polycylindric projection, mapping equations).
The conformal mapping of the ellipsoid-of-revolution (I.32) with semi-major axis A 1 , semi-minor axis
A 2 , and relative eccentricity squared E
2 := (A
2
1 − A
2
2 )/A
2
1 onto the developed circular cylinder C
2
R
of radius (I.31) is called a generalized polycylindric projection if the parallel circle-of-reference Φ 0 is
mapped equidistantly except for a dilatation factor ρ 0 such that the mapping equations (I.33) hold.
R = ρ 0 A 1 cos Φ 0 /
1 − E 2 sin
2 Φ 0 ,
(I.31)
X ∈ E
2
A 1 ,A 2
:=
X ∈ R
3
X
2 + Y
2
A 2
1
+
Z
2
A 2
2
= 1, A 1 ∈ R
+ , A 2 ∈ R
+ , A 1 > A 2
,
(I.32)
x = ρ 0 A 1
cos Φ 0 (Λ − Λ 0 )
1 − E 2 sin
2 Φ 0
,
y = ρ 0 A 1
cos Φ 0
1 − E 2 sin
2 Φ 0
ln
tan
π
4
+
Φ
2
1 − E sin Φ
1 + E sin Φ
E/2
.
(I.33)
The plane covered by the chart (x, y). Cartesian coordinates, with an Euclidean metric, namely
{R
2 , δ kl } (Kronecker delta, unit matrix) is generated by developing the circular cylinder C
2
R of radius
(I.31) with respect to the surface normal latitude Φ 0 of reference.
End of Definition.
Lemma I.9 (Generalized polycylindric projection, principal stretches).
With respect to the left Tissot distortion measure represented by the matrix C l G
−1
l
of the left Cauchy–
Green deformation tensor C l = J
T
l G r J l multiplied by the inverse of the left metric tensor G l , the matrix
of the metric tensor of E
2
A 1 ,A 2
, the left principal stretches of the generalized polycylindric projection
are given by
Λ 1 = Λ 2 = ρ 0
cos Φ 0
1 − E 2 sin
2 Φ 0
1 − E 2 sin
2 Φ
cos Φ
.
(I.34)
The eigenvalues {Λ 1 , Λ 2 } cover the eigenspace of the left Tissot matrix C l G
−1
l . Due to conformality,
they are identical, Λ 1 = Λ 2 = Λ S . J l denotes the left Jacobi map (dx, dy) → (dΛ, dΦ), G r the matrix of
the right metric tensor of the plane generated by developing the circular cylinder C
2
R of radius (I.31),
namely the unit matrix G r = I 2 .
End of Lemma.
Definition I.10 (Generalized polycylindric projection, Airy optimum).
The generalized polycylindric projection of conformal type of (I.33) and (I.34) of the ellipsoid-ofrevolution E
2
A 1 ,A 2
onto the developed circular cylinder C
2
R of radius (I.31) is called Airy optimal if the
deviation from an isometry (I.35) in terms of the left principal stretches {Λ 1 , Λ 2 }, in particular (I.34),
averaged over a mapping area of interest, namely the surface integral (I.36), is minimal with respect
to the unknown dilatation factor ρ 0 .
[(Λ 1 − 1)
2 + (Λ 2 − 1)
2 ]/2 ,
(I.35)
J lA :=
1
2S
area
[(Λ 1 − 1)
2 + (Λ 2 − 1)
2 ]dS = min
ρ 0
.
(I.36)
End of Definition.