17-2 Special mapping equations 383
17-211 Equidistance and conformality on the circle-of-contact (C. Ptolemy, 85–150 AD),
compare with Fig. 17.5
We require the circle-of-contact to be mapped equidistantly and thus state (17.8) from which – together
with the cone constant n = sin Φ 0 – the integration constant c is determined as (17.9).
Λ 1 | Φ=Φ 0 =
n[R(
π
2 − Φ 0 ) + c]
R cos Φ 0
= 1 ,
(17.8)
c = R(
cos Φ 0
n
−
π
2
+ Φ 0 ) = R(cot Φ 0 −
π
2
+ Φ 0 ) .
(17.9)
Since c = 0, the image of the North Pole is a circular arc. The final mapping equations now result to
(17.10) or (17.11) with the left principal stretches (17.12). The circle-of-contact, Φ = Φ 0 , is mapped
equidistantly and conformally, i. e. Λ 1 | Φ=Φ 0 = Λ 2 | Φ=Φ 0 = 1.
α
r
=
Λ sin Φ 0
R(Φ 0 − Φ + cot Φ 0 )
,
(17.10)
x
y
= R(Φ 0 − Φ + cot Φ 0 )
cos(Λ sin Φ 0 )
sin(Λ sin Φ 0 )
,
(17.11)
Λ 1 =
sin Φ 0 (Φ 0 − Φ + cot Φ 0 )
cos Φ
,
Λ 2 = 1 .
(17.12)
Fig. 17.5. Mapping the sphere to a cone. Polar aspect, equidistant mapping of the set of meridians, equidistant
and conformal on the standard parallel Φ = Φ 0 = 30
◦ (Ptolemy projection).