17-1 General mapping equations 381
17-1 General mapping equations
Setting up general equations of the mapping “sphere to cone”: projections in the polar aspect. Jacobi
matrix, Cauchy–Green matrix, principal stretches.
The axis of the cone coincides with the polar axis of the Earth, i. e. the straight line passing through
the North Pole N and the center O of the sphere. The main construction principals are that first two
points of equal spherical latitude Φ have the same distance r 0 from the map center, which is the image
of the apex. Second, the cone is sliced along the image of that meridian which is diametrically opposed
to the image of the central meridian (compare with Fig. 17.4). Third, the cone can be developed into
the plane. The circle-of-contact is that parallel circle Φ = Φ 0 where the cone touches the sphere. If
necessary, the cone is shifted along the polar axis until the touching position is reached. The radius
R 0 of the circle-of-contact is given by R 0 = R cos Φ 0 . The slant height r 0 , which is the radius of the
map image of the circle-of-contact, is r 0 = R 0 / sin Φ 0 = R cot Φ 0 .
r 0 = R 0 / sin Φ 0
(slant height)
image of the
meridian
Λ = Λ P
radius r 0 = R cot Φ 0
(circle-of-contact)
image of the
central meridian
Λ = Λ 0
α
r
p
r 0 α
Fig. 17.4. Image of the developed cone.