Geography Reference
In-Depth Information
J. H. Lambert (1726-1777) was probably the first cartogra-
pher who compared different mappings and projections on
a mathematical basis: in order to make the mapping of the
sphere onto the plane locally similar (“in kleinsten Teilen
ahnlich”) he considered similar triangles on the sphere
and the plane, which J. H. Lambert tested with respect
to the stereographic projection as well as to the Mercator
projection:
d x = a d Q
cos Φ + b d Λ
(spherical longitude Λ, spherical latitude) Φ ) ,
d y = b d Q
cos Φ + a d Λ
(1.262)
(righthand rectangular coordinates
{
Λ, Φ
}
of the plane) .
In support of J.L. Lagrange (1736-1813), he sets d Φ/
cos Φ =d Q , which leads to the famous differential equa-
tions for two-dimensional conformal mapping ,namely
d x = −a d Q + b d Λ,
+=conformal
y +i x = f ( Q
±
i Λ )
,
=anticonformal
d y =+ b d Q + a d Λ,
(1.263)
with special reference to de Bougainville's “Traite du calcul
integral” (Paris 1756, p. 140), who in turn gave reference to
d'Alembert. It was only de Lagrange ( 1779a , b ) who could
work with the fundamental solution y +i x = f ( Q ± i Λ ).
Meanwhile L. Euler (1777) had published the same result,
finally leading to the notation of d ' Alembert-Euler equa-
tions for two-dimensional conformal mapping . Addition-
ally, note that the fundamental equations which govern
infinitesimal conformality have been written as differential
one-forms.
1-11 Areal Distortion
It isn't that they can't see the solution. It is that they can't see the problem.” (G.
K. Chesterton, The Scandal of Father Brown. The Point of a Pin.)
Fourth multiplicative and additive measures of deformation, dual deformation measures,
areomorphism, equiareal mapping.
 
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