72
1 From Riemann manifolds to Riemann manifolds
Historical aside.
According to the documents of Synesius (378–430), bishop of Ptolemaios, as well as of
Prokius Diadochus (412–485), a philosopher in Athens, the stereographic projection originates from Hipparch (180–125 B. C.), astronomer in Nicaea (Bythinia). His planisphere
shows the celestial sphere in a polar stereographic projection. For the use of terrestrial charts
the stereographic projection has been used for the first time by Walter Lude (1507), canonicus
in Lothringen. While his choice was polar projection, J. Stab and J. Werner (1514), respectively, used an arbitrary placement of the projection plane, finally Gemma Frisius (1540)
its equatorial placement. The particular properties of the stereographic projection, namely
conformality and the circular map of parallel circles of the sphere, has been recognized only
later: Jordanius Nemorarius (1507) mentioned the circularity of transformal parallel circles.
Gerhard Mercator (1587) invented conformality in his Duisburg map of the eastern and
western half spheres in stereographic projection. At the bottom line of his map he writes:
“. . . Etsi enim gradus a centro versus circumferentiam crescant, uti in gradibus aeqhimoctialibus vides, tamem latitudinis longitudinisque gradus in eadem a centro distantia eandem
ad invicem proportionem servant quam in sphaera et quadranguli inter duos proximos parallelos dusque meridianos rectangulam figuram habent quemadmodum in sphaera, ita ut
regiones undiquaque omnes motivam figuram obtineant sine omni tortuosa distractione.”
(Indeed though the distances grow from the center to the periphery as to be seen from the
lines of constant aequinoctium, they preserve the lengths of longitude and latitude arcs in
relative proportion with respect to the sphere. Quadrangles between to nearly parallels and
two meridians are represented by a rectangular figure like on the sphere such that all areas
keep their natural figure without distortions.) The name stereographic projection originates
from the mathematician Aguilonius (1566–1617) of Belgium. Compare with Fig. 1.28, which
gives an impression of a typical ancient map.
Historical aside.
J. H. Lambert (1726–1777) was probably the first cartographer 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:
dx = a
dQ
cos Φ
+ b dΛ
(spherical longitude Λ, spherical latitude Φ) ,
dy = b
dQ
cos Φ
+ a dΛ
(righthand rectangular coordinates {Λ, Φ} of the plane) .
(1.262)
In support of J. L. Lagrange (1736–1813), he sets dΦ/cos Φ = dQ, which leads to the famous
differential equations for two-dimensional conformal mapping, namely
dx = −a dQ + b dΛ ,
dy = +b dQ + a dΛ ,
y + ix = f (Q ± iΛ)
⎧
⎨
⎩
+ = conformal
− = anticonformal
,
(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 J. L. Lagrange (1779) who could
work with the fundamental solution y + ix = f (Q ± iΛ). Meanwhile L. Euler (1777) had
published the same result, finally leading to the notation of d’Alembert-Euler equations
for two-dimensional conformal mapping. Additionally, note that the fundamental equations
which govern infinitesimal conformality have been written as differential one-forms.
1 From Riemann manifolds to Riemann manifolds
Historical aside.
According to the documents of Synesius (378–430), bishop of Ptolemaios, as well as of
Prokius Diadochus (412–485), a philosopher in Athens, the stereographic projection originates from Hipparch (180–125 B. C.), astronomer in Nicaea (Bythinia). His planisphere
shows the celestial sphere in a polar stereographic projection. For the use of terrestrial charts
the stereographic projection has been used for the first time by Walter Lude (1507), canonicus
in Lothringen. While his choice was polar projection, J. Stab and J. Werner (1514), respectively, used an arbitrary placement of the projection plane, finally Gemma Frisius (1540)
its equatorial placement. The particular properties of the stereographic projection, namely
conformality and the circular map of parallel circles of the sphere, has been recognized only
later: Jordanius Nemorarius (1507) mentioned the circularity of transformal parallel circles.
Gerhard Mercator (1587) invented conformality in his Duisburg map of the eastern and
western half spheres in stereographic projection. At the bottom line of his map he writes:
“. . . Etsi enim gradus a centro versus circumferentiam crescant, uti in gradibus aeqhimoctialibus vides, tamem latitudinis longitudinisque gradus in eadem a centro distantia eandem
ad invicem proportionem servant quam in sphaera et quadranguli inter duos proximos parallelos dusque meridianos rectangulam figuram habent quemadmodum in sphaera, ita ut
regiones undiquaque omnes motivam figuram obtineant sine omni tortuosa distractione.”
(Indeed though the distances grow from the center to the periphery as to be seen from the
lines of constant aequinoctium, they preserve the lengths of longitude and latitude arcs in
relative proportion with respect to the sphere. Quadrangles between to nearly parallels and
two meridians are represented by a rectangular figure like on the sphere such that all areas
keep their natural figure without distortions.) The name stereographic projection originates
from the mathematician Aguilonius (1566–1617) of Belgium. Compare with Fig. 1.28, which
gives an impression of a typical ancient map.
Historical aside.
J. H. Lambert (1726–1777) was probably the first cartographer 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:
dx = a
dQ
cos Φ
+ b dΛ
(spherical longitude Λ, spherical latitude Φ) ,
dy = b
dQ
cos Φ
+ a dΛ
(righthand rectangular coordinates {Λ, Φ} of the plane) .
(1.262)
In support of J. L. Lagrange (1736–1813), he sets dΦ/cos Φ = dQ, which leads to the famous
differential equations for two-dimensional conformal mapping, namely
dx = −a dQ + b dΛ ,
dy = +b dQ + a dΛ ,
y + ix = f (Q ± iΛ)
⎧
⎨
⎩
+ = conformal
− = anticonformal
,
(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 J. L. Lagrange (1779) who could
work with the fundamental solution y + ix = f (Q ± iΛ). Meanwhile L. Euler (1777) had
published the same result, finally leading to the notation of d’Alembert-Euler equations
for two-dimensional conformal mapping. Additionally, note that the fundamental equations
which govern infinitesimal conformality have been written as differential one-forms.