In the normal aspect for the artificial group of projections
known as pseudoconic, all parallels are circular arcs
with a common central point; however, meridians are not constrained to be
straight lines, in contrast to true conic projections. The concept is
quite old and was used by Ptolemy.
Hemispheres in first Stabius-Werner projection
Cordiform Maps
Stabius-Werner Projections
The shape of maps is not constrained to rectangles, discs or
ellipses. Some are not even convex, like those created
by the beautiful projections devised (ca. 1500)
by Johann Stabius of Vienna, also known as Stab.
His three cordiform ("heart-shaped") projections
were so popularized in treatises by Johannes Werner that they
usually bear the latter's name. They all share some features in
the normal aspect:
parallels are arcs of circle centered on a pole (commonly
North), all with identical linear scale
Stabius-Werner's third projection including
the whole world but the very Eastern end of Siberia
Only parallel scaling distinguishes the three projections.
On a worldwide map drawn using the
first one, the Equator is a circle and
boundary meridians would significantly overlap, therefore the map
is normally clipped to one hemisphere. On
the third one, the equatorial scale is
slightly larger than the central meridian's, so there is a small
overlap north of the 60° parallel.
The Werner Projection
Werner (second Stabius) map,
polar aspect
The three Stabius-Werner projections are
equal-area and clearly
suggest the Earth's roundness, much like as its crust were cut at
a meridian and peeled off. However, only the second version -
known as the Werner projection - was widely used. It has
the Equator twice as long as the central meridian, therefore all
parallels are standard lines and there is no overlap.
Works by Oronce Finé (1531) and Mercator (1538) employed
a butterfly-shaped Werner map interrupted at the Equator,
with a central meridian emphasizing the Eastern hemisphere.
This projection is seldom used today in its original shape,
but it does appear in some specialized forms, notably interrupted in
irregular petals around Antarctica as an inverted
star-like
map emphasizing oceans, and combined with part of an
azimuthal
hemisphere for a more conventional star in the
"tetrahedral" projection.
A
butterfly-shaped Werner
map, interrupted at the Equator.
Once very popular for large-scale topographic maps,
the "Bonne" pseudoconic projection has generally fallen in disuse,
usually replaced by transverse Mercator maps. Although
named after the French R. Bonne (1727-1795), it was used much
earlier, ca. 1500. It preserves areas, and its shape distortion is
acceptable except far from the center.
In a Bonne map, except in one special case mentioned below all
parallels are concentric arcs of circle, all equally spaced and
all standard lines. Scale is also correct along the straight
vertical central meridian. For construction, one parallel at the
sphere is chosen, and a cone tangent at that central parallel
is built.
That parallel's radius at the map is the same as the radius along
the cone. All other parallels's radii are marked accordingly.
Bonne map, central parallel 15°S
As a consequence, each central parallel creates a
different Bonne map. Two special cases are well-known:
the Werner
projection,
with central parallel 90°N, zero radius and a
zero-sized flat "cone"
the sinusoidal
or Sanson-Flamsteed projection, with central parallel 0°.
In this particular case, the radius is infinite, the "cone"
cylindrical and every parallel just a straight line.
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![Map in Bonne's projection](javascript:showImage('Map in "Bonne"'s projection, reference parallel 45°N', '2008,2011 C.A.Furuti', 'Img/Z1/mp2_Bonne-s75-p45.png', 648, 616))
![Map in Bonne's projection](javascript:showImage('Map in "Bonne"'s projection, reference parallel 15°S', '2008,2011 C.A.Furuti', 'Img/Z1/mp2_Bonne-s75-p-15.png', 858, 480))