Map Projections: Pseudoconic Projections

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Map Projections

Pseudoconic Projections

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 the north one), all with identical linear scale
the central meridian is a straight standard line
all other meridians are curves

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

Map in Stabius's second projection, best know as Werner's

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.

The Schjerning V projection

This projection is seldom seen today in its original shape, but it has been used in some specialized forms, notably in Wilhelm Schjerning's sixth projection (interrupted in irregular petals around Antarctica as an inverted star-like map emphasizing oceans), Goode's polar equal-area (similar, north polar privileging land masses, multiple central meridians) and combined with part of an azimuthal hemisphere for a more conventional star, with rescaled parallels in the "tetrahedral" and William-Olsson's projections.

In the same 1904 paper, Schjerning proposed five other projections — one equidistant conic, two modified azimuthals, and two variations of the second Stabius-Werner: Schjerning IV, an oblique aspect, and Schjerning V, a normal aspect with parallels shortened by 50%. After a proportional rescaling, the fifth design regains Werner's original area, but not the distortion-free central meridian.


A butterfly-shaped Werner map, interrupted at the Equator.	Transverse Werner map, interrupted at 25°E

The "Bonne" Projection

Bonne map, central parallel 45°N

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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