A hypothetical map projection faithfully reproducing all features of the original
sphere would be perfectly equidistant, i.e., distances between
*every* two points would keep the same ratio on both map and sphere;
therefore, all shapes would also be preserved.
On a flat map this property is simply not possible, as easily seen at
points at the map's edge.

Where conformality fails: singularities

In nearly every whole-world map classified as "conformal",
at least one point is
a - it may be mapped at infinity, therefore can not be included, or
- lines converge there at a different angle than on Earth, or
- along a closed curve encircling the point, the sum of angles is not 360° as it happens on Earth

Consider the polar conic conformal map above on the left; its central pole is nonconformal owing to the fact that a curve around it (green) completes a 254°33′ circuit instead of 360° like in the globe, which is represented by the partially transparent oblique azimuthal orthographic map on the right. The opposite pole (purple) is the second singular point, necessarily absent on the map. Everywhere else, the graticule lines intersect at right angles, a necessary but not sufficient condition for conformality.

Notice- how points on this map's border (e.g., at the blue disc) are technically conformal despite the unavoidable gross distance distortion relative to its neighbors near the opposite border
- on a conformal projection, scale is deformed homogeneously in all directions, therefore "small" circles are preserved. Diameters of the colored discs are the same on Earth, but vary greatly on the map, as typical of conformal projections

For many mapping applications, like topography and certain kinds
of navigation, a lesser constraint, conformality or
fidelity of shape, is the most fundamental requisite: at the
intersection of *any* two lines on the map, the angle
between is the same as between their counterparts on the sphere;
in particular, each parallel must cross every meridian at right
angles. Also, at any point the scale distortion, either compression or
exaggeration, must be the same in *all* directions.
Conformality is a strictly *local* property: angles,
consequently shapes, are not expected to be preserved much beyond
the intersection point; in fact, straight lines on the sphere are
usually curved along the plane, and vice versa.

Conformal map projections are frequently employed in large-scale applications, and seldom used for continental or world maps (those shown here are included for comparison only), although interruption may alleviate this issue. Since no conformal map can be equal-area — most in fact grossly distort dimensions far from the center of the map — conformal projections are almost never applied to thematic and statistical mapping, where comparisons based on size are common.

Albeit one very important conformal projection is one of the oldest map projections still in use, systematic understanding of requisites and properties of conformality had to wait for the development of sophisticated mathematical tools like differential calculus and complex analysis in the 18th and 19th centuries. Conversely, conformal mapping grew into an important branch of modern mathematics. It is also a remarkable engineering tool: for instance, since several flow problems can be more easily solved in regular shapes like a circle or a square, a complex region can be conformally mapped to the simple shape, the problem can be solved, then the solution transformed back to the original context via the inverse conformal mapping.

Lagrange's variations on Lambert's projection

A few maps using spherical cases of the conformal “Lagrange” projection with graticule comprising arcs of circles — from top to bottom:

- the basic projection published by J.Lambert in 1772, with the Equator as a straight line and a 1/2 factor which contracts the equatorial azimuthal stereographic projection from an infinite plane into a circular frame
- the same map with 1/2 factor but straight parallel at 65°S reduces the range of areal distortion at the inhabited lands in the north
- a reconstruction of Lagrange's map of Europe parameterized to reduce the rate of scale change near Berlin: straight parallel 18°30'S, factor 1.17 and central meridian at Berlin
- a map with straight Equator and factor 2/3, with one and a half hemispheres bounded by a circle.

- the ancient azimuthal stereographic, which has the unique additional property of preserving the shape of any circle on the sphere, no matter how large
- Mercator's, a cylindrical projection which in the normal aspect has straight vertical meridians, therefore enabling direct bearing measurements
- Lambert's conformal conic, the orthomorphic, a general case of the other two projections

Like most conformal projections, those three suffer from singularities; in particular (descriptions hold for the normal aspects), the azimuthal stereographic cannot include the point antipodal to the center of projection; Mercator's projection excludes both poles, and the conformal conic shows a single pole, which is nonconformal since the sum of angles of all meridians is less than 360°.

In a short section of his manuscript of 1772 describing two other seminal conformal projections, the conic orthomorphic and the transverse Mercator, Lambert described a relatively simple but very interesting approach:

- on the sphere, compress (less usually, expand) every meridian by multiplying its longitude by a constant factor
- still on the sphere, move the parallels along to restore conformality
- apply an azimuthal stereographic projection in the equatorial aspect

The result is a class of conformal — because the composition of sucessive conformal mappings is itself conformal— projections with graticules comprising circular arcs; the only straight lines are the central meridian and one base parallel, which always lies midway between the poles (there are two special cases noted below). Scale distortion is large near the poles, which are nonconformal.

Lambert illustrated his paper with a map using a factor 0.5 and the Equator as the straight parallel; this maps the whole Earth into a disc which of course coincides with the inner hemisphere of the original equatorial azimuthal stereographic projection, with total meridian angles at poles 180° instead of 360°. He emphasized the effect of changing the factor and base parallel, but noted other alternatives as inferior, except two special cases: the factor 1 which yields the original stereographic and 0 for the equatorial Mercator.

This class of projections, even the basic case preferred by Lambert, is usually known as the “Lagrange” projection after a famous promoter, the accomplished mathematician Joseph Lagrange who developed its ellipsoidal case and thoroughly studied its properties (1779). For instance, he proposed parameters which minimized the rate of change of scale near any given place, illustrating his point with a map for Berlin.

Today, the “Lagrange” projection is almost never used per se, but it became a fundamental step for the mathematical development of projections, since the sphere conformally mapped to a unit disc is a convenient basis for further transformations. Lambert's method of compressing longitudes was later applied by Aitoff and Hammer on the azimuthal equidistant and azimuthal equal-area projections respectively. A comparison of longitude factors on a finite base projection is instructive and maybe easier to visualize than for the azimuthal stereographic.

Avoiding singularities was a requisite of two superficially very
similar designs from Germany. Both were developed for the equatorial
aspect; the Equator and central meridian are straight lines, and
the poles are prominent cusps. Area distortion is evident near the
boundary meridians but, uniquely, both projections are conformal at
*every* point, even the poles.

The design published by Friedrich Eisenlohr in 1870 has two additional features: the scale is constant along the boundary meridians; and more remarkably, scale range is the narrowest of any conformal projection, — compare with the range for the three classic projections. Relatively complex calculations initially restricted its use, and even today it is rarely employed.

The projection designed by Friedrich August and co-developed by Bellermann was published in 1874 as an alternative to Eisenlohr's design: the range of scale distortion is wider (1 : 8) and not constant at the boundary meridians, but its construction is somewhat simpler. A world map is bounded by an epicycloid, the shape defined by a point on a circle rolling without sliding around another, fixed, circle.

Neither Eisenlohr's nor August's projection should be confused with similar-looking designs like the generalized “Lagrange” and the nonconformal American polyconic, rectangular polyconic and van der Grinten's IV.

The maturation of complex analysis led to general techniques for conformal mapping, where points of a flat surface are handled as numbers on the complex plane. In particular, three notable cartographers developed aspects of a conformal projection of one hemisphere (or the whole world, after a suitable rearrangement) on a square. All three approaches require evaluation of elliptic integrals of the first kind.

While working at the U.S. Coast and Geodetic Survey, the American
philosopher and polymath Charles Sanders Peirce disclosed
his conformal projection in 1879. In the normal aspect, it presents the
northern hemisphere in a square; the other hemisphere is split
into right four triangles symmetrically surrounding the square, akin
to star-like projections. In
effect, the whole map is a larger square, inspiring Peirce to call his
projection *quincuncial*, after the arrangement of five items in a
cross.

Peirce's projection is conformal everywhere except at the corners of the inner hemisphere — thus the midpoints of edges in the whole map — where the Equator breaks abruptly. Scale is highly stretched near those four points; conversely, polar regions are rather compressed. The Equator and four meridians are straight but broken lines; all other graticule lines are complex curves.

The eight triangles of a quincuncial map can evidently be
rearranged as a rectangle, or in a south polar aspect.
Also, the map *tessellates* the plane; i.e., with a trivial rotation,
repeated copies can completely cover (i.e., *tile*) an
arbitrary area, each copy's features exactly matching those of
its neighbors. However, points in the vicinity of the singularities
appear twice on the tesselation; with Peirce's choice of aspect,
they all fall at sea so are hardly noticeable.

Only a few years after Peirce, Émile Guyou from France presented his conformal projection (1886-1887). In its original form, it comprises the western and eastern hemispheres, each in a square; the Equator and four meridians are straight lines, two of the later broken along the squares' edges. Other meridians and parallels are complex curves.

Again, there's a wide range of scale distortion, and conformality is absent at the corners of each square, where parallels 45°N/S meet the straight meridians.

Actually, Peirce's and Guyou's projections are transverse cases of each other, emphasizing polar and equatorial aspects, respectively. With a simple rotation, identical copies of Guyou maps can also tile the plane.

Yet another development of the square theme is an oblique aspect where the poles are placed at two of the square corners. Oscar S. Adams, also a prolific member of the U.S. Coast and Geodetic Survey, presented his conformal world map in two square hemispheres em 1925.

Exactly as in the other aspects by Peirce and Guyou, at the square's corners, where conformality fails, scale distortion is extreme. Only the Equator and the central meridian are straight lines; the boundary meridians are also straight but broken at the Equator.

Despite interest due to their mathematical development, the conformal projections in squares of Peirce, Guyou and Adams have seldom been used.

Further development on conformal projections mainly relied on notable results of complex analysis:

- Riemann's theorem on conformal mapping (1851), which states conditions necessary for conformally converting between two connected plane regions, but does not describe how to achieve this mapping
- Hermann A. Schwarz's integral on the complex plane
for mapping a circle with radius 1
(called the
*unit disk*) to any regular polygon - the Schwarz-Christoffel transformation, another complex integral
independently demonstrated
by Elwin B. Christoffel in 1867 and Schwarz in 1869; it expresses
how to map between half a plane (or the unit disk) and
*any*simply connected (i.e., not self-intersecting) polygon.

Although the works of Schwarz and Christoffel realized a constructive proof of Riemann's theorem, their application to cartography (other than simple, particular instances) remained impractical for nearly one century; for most cases, they do not result in closed formulas and require solving a system of nonlinear equations. Actual mapping involves lengthy numerical evaluation by successive approximations.

Even after digital computers became generally available, results were far from uniform. Many algorithms for Schwarz-Christoffel mapping suffered from low efficiency, limited precision, or instability, i.e., failure to converge to a result, or poor handling of singularities (usually present at polygon vertices).

After presenting his conformal hemispheres in squares, Oscar S.Adams proposed two projections with a world map in a single square.

The first version (1929) has poles in opposite corners; scale distortion is extreme at each corner, which lacks conformality. The second one (1936) has poles at midpoints of opposite edges; again, there's strong scale distortion at the vertices. This projection is not conformal at each corner and at edge midpoints.

Other less-known conformal projections by Adams were based on an ellipse and several other polygons.

Laurence P. Lee, distinguished cartographer and senior officer at a national mapping agency in New Zealand, further generalized and improved the accuracy of methods for arbitrary conformal mapping. His projections included maps of the world on rectangles, ellipses, the equilateral triangle, the regular tetrahedron (1965) and other regular polyhedra (1976).

Like Adams's, Lee's designs attracted academic interest and paved the way for new mathematical achievements, but found limited usage in common maps.

Constant Xarax from Greece, influenced by Lee's conformal maps on polygons and polyhedra, Briesemeister's oblique projection and polyhedral maps in a butterfly arrangement, proposed a conformal map of the world in half a regular hexagon (2004). Essentially, Lee's tetrahedral conformal projection is drawn in the original south polar aspect, then split into three lobes, which are rearranged around the north pole; the result balances legibility, low interruption count and easily recognizable shapes. Like in the original design, conformality fails at the north pole and at the four points on the edge of the map where the three straight meridians are broken.

Assorted orthographic views of Gilbert's
double-world projection

Gilbert's projection of the world on hemispheres is conformal on the sphere, not on flat maps

Using the same approach of Lambert's compression of the equatorial azimuthal stereographic which originated the "Lagrange" projection, Edgar N. Gilbert (ca. 1970) shrunk longitudes of the sphere itself, with a multiplier 0.5, then moved the latitudes to restore conformality. The whole Earth is remapped into a hemisphere, which can be duplicated in the final “map”. Gilbert had at least one instance of this “double globe” actually assembled where each point on Earth is represented twice. The whole sphere is conformal except at either pole.

What's wrong with this globe — seriously?

In his *Mathematical Games* column
for *Scientific American* magazine, Martin Gardner
quoted Gilbert as mentioning that few people notice
anything odd about the repeating globe at his office. In
a lighthearted presentation on unusual maps, cartographer
Waldo Tobler suggests that hardly anybody objects to an
equatorial orthographic aspect of a globe with simple
meridian compression and no conformal correction.

Do such anedoctal references reflect some innate tendency to ignore geographic details, a general lack of familiarity with globes, or merely the authors's facetiousness? In 2015, Gilbert's experiment was performed by artist David Swart with his own double globe, with similar results.

Somewhat like Raisz's orthoapsidal proposals, Alan DeLucia and John Snyder applied (1986) an orthographic transformation to Gilbert's “two-world” sphere, creating a flat map centered at 5°N 5°E. As typical of orthographic views, the eastern and western borders of the map are considerably compressed; a thin lune of the Arctic Ocean appears twice and a corresponding slice of Antarctica is omitted. Visually the result resembles the globular appearance of the Lagrange and van der Grinten projections, with the polar area exaggeration reminiscent of Mercator maps. Like other orthographic representations of the sphere, the graticule comprises elliptical arcs and the map is neither conformal nor equal-area.

www.progonos.com/furuti/MapProj/Normal/ProjConf/projConf.html — August 5, 2018

Copyright © 2005 Carlos A. Furuti