For many mapping applications (like topography and certain kinds of navigation), a lesser constraint - fidelity of shape, or conformality, is the most fundamental requisite: the angle between any two lines on the sphere must be the same between their projected counterparts on the map; in particular, each parallel must cross every meridian at right angles. Also, scale at any point must be the same in all directions. Conformality is a strictly local property: angles (therefore shapes) are not expected to be preserved much beyond the intersection point; in fact, straight lines on the sphere are usually curved in 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). Since no conformal map can be equal-area (most in fact grossly distort dimensions far from the center of the map), conformal projections are not frequently applied to statistical mapping, where comparisons based on size are common.
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 became an important branch of modern mathematics.
Even though Lambert developed equations for general n and central parallel, the common equatorial case for n = 0.5 is better known as the "Lagrange" projection. Another accomplished mathematician, Joseph Lagrange further generalized Lambert's idea for the ellipsoid.
A "Lagrange" map can show the whole world in a circle. Also, as a consequence of the stereographic step, all meridians and parallels are circular arcs (the central meridian and central parallel are straight lines). Scale is extremely exaggerated near the poles; conformality also fails at these two points.
This projection is seldom used for actual maps. However, it is the base for many designs, because the sphere mapped on a circle is a fundamental step for conformal mapping.
The design published by Friedrich Eisenlohr in 1870 has two additional features: the scale is constant along the boundary meridians; more remarkably, scale range is the narrowest of any conformal projection: 1 to 3 + 81/2. Relatively complex calculations initially restricted its use.
The projection designed by Friedrich August and co-developed by Bellermann was published in 1874 as an alternative to Eisenlohr's design: scale range is wider and not constant at the boundary meridians, but 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 projections should be confused with
other nonconformal, similar-looking designs like the
van der Grinten's IV.
|Peirce's Quincuncial conformal map in a square|
|Tiled Peirce Quincuncial maps|
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.
Pieces of a quincuncial map can evidently be rearranged as a 2:1 rectangle. Also, the map tessellates the plane; i.e., with a trivial rotation, repeated copies can completely cover (tile) an arbitrary area, each copy's features exactly matching those of its neighbors.
|Peirce's projection in a transverse aspect|
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.
|Guyou map, central meridian 20°E|
|Oblique Guyou map, or two Adams hemispheres; central meridian 25°W|
Actually, Peirce's and Guyou's projections are transverse cases of each other, emphasizing polar and equatorial aspects, respectively. Guyou maps can also tile the plane.
|Tiled Guyou maps|
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 were seldom used.
|Adams's world in a square (1929)|
|Adams's world in a square (1936)|
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, O.S.Adams proposed two projections with a world map in a single square.
The first one (1929) has poles in opposite corners; scale distortion is extreme at each corner, which lacks conformality. The second version (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 the two poles.
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.
|Xarax's world in half of a regular hexagon. Redrawn after Xarax.|
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.