Map Projections: Conformal Projections

Map Projections

Conformal Projections

Introduction

A 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 proved by points at the map's edge).

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.

"Classic" Conformal Projections

For each of the three major projection groups, there is a single conformal design, better presented elsewhere:

the azimuthal stereographic, which has the unique property of preserving the shape of any circle on the sphere
the Mercator, a cylindrical projection which, in the normal aspect has straight vertical meridians, therefore enabling direct bearing measurements
Lambert's conformal conic, a general case of the other two projections

Like most conformal projections, those three suffer from singularities, which are points either

projected at infinity, so cannot be included in the map, or
where conformality is absent

"Lagrange" map

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°).

The "Lagrange" Projection

In the same work (1772) presenting the conic conformal, Johann Lambert included another conformal design. Its development is relatively simple but very interesting:

on the sphere, reduce meridian spacing by a factor n
still on the sphere, change parallel spacing in order to restore conformality to the compressed surface
apply an azimuthal stereographic projection in the equatorial aspect

Fortunately, the result of successive conformal transformations is itself conformal.

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.

Conformal Projections by Eisenlohr and August

Eisenlohr map

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. Scale distortion is strong near the boundary meridians, but both projections are conformal at every point, even the poles.

August map

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 + 8^1/2. 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 American polyconic, rectangular polyconic and van der Grinten's IV.

Peirce Quincuncial map

Peirce's Quincuncial conformal map in a square

Conformal Hemispheres in Squares

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.

Peirce's Quincuncial Projection

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

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.

Transverse Peirce Quincuncial map

Peirce's projection in a transverse aspect

Guyou's Projection

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

Again, scale distortion is great 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. Guyou maps can also tile the plane.

Tiled Guyou maps

Tiled Guyou maps

Hemispheres by Adams

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 scale distortion is extreme and the map is not conformal. 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 map in a square, poles at corners

Adams's world in a square (1929)

Conformal World Maps in Other Polygons

New Complex Tools

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.

Adams's world map in a square, poles at edges

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).

World Maps by Adams

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.

Lee's Conformal Maps

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, a regular tetrahedron (1965), an equilateral triangle and on 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 map in half of a regular hexagon

Xarax's world in half of a regular hexagon. Map outline by Xarax.

Newer Conformal Maps

Constant Xarax from Greece, influenced by Lee's conformal maps on polygons, 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 a three-lobed design, the result balances legibility, low interruption count and easily recognizable shapes.