Conformal map projection

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In cartography, a map projection is called conformal if any angle at anywhere on the earth (a sphere or an ellipsoid) is preserved in the image of the projection, i.e. the projection is a conformal map in the mathematical meaning.

Local conformality[edit]

Let's call a map projection locally conformal in a small domain on the earth if any angle in the domain is preserved in the image of the projection. Then any figure in the domain are nearly similar to the image on the map. This means that same lengths in the small domain are drawn as same lengths on the map (regardless of directions). Thus the projection in the small domain can be approximated by an isometric transformation. The Tissot's indicatrix of the projection around the domain is a circle.

Many map projections are locally conformal around the center points or lines, but some of these projections are "shape-distorted" away from the centers. "Shape-distortions" mean breadth-wise expansions or distortions from a square to a parallelogram. Simple magnifications or rotations are not "shape-distorted" but similar.

Azimuthal equidistant projection, Lambert cylindrical equal-area projection, sinusoidal projection, Mercator projection and stereographic projection with the center points x of projections respectively (of oblique aspects). All images look nearly same.
The same projections in the above figure, but with the center point 180°E, 0°N of projections respectively (of normal or transverse aspects). 3 projections are clearly "shape-distorted". Mercator and stereographic projections are conformal, so small figures are nearly similar, but overall figures are slightly distorted.

Properties of conformal projection[edit]

We can reword that a conformal projection is locally conformal at any point on the earth. Thus any small figure on the earth is nearly similar to the image on the map. The projection preserves the ratio of two length in the small domain. All Tissot's indicatrices of the projection are circles.

You must remark conformal projections preserve only small figures. Large figures are distorted even by conformal projections.

In a conformal projection, any small figure is similar to the image, but the ratio of similarity (scale) vary by the location. This causes the distortion of the conformal projection.

In a conformal projection, parallels and meridians cross rectangularly on the map. But the converse is not necessarily true. The counter examples are equirectangular and equal-area cylindrical projections (of normal aspects). These projections expand meridian-wise and parallel-wise by different ratios respectively. Thus parallels and meridians cross rectangularly on the map, but these projections do not preserve other angles, i.e. these projections are not conformal.

List of conformal projections[edit]

Usage of conformal projections[edit]

For large scale[edit]

Many large scale maps use conformal projections, because figures in large scale maps can be regarded as small enough. Then figures on the maps are nearly similar to themselves on the earth.

You can use a non-conformal projection in a limited domain such that the projection is locally conformal. But when you glue many maps together, you restore the roundness of the earth. If you want to make a new sheet from many maps or to change the center, you must re-project the earth to a paper.

On the other hand, seamless online maps are very large Mercator map in effect, so you can scroll and put any place to the center of the map, then the map remains conformal. But you must remark that it is difficult to compare lengths or areas of two far-off figures.

Universal Transverse Mercator coordinate system and Lambert system in France are projections trading-off between seamlessnesses and variablities of scales.

For small scale[edit]

A contour chart of scale factors of GS50 projection

Maps treating directions, like as nautical chart and aeronautical chart, are projected by conformal projections. Maps treating values whose gradients are important, like as weather map with atmospheric pressure, are also projected by conformal projections.

Very small scale maps as world map have very large variations of scale in a conformal projection, so recent world maps use other projections than conformal projections. But historically many world maps are drawn by conformal projections, like as Mercator maps or hemisphere maps by stereographic projection.

Conformal maps containing large regions vary scales by locations, so it is difficult to compare lengths or areas. But there are some techniques such that a length of 1 degree on a meridian = 111 km = 60 nautical mile. In non-conformal maps, you can not use the techniques because same lengths at a point vary the lengths on the map by the directions.

In Mercator or stereographic projection of normal aspect, scales vary by latitudes, so bar scales by latitudes are often appended. In complex projections like as of oblique aspect, contour charts of scale factors are sometimes appended.

References[edit]

Snyder, John P. (1989). An Album of Map Projections, Professional Paper 1453 (PDF). US Geological Survey.