Conformal map projection

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.

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.

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.

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