Conformal map projection

In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth (a sphere or an ellipsoid) is preserved in the image of the projection; that is, the projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, their images on a map with a conformal projection cross at a 39° angle. A conformal projection can be defined as one that is locally conformal at every point on the map, albeit possibly with singular points where conformality fails. Thus, every small figure is nearly similar to its image on the map. The projection preserves the ratio of two lengths in the small domain. All of the projection's Tissot's indicatrices are circles. Conformal projections preserve only small figures. Large figures are distorted by even conformal projections. In a conformal projection, any small figure is similar to the image, but the ratio of similarity (scale) varies by location, which explains the distortion of the conformal projection. In a conformal projection, parallels and meridians cross rectangularly on the map. The converse is not necessarily true. The counterexamples 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. As proven by Leonhard Euler in 1775, a conformal map projection cannot be equal-area, nor can an equal-area map projection be conformal. This is also a consequence of Carl Gauss's 1827 Theorema Egregium [Remarkable Theorem]. Mercator projection (conformal cylindrical projection) Mercator projection of normal aspect (Every rhumb line is drawn as a straight line on the map.