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The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with the Collignon projection in polar areas. Conic Albers conic. The term "conic projection" is used to refer to any projection in which meridians are mapped to equally spaced lines radiating out from the apex and circles of latitude (parallels) are mapped to circular arcs centered on the apex.[23] When making a conic map, the map maker arbitrarily picks two standard parallels. Those standard parallels may be visualized as secant lines where the cone intersects the globe—or, if the map maker chooses the same parallel twice, as the tangent line where the cone is tangent to the globe. The resulting conic map has low distortion in scale, shape, and area near those standard parallels. Distances along the parallels to the north of both standard parallels or to the south of both standard parallels are stretched; distances along parallels between the standard parallels are compressed. When a single standard parallel is used, distances along all other parallels are stretched. The most popular conic maps include: Equidistant conic, which keeps parallels evenly spaced along the meridians to preserve a constant distance scale along each meridian, typically the same or similar scale as along the standard parallels. Albers conic, which adjusts the north-south distance between non-standard parallels to compensate for the east-west stretching or compression, giving an equal-area map. Lambert conformal conic, which adjusts the north-south distance between non-standard parallels to equal the east-west stretching, giving a conformal map. Pseudoconic Bonne Werner cordiform, upon which distances are correct from one pole, as well as along all parallels. Continuous American polyconic Azimuthal (projections onto a plane) See also: List of map projections § azimuthal An azimuthal equidistant projection shows distances and directions accurately from the center point, but distorts shapes and sizes elsewhere. Azimuthal projections have the property that directions from a central point are preserved and therefore great circles through the central point are represented by straight lines on the map. Usually these projections also have radial symmetry in the scales and hence in the distortions: map distances from the central point are computed by a function r(d) of the true distance d, independent of the angle; correspondingly, circles with the central point as center are mapped into circles which have as center the central point on the map. The mapping of radial lines can be visualized by imagining a plane tangent to the Earth, with the central point as tangent point. The radial scale is r'(d) and the transverse scale r(d)/(R sin(d/R)) where R is the radius of the Earth. Some azimuthal projections are true perspective projections; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a point of perspective (along an infinite line through the tangent point and the tangent point's antipode) onto the plane: The gnomonic projection displays great circles as straight lines. Can be constructed by using a point of perspective at the center of the Earth. r(d) = c tan(d/R); a hemisphere already requires an infinite map,[24][25] The General Perspective projection can be constructed by using a point of perspective outside the earth. Photographs of Earth (such as those from the International Space Station) give this perspective. The orthographic projection maps each point on the earth to the closest point on the plane. Can be constructed from a point of perspective an infinite distance from the tangent point; r(d) = c sin(d/R).[26] Can display up to a hemisphere on a finite circle. Photographs of Earth from far enough away, such as the Moon, give this perspective. The azimuthal conformal projection, also known as the stereographic projection, can be constructed by using the tangent point's antipode as the point of perspective. r(d) = c tan(d/2R); the scale is c/(2R cos²(d/2R)).[27] Can display nearly the entire sphere's surface on a finite circle. The sphere's full surface requires an infinite map. Other azimuthal projections are not true perspective projections: Azimuthal equidistant: r(d) = cd; it is used by amateur radio operators to know the direction to point their antennas toward a point and see the distance to it. Distance from the tangent point on the map is proportional to surface distance on the earth (;[28] for the case where the tangent point is the North Pole, see the flag of the United Nations) Lambert azimuthal equal-area. Distance from the tangent point on the map is proportional to straight-line distance through the earth: r(d) = c sin(d/2R)[29] Logarithmic azimuthal is constructed so that each point's distance from the center of the map is the logarithm of its distance from the tangent point on the Earth. r(d) = c ln(d/d0); locations closer than at a distance equal to the constant d0 are not shown (,[30] figure 6-5) Projections by preservation of a metric property A stereographic projection is conformal and perspective but not equal area or equidistant. Conformal, or orthomorphic, map projections preserve angles locally, implying that they map infinitesimal circles of constant size anywhere on the Earth to infinitesimal circles of varying sizes on the map. In contrast, mappings that are not conformal distort most such small circles into ellipses of distortion. An important consequence of conformality is that relative angles at each point of the map are correct, and the local scale (although varying throughout the map) in every direction around any one point is constant. These are some conformal projections: Mercator: Rhumb lines are represented by straight segments Transverse Mercator Stereographic: Any circle of a sphere, great and small, maps to a circle or straight line. Roussilhe Lambert conformal conic Peirce quincuncial projection Adams hemisphere-in-a-square projection Guyou hemisphere-in-a-square projection Equal-area "Area preserving maps" redirects here. For the mathematical concept, see Measure-preserving dynamical system. The equal-area Mollweide projection Equal-area maps preserve area measure, generally distorting shapes in order to do that. Equal-area maps are also called equivalent or authalic. These are some projections that preserve area: Gall orthographic (also known as Gall–Peters, or Peters, projection) Albers conic Lambert azimuthal equal-area Lambert cylindrical equal-area Mollweide Hammer Briesemeister Sinusoidal Werner Bonne Bottomley Goode's homolosine Hobo–Dyer Collignon Tobler hyperelliptical Snyder’s equal-area polyhedral projection, used for geodesic grids. Equidistant A two-point equidistant projection of Asia These are some projections that preserve distance from some standard point or line: Equirectangular—distances along meridians are conserved Plate carrée—an Equirectangular projection centered at the equator Azimuthal equidistant—distances along great circles radiating from centre are conserved Equidistant conic Sinusoidal—distances along parallels are conserved Werner cordiform distances from the North Pole are correct as are the curved distance on parallels Soldner Two-point equidistant: two "control points" are arbitrarily chosen by the map maker. Distance from any point on the map to each control point is proportional to surface distance on the earth. Gnomonic The Gnomonic projection is thought to be the oldest map projection, developed by Thales in the 6th century BC Great circles are displayed as straight lines: Gnomonic projection Retroazimuthal Direction to a fixed location B (the bearing at the starting location A of the shortest route) corresponds to the direction on the map from A to B: Littrow—the only conformal retroazimuthal projection Hammer retroazimuthal—also preserves distance from the central point Craig retroazimuthal aka Mecca or Qibla—also has vertical meridians Compromise projections The Robinson projection was adopted by National Geographic Magazine in 1988 but abandoned by them in about 1997 for the Winkel Tripel. Compromise projections give up the idea of perfectly preserving metric properties, seeking instead to strike a balance between distortions, or to simply make things "look right". Most of these types of projections distort shape in the polar regions more than at the equator. These are some compromise projections: Robinson van der Grinten Miller cylindrical Winkel Tripel Buckminster Fuller's Dymaxion B.J.S. Cahill's Butterfly Map Kavrayskiy VII Wagner VI projection Chamberlin trimetric Oronce Finé's cordiform |
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