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The south pole is the point on a sphere with minimum -coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is defined by the rotation axis, with the south pole given by the point in the southern hemisphere that is farthest from the equator (i.e., midplane of the sphere).The coordinate in spherical coordinates is measured from the north pole and takes on the value at the south pole.

Let be the longitude, the reference longitude, the latitude, the reference latitude, and and the standard parallels. Then the transformation of spherical coordinates to the plane via the Lambert conformal conic projection is given by(1)(2)where(3)(4)(5)(6)The inverse formulas are(7)(8)where(9)(10)with , , and as defined above.

The Bonne projection is a map projection that resembles the shape of a heart. Let be the standard parallel, the central meridian, be the latitude, and the longitude on a unit sphere. Then(1)(2)where(3)(4)The illustrations above show Bonne projections for two different standard parallels.The inverse formulas are(5)(6)where(7)The Werner projection is a special case of theBonne projection.

The sinusoidal projection is an equal-area projectiongiven by the transformation(1)(2)The inverse formulas are(3)(4)

The gnomonic projection is a nonconformal map projection obtained by projecting points (or ) on the surface of sphere from a sphere's center to point in a plane that is tangent to a point (Coxeter 1969, p. 93). In the above figure, is the south pole, but can in general be any point on the sphere. Since this projection obviously sends antipodal points and to the same point in the plane, it can only be used to project one hemisphere at a time. In a gnomonic projection, great circles are mapped to straight lines. The gnomonic projection represents the image formed by a spherical lens, and is sometimes known as the rectilinear projection.In the projection above, the point is taken to have latitude and longitude and hence lies on the equator. The transformation equations for the plane tangent at the point having latitude and longitude for a projection with central longitude and central latitude are given by(1)(2)and is the angular distance of the point..

An azimuthal projection which is neither equal-area nor conformal. Let and be the latitude and longitude of the center of the projection, then the transformation equations are given by(1)(2)Here,(3)and(4)where is the angular distance from the center. The inverse formulas are(5)and(6)with the angular distance from the center given by(7)

A class of map projections in which the parallels are represented by a system of non-concentric circular arcs with centers lying on the straight line representing the central meridian (Lee 1944). The term was first applied by Hunt, and later extended by Tissot (1881).(1)(2)where(3)The inverse formulas are(4)and is determined from(5)starting with the initial vale and defining(6)(7)

The Peters projection is a cylindrical equal-area projection that de-emphasizes the exaggeration of areas at high latitudes by shifting the standard latitude to (or sometimes or ; Dana).

The orthographic projection is a projection from infinity that preserves neitherarea nor angle. It is given by(1)(2)where is the latitude, is the longitude, and and are reference longitudes and latitudes, respectively.The inverse transformations are(3)(4)where(5)(6)and the two-argument form of the inverse tangentfunction is best used for this computation.

A projection of a figure by parallel rays. In such a projection, tangencies are preserved. Parallel lines project to parallel lines. The ratio of lengths of parallel segments is preserved, as is the ratio of areas.Any triangle can be positioned such that its shadow under an orthogonal projection is equilateral. Also, the triangle medians of a triangle project to the triangle medians of the image triangle. Ellipses project to ellipses, and any ellipse can be projected to form a circle. The center of an ellipse projects to the center of the image ellipse. The triangle centroid of a triangle projects to the triangle centroid of its image. Under an orthogonal transformation, the Steiner inellipse can be transformed into a circle inscribed in an equilateral triangle.Spheroids project to ellipses (or circles in the degenerate case).In an orthogonal projection, any vector can be written , soand the projection matrix is a symmetric matrix iff the vector..

The equations are(1)(2)where is the solution to(3)This can be solved iteratively using Newton's method with to obtain(4)The inverse formulas are(5)(6)where(7)

A parametric latitude which gives a sphere equal surface area relative to an ellipsoid. The authalic latitude is defined by(1)where(2)and is evaluated at the north pole (). Let be the radius of the sphere having the same surface area as the ellipsoid, then(3)The series for is(4)The inverse formula is found from(5)where(6)and . This can be written in series form as(7)

The north pole is the point on a sphere with maximum -coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is defined by the rotation axis, with the north pole given by the point in the northern hemisphere that is farthest from the equator (i.e., midplane of the sphere).The coordinate in spherical coordinates is measured from the north pole.

The equations are(1)(2)where is the solution to(3)This can be solved iteratively using Newton's method with to obtain(4)The inverse formulas are(5)(6)where(7)

The Mollweide projection is a map projection also called the elliptical projection or homolographic equal-area projection. The forward transformation is(1)(2)where is given by(3)Newton's method can then be used to compute iteratively from(4)where(5)(Snyder 1987, p. 251) or, better yet,(6)can be used as a first guess.The inverse formulas are(7)(8)where(9)

A cylindrical projection of points on a unit sphere centered at consists of extending the line for each point until it intersects a cylinder tangent to the sphere at its equator at a corresponding point . If the sphere is tangent to the cylinder at longitude , then a point on the sphere with latitude and longitude is mapped to a point on the cylinder with height . Unwrapping and flattening out the cylinder then gives the Cartesian coordinates(1)(2)The cylindrical projection of the Earth is illustrated above.This form of the projection, however, is seldom used in practice, and the term "cylindrical projection" is used instead to refer to any projection in which lines of longitude are mapped to equally spaced parallel lines and lines of latitude (parallels) are mapped to parallel lines with arbitrary mathematically spaced separations (Snyder 1987, p. 5). For example, the common Mercator projection uses the complicated transformation(3)instead..

Several cylindrical equidistant projections were devised by R. Miller. Miller's projections have standard parallels of (giving minimal overall scale distortion), (giving minimal scale distortion over continents), and (Miller 1949).

The map projection having transformation equations(1)(2)and the inverse formulas are(3)(4)The following table gives special cases of the cylindrical equidistant projection. projection nameequirectangular projectionMiller equidistant projectionMiller equidistant projectionMiller equidistant projection

A map projection given by the following transformation,(1)(2)(3)Here, and are the plane coordinates of a projected point, is the longitude of a point on the globe, is central longitude used for the projection, and is the latitude of the point on the globe.The inverse formulas are(4)(5)(6)

A map projection. The inverse equations for are computed by iteration. Let the angle of the projection plane be . Define(1)For proper convergence, let and compute the initial point by checking(2)As long as , take and iterate again. The first value for which is then the starting point. Then compute(3)until the change in between evaluations is smaller than the acceptable tolerance. The (inverse) equations are then given by(4)(5)

A conic projection of points on a unit sphere centered at consists of extending the line for each point until it intersects a cone with apex which tangent to the sphere along a circle passing through a point in a point . For a cone with apex a height above , the angle from the z-axis at which the cone is tangent is given by(1)and the radius of the circle of tangency and height above at which it is located are given by(2)(3)Letting be the colatitude of a point on a sphere, the length of the vector along is(4)The left figure above shows the result of re-projecting onto a plane perpendicular to the z-axis (equivalent to looking at the cone from above the apex), while the figure on the right shows the cone cut along the solid line and flattened out. The equations transforming a point on a sphere to a point on the flattened cone are(5)(6)This form of the projection, however, is seldom used in practice, and the term "conic projection" is used instead to refer..

The Mercator projection is a map projection that was widely used for navigation since loxodromes are straight lines (although great circles are curved). The following equations place the x-axis of the projection on the equator and the y-axis at longitude , where is the longitude and is the latitude.(1)(2)(3)(4)(5)(6)The inverse formulas are(7)(8)(9)(10)where is the Gudermannian.An oblique form of the Mercator projection is illustrated above. It has equations(11)(12)(13)where(14)(15)(16)The inverse formulas are(17)(18)There is also a transverse form of the Mercator projection, illustrated above (Deetz and Adams 1934, Snyder 1987). It is given by the equations(19)(20)(21)(22)(23)where(24)(25)Finally, the "universal transverse Mercator projection" is a map projection which maps the sphere into 60 zones of each, with each zone mapped by a transverse Mercator projection with central meridian in the center of the..

A map projection with transformation equations(1)(2)where(3)(4)(5)(6)(7)The inverse formulas are given by(8)(9)where(10)(11)

A projection which maps a sphere (or spheroid) onto a plane. Map projections are generally classified into groups according to common properties (cylindrical vs. conical, conformal vs. area-preserving, , etc.), although such schemes are generally not mutually exclusive. Early compilers of classification schemes include Tissot (1881), Close (1913), and Lee (1944). However, the categories given in Snyder (1987) remain the most commonly used today, and Lee's terms authalic and aphylactic are not commonly encountered.No projection can be simultaneously conformaland area-preserving.

A map projection which is a conformal mapping, i.e., one for which local (infinitesimal) angles on a sphere are mapped to the same angles in the projection. On maps of an entire sphere, however, there are usually singular points at which local angles are distorted.The term conformal was applied to map projections by Gauss in 1825, and eventually supplanted the alternative terms "orthomorphic" (Lee 1944; Snyder 1987, p. 4) and "autogonal" (Tissot 1881, Lee 1944).No projection can be both equal-area and conform, and projections which are neither equal-area nor conformal are sometimes called aphylactic (Lee 1944; Snyder 1987, p. 4).

The vertical perspective projection is a map projection that corresponds to the appearance of a globe when directly viewed from some distance away with the -axis of the viewer aligned parallel to the positive -axis of the globe. It is given by the transformation equations(1)(2)where is the distance of the point of perspective in units of sphere radii and(3)(4)Note that points corresponding to are on the back side of the globe and so should be suppressed when making the projection.

A path, also known as a rhumb line, which cuts a meridian on a given surface at any constant angle but a right angle. If the surface is a sphere, the loxodrome is a spherical spiral. The loxodrome is the path taken when a compass is kept pointing in a constant direction. It is a straight line on a Mercator projection or a logarithmic spiral on a polar projection (Steinhaus 1999, pp. 218-219). The loxodrome is not the shortest distance between two points on a sphere.

Conformal latitude is defined by(1)(2)(3)The inverse is obtained by iterating the equation(4)using as the first trial. A series form is(5)The conformal latitude was called the isometriclatitude by Adams (1921), but this term is now used to refer to a different quantity.

The van der Grinten projection is a map projectiongiven by the transformation(1)(2)where(3)(4)(5)(6)(7)The inverse formulas are(8)(9)where(10)(11)(12)(13)(14)(15)(16)(17)(18)

The azimuthal coordinate on the surface of a sphere ( in spherical coordinates) or on a spheroid (in prolate or oblate spheroidal coordinates). Longitude is defined such that . Lines of constant longitude are generally called meridians. The other angular coordinate on the surface of a sphere is called the latitude.The shortest distance between any two points on a sphere is the so-called great circle distance, which can be directly computed from the latitude and longitudes of two points.

The latitude of a point on a sphere is the elevation of the point from the plane of the equator. The latitude is related to the colatitude (the polar angle in spherical coordinates) by . More generally, the latitude of a point on an ellipsoid is the angle between a line perpendicular to the surface of the ellipsoid at the given point and the plane of the equator (Snyder 1987).The equator therefore has latitude , and the north and south poles have latitude , respectively. Latitude is also called geographic latitude or geodetic latitude in order to distinguish it from several subtly different varieties of authalic latitudes.The shortest distance between any two points on a sphere is the so-called great circle distance, which can be directly computed from the latitudes and longitudes of the two points...

A map projection defined by(1)(2)The inverse formulas are(3)(4)where(5)

A map projection obtained by projecting points on the surface of sphere from the sphere's north pole to point in a plane tangent to the south pole (Coxeter 1969, p. 93). In such a projection, great circles are mapped to circles, and loxodromes become logarithmic spirals.Stereographic projections have a very simple algebraic form that results immediately from similarity of triangles. In the above figures, let the stereographic sphere have radius , and the -axis positioned as shown. Then a variety of different transformation formulas are possible depending on the relative positions of the projection plane and -axis.The transformation equations for a sphere of radius are given by(1)(2)where is the central longitude, is the central latitude, and(3)The inverse formulas for latitude and longitude are then given by(4)(5)where(6)(7)and the two-argument form of the inverse tangentfunction is best used for this computation.For an oblate..

The polar angle on a sphere measured from the north pole instead of the equator. The angle in spherical coordinates is the colatitude. It is related to the latitude by .

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