Stereographic projection of a circle of radius R onto the x axis. [x,y] is the projected point, and [x',0] is the projection.

In cartography and geometry, the stereographic projection is a mapping that projects each point on a sphere onto a tangent plane along a straight line from the antipode of the point of tangency (with one exception: the center of projection, antipodal to the point of tangency, is not projected to any point in the Euclidean plane; it is thought of as corresponding to a "point at infinity"). One approaches that point at infinity by continuing in any direction at all; in that respect this situation is unlike the projective plane, which has many points at infinity. Jump to: navigation, search Image File history File links Stereographic. ... Jump to: navigation, search Image File history File links Stereographic. ... Cartography or mapmaking (in Greek chartis = map and graphein = write) is the study and practice of making maps or globes. ... Jump to: navigation, search Geometry (Greek Γεωμετρια, geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... A sphere is a perfectly symmetrical geometrical object. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... Antipodal points on the surface of a sphere are diametrically opposite; on the other side of a globe. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

Notable properties

Two notable properties of this projection were demonstrated by Hipparchus: Jump to: navigation, search Hipparchus (Greek Ἳππαρχος) (ca. ...

• this mapping is conformal, i.e., it preserves the angles at which curves cross each other, and

• this mapping transforms those circles on the surface of the sphere that do not pass through the center of projection to circles on the plane. It transforms circles on the sphere that do pass through the center of projection to straight lines on the plane (these are sometimes thought of as circles through a point at infinity).

A stereographic projection is conformal and perspective but not equal area or equidistant.

In mathematics, a conformal map is a function which preserves angles. ... Image File history File links World map projection (source) File links The following pages link to this file: Map projection Stereographic projection ...

Formula

Polar coordinates

Stereographic projection of the Northern hemisphere

On a sphere, let φ be azimuth and θ be co-latitude (angular distance from the pole). Let R be the radius of the sphere. Let the points of the sphere be projected stereographically onto a plane which is tangent to the pole. Let the points of the projection have coordinates ρ_P (radial distance away from origin) and θ_P. Then the projection is This article describes some of the common coordinate systems that appear in elementary mathematics. ... Jump to: navigation, search Image File history File links Stereographic_Projection_Northern_Hemisphere. ... Jump to: navigation, search Image File history File links Stereographic_Projection_Northern_Hemisphere. ... Azimuth is the horizontal component of a direction (compass direction), measured around the horizon toward the East, i. ...

If θ_L is, instead, the latitude, then the equation for ρ_P changes to Latitude, denoted by the Greek letter φ, gives the location of a place on Earth north or south of the Equator. ...

or, equivalently,

Cartesian coordinates

There are a number of ways to perform stereographic projection onto a sphere, based on your choice of where you put the plane and the sphere. One can treat the plane as being complex numbers, and use the following pair of transformations: Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

Loxodromes on a stereographic projection

It is possible to find the equations of loxodromes on the stereographic projection. A loxodrome on a sphere is described by Line crossing all meridians at the same angle. ...

Substituting equation (1) we obtain

Equation (3) can be solved for θ_L:

Substitute equation (5) into equation (4), then simplify,

Apply the following trigonometric identity Jump to: navigation, search In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

to equation (6), yielding

Let b = −1/a; then

therefore a loxodrome on a stereographic projection is a equiangular spiral. Line crossing all meridians at the same angle. ... Logarithmic spiral (pitch 10°) Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ...

Loxodromes may also found by transforming any point with a Möbius transformation, in particular one with a "characteristic constant" that has an nonzero argument and a modulus not equal to one, and which has fixed points that map to diametrically opposite points on the sphere. Continuous iteration may be done by scaling the log of the characteristic constant. These loxodromes are a family of S-shaped Spirals with varying degrees of "tightness". In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ... In mathematics, a spiral is a curve which turns around some central point or axis, getting progressively closer to or farther from it, depending on which way you follow the curve. ...

See also

• map projection
• Riemann sphere
• projective geometry

Jump to: navigation, search The Mercator projection shows courses of constant bearing as straight lines. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ...

External link

• http://mathworld.wolfram.com/StereographicProjection.html