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Stereographic projection

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	Topics Stereographic projection Topic Home In geometryGeometry Geometry arose as the field of knowledge dealing with spatial relationships.... , the stereographic projection is a particular mapping\|function]]) that projects a sphereSphere A sphere is a perfectly symmetrical geometrical object.... onto a planePlane (mathematics) In mathematics, a plane is a fundamental two-dimensional object.... . The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smoothSmooth function In mathematics, a smooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders:... and bijectiveBijection In mathematics, a function f from a set X to a set Y is said to be bijective if for every y in Y there i... . It is also conformalConformal map In mathematics, a conformal map is a function which preserves angles.... , meaning that it preserves angleAngle An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.... s. On the other hand, it does not preserve areaArea Area is a physical quantity expressing the size of a part of a surface.... , especially near the projection point. Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysisComplex analysis Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use i... , cartographyCartography Cartography or mapmaking is the study and practice of making maps or globes.... , geologyGeology Geology anetary geology]] refers to the application of geologic principles to other bodies of the solar system.... , and photographyPhotography Summary Photography is the process of making pictures by means of the action of light.... . In practice, the projection is carried out by computerComputer A computer is a machine for manipulating data according to a list of instructions known as a program.... or by hand using a special kind of graph paperGraph paper Graph paper is paper that is printed with fine lines making up a grid.... called a Wulff net or stereonet. Definition This section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections. The unit sphere in three-dimensional space R³ is the set of points (x, y, z) such that x² + y² + z² = 1. Let N = (0, 0, 1) be the "north pole", and let M be the rest of the sphere. The plane z = 0 runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane. For any point P on M, there is a unique line through N and P, and this line intersects the plane z = 0 in exactly one point P'. Define the stereographic projection of P to be this point P' in the plane. For the stereographic projection to be performed on a computer, it must be expressed by explicit formulas. In Cartesian coordinates (x, y, z) on the sphere and (X, Y) on the plane, the projection and its inverse are given by the formulas In spherical coordinates (f, ?) on the sphere (with f the zenith and ? the azimuth) and polar coordinates (R, T) on the plane, the projection and its inverse are Here, f is understood to have value p when R = 0. Also, there are many ways to rewrite these formulas using trigonometric identitiesList of trigonometric identities In mathematics, trigonometric identities are equalities involving trigonometric functions that are true for all values of th... . In cylindrical coordinates (r, ?, z) on the sphere and polar coordinates (R, T) on the plane, the projection and its inverse are Properties The stereographic projection defined in the preceding section sends the "south pole" (0, 0, −1) to (0, 0), the equator to the unit circleUnit circle In mathematics, a unit circle is a circle with unit radius, i.e., a circle whose radius is 1.... , the southern hemisphere to the region inside the circle, and the northern hemisphere to the region outside the circle. The projection is not defined at the projection point N = (0, 0, 1). Small neighborhoods of this point are sent to subsets of the plane far away from (0, 0). The closer P is to (0, 0, 1), the more distant its image is from (0, 0) in the plane. For this reason it is common to speak of (0, 0, 1) as mapping to "infinity" in the plane, and of the sphere as completing the plane by adding a "point at infinity". This notion finds utility in projective geometryProjective geometry Summary Projective geometry is a non-metrical form of geometry that emerged in the early 19th century.... and complex analysis. On a merely topologicalTopology Topology is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation ; these are ... level, it illustrates how the sphere is homeomorphicFacts About Homeomorphism In the mathematical field of topology a homeomorphism or topological isomorphism is a special isomorphism between top... to the one point compactification of the plane. Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other (see figures). On the other hand, stereographic projection does not preserve area; in general, the area of a region of the sphere does not equal the area of its projection onto the plane. The area element is given in (X, Y) coordinates by Along the unit circle, where X² + Y² = 1, there is no infinitesimal distortion of area. Near (0, 0) areas are distorted by a factor of 4, and near infinity areas are distorted by arbitrarily small factors. No map from the sphere to the plane can be both conformal and area-preserving. If it were, then it would be a local isometryIsometry In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism be... and would preserve Gaussian curvatureGaussian curvature In mathematics, the Gaussian curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2... . The sphere and the plane have different Gaussian curvatures, so this is impossible. The conformality of the stereographic projection implies a number of convenient geometric properties. Circles on the sphere that do not pass through the point of projection are projected to circles on the plane. Circles on the sphere that do pass through the point of projection are projected to straight lines on the plane. These lines are sometimes thought of as circles through the point at infinity, or circles of infinite radius. All lines in the plane, when transformed to circles on the sphere by the inverse of stereographic projection, intersect each other at infinity. Parallel lines, which do not intersect in the plane, are tangent at infinity. Thus all lines in the plane intersect somewhere in the sphere — either transversallyFacts About Transversality Transversality in mathematics is a notion that describes how spaces can intersect; transversality can be seen as the "opposi... at two points, or tangently at infinity. (Similar remarks hold about the real projective planeReal projective plane In mathematics, the real projective plane is the space of lines in R3 passing through the origin.... , but the intersection relationships are different there.) The loxodromes of the sphere map to curves on the plane of the form where the parameter a measures the "tightness" of the loxodrome. Thus loxodromes correspond to equiangular spirals. These spirals intersect radial lines in the plane at equal angles, just as the loxodromes intersect meridians on the sphere at equal angles. Wulff net Stereographic projection plots can be carried out by a computer using the explicit formulas given above. However, for graphing by hand these formulas are unwieldy; instead, it is common to use graph paper designed specifically for the task. To make this graph paper, one places a grid of parallelsCircle of latitude On the Earth, a circle of latitude or parallel is an imaginary east-west circle that connects all locations with a giv... and meridians on the hemisphere, and then stereographically projects these curves to the disk. The result is called a stereonet or Wulff net (named for the Russian mineralogistMineralogy Mineralogy is an earth science focused around the chemistry, crystal structure, and physical properties of minerals.... George (Yuri Viktorovich) Wulff ). In the figure, the area-distorting property of the stereographic projection can be seen by comparing a grid sector near the center of the net with one at the far right of the net. The two sectors have equal areas on the sphere. On the disk, the latter has nearly four times the area as the former; if one uses finer and finer grids on the sphere, then the ratio of the areas approaches exactly 4. The angle-preserving property of the projection can be seen by examining the grid lines. Parallels and meridians intersect at right angles on the sphere, and so do their images on the Wulff net. For an example of the use of the Wulff net, imagine that we have two copies of it on thin paper, one atop the other, aligned and tacked at their mutual center. Suppose that we want to plot the point (0.321, 0.557, -0.766) on the lower unit hemisphere. This point lies on a line oriented 60° counterclockwise from the positive x-axis (or 30° clockwise from the positive y-axis) and 50° below the horizontal plane z = 0. Once these angles are known, there are four steps: Using the grid lines, which are spaced 10° apart in the figures here, mark the point on the edge of the net that is 60° counterclockwise from the point (1, 0) (or 30° clockwise from the point (0, 1)). Rotate the top net until this point is aligned with (1, 0) on the bottom net. Using the grid lines on the bottom net, mark the point that is 50° toward the center from that point. Rotate the top net oppositely to how it was rotated before, to bring it back into alignment with the bottom net. The point marked in step 3 is then the projection that we wanted. To plot other points, whose angles are not such round numbers as 60° and 50°, one must visually interpolate between the nearest grid lines. It is helpful to have a net with finer spacing than 10°; spacings of 2° are common. To find the central angleCentral angle External links * With interactive animation... between two points on the sphere based on their stereographic plot, overlay the plot on a Wulff net and rotate the plot about the center until the two points lie on or near a meridian. Then measure the angle between them by counting grid lines along that meridian. Other formulations and generalizations Some authors define stereographic projection from the north pole (0, 0, 1) onto the plane z = −1, which is tangent to the unit sphere at the south pole (0, 0, −1). The values X and Y produced by this projection are exactly twice those produced by the equatorial projection described in the preceding section. For example, this projection sends the equator to the circle of radius 2 centered at the origin. While the equatorial projection produces no infinitesimal area distortion along the equator, this pole-tangent projection instead produces no infinitesimal area distortion at the south pole. In general, one can define a stereographic projection from any point Q on the sphere onto any plane E such that E is perpendicular to the diameter through Q, and E does not contain Q. As long as E meets these conditions, then for any point P other than Q the line through P and Q meets E in exactly one point P^′, which is defined to be the stereographic projection of P onto E. All of the formulations of stereographic projection described thus far have the same essential properties. They are smooth bijections defined everywhere except at the projection point. They are conformal and not area-preserving. More generally, stereographic projection may be applied to the n-sphere Sⁿ in (n + 1)-dimensional Euclidean spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... E^n + 1. If Q is a point of Sⁿ and E a hyperplaneHyperplane A hyperplane is a concept in geometry.... in E^n + 1, then the stereographic projection of a point P ? Sⁿ − is the point P^′ of intersection of the line with E. Still more generally, suppose that S is a (nonsingular) quadric hypersurfaceQuadric Overview In mathematics a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of zeros of ... in the projective spaceProjective space In mathematics, a projective space is a fundamental construction, obtained from a vector space over an arbitrary division ri... P^n + 1. By definition, S is the locus of zeros of a non-singular quadratic form f(x₀, ..., x_n + 1) in the homogeneous coordinatesHomogeneous coordinates In mathematics, homogeneous coordinates, introduced by August Ferdinand Mbius, allow affine transformations to be easily rep... x_i. Fix any point Q on S and a hyperplane E in P^n + 1 not containing Q. Then the stereographic projection of a point P in S − is the unique point of intersection of with E. As before, the stereographic projection is conformal and invertible outside of a "small" set. The stereographic projection presents the quadric hypersurface as a rational hypersurfaceRational variety In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to pr... . This construction plays a role in algebraic geometryAlgebraic geometry Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative... and conformal geometryConformal geometry In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a Riemannian manifold or p... . Applications within mathematics Complex analysis Although any stereographic projection misses one point on the sphere (the projection point), the entire sphere can be mapped using two projections from distinct projection points. In other words, the sphere can be covered by two stereographic parametrizationParametrization Parametrization may refer to:* A mathematical concept related to coordinate system... s (the inverses of the projections) from the plane. The parametrizations can be chosen to induce the same orientationOrientation (mathematics) In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented and which... on the sphere. Together, they describe the sphere as an oriented surfaceSurface In mathematics, specifically in topology, a surface is a two-dimensional manifold.... (or two-dimensional manifoldManifold A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in... ). This construction has special significance in complex analysis. The point (X, Y) in the real plane can be identified with the complex numberComplex number In mathematics, a complex number is a number of the form ... ? = X + iY. The stereographic projection from the north pole onto the equatorial plane is then Similarly, letting ? = X − iY be another complex coordinate, the functions define a stereographic projection from the south pole onto the equatorial plane. The transition maps between the ?- and ?-coordinates are then ? = 1 / ? and ? = 1 / ?, with ? approaching 0 as ? goes to infinity, and vice versa. This facilitates an elegant and useful notion of infinity for the complex numbers and indeed an entire theory of meromorphic functionMeromorphic function In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic o... s mapping to the Riemann sphereRiemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is the unique way of viewing the extended complex plan... . The standard metric on the unit sphere agrees with the Fubini-Study metricFubini-Study metric In mathematics, the Fubini-Study metric is a Khler metric on projective Hilbert space, that is, complex projective space ... on the Riemann sphere. Visualization of lines and planes The set of all lines through the origin in three-dimensional space forms a space called the real projective planeReal projective plane In mathematics, the real projective plane is the space of lines in R3 passing through the origin.... . This space is difficult to visualize, because it cannot be embeddedEmbedding Overview In mathematics, an embedding is one instance of some mathematical object contained within another instance, such as a group... in three-dimensional space. However, one can "almost" visualize it as a disk, as follows. Any line through the origin intersects the southern hemisphere in a point, which can then be stereographically projected to a point on a disk. Horizontal lines intersect the southern hemisphere in two antipodal pointAntipodal point In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it s... s along the equator, either of which can be projected to the disk; it is understood that antipodal points on the boundary of the disk represent a single line. (See quotient topology.) So any set of lines through the origin can be pictured, almost perfectly, as a set of points in a disk. Also, every plane through the origin intersects the unit sphere in a great circle, called the trace of the plane. This circle maps to a circle under stereographic projection. So the projection lets us visualize planes as circular arcs in the disk. Prior to the availability of computers, stereographic projections with great circles often involved drawing large-radius arcs that required use of a beam compass. Computers now make this task much easier. Further associated with each plane is a unique line, called the plane's pole, that passes through the origin and is perpendicular to the plane. This line can be plotted as a point on the disk just as any line through the origin can. So the stereographic projection also lets us visualize planes as points in the disk. For plots involving many planes, plotting their poles produces a less-cluttered picture than plotting their traces. This construction is used to visualize directional data in crystallography and geology, as described below. Other visualization Stereographic projection is also applied to the visualization of polytopePolytope In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in thre... s. In a Schlegel diagramSchlegel diagram In geometry, a Schlegel diagram is a special projection of a polytope down one dimension.... , an n-dimensional polytope in R^n + 1 is projected onto an n-dimensional sphere, which is then stereographically projected onto Rⁿ. The reduction from R^n + 1 to Rⁿ can make the polytope easier to visualize and understand. Applications to other disciplines Cartography The fact that no map from the sphere to the plane can accurately represent both angles (and thus shapes) and areas is the fundamental problem of cartography. In general, area-preserving map projectionMap projection A map projection is any method used in cartography to represent the two-dimensional curved surface of the earth or other bo... s are preferred for statisticalStatistics Statistics is a mathematical science pertaining to the collection, analysis, interpretation, and presentation of data.... applications, because they behave well with respect to integrationIntegral In calculus, the integral of a function is an extension of the concept of a sum.... , while angle-preserving (conformal) map projections are preferred for navigationNavigation There are several traditions of navigation.... . Stereographic projection falls into the second category. When the projection is centered at the Earth's north or south pole, it has additional desirable properties: It sends meridianMeridian (geography) A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations wit... s to rays emanating from the origin and parallelCircle of latitude On the Earth, a circle of latitude or parallel is an imaginary east-west circle that connects all locations with a giv... s to circles centered at the origin. Crystallography In crystallographyCrystallography Summary Crystallography is the experimental science of determining the arrangement of atoms in solids.... , the orientations of crystalCrystal In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regular... axes and faces in three-dimensional space are a central geometric concern, for example in the interpretation of X-ray and electron diffractionElectron diffraction Electron diffraction is a technique used to study matter by firing electrons at a sample and observing the resulting interfe... patterns. These orientations can be visualized as in the section Visualization of lines and planesStereographic projection In cartography and geometry, the stereographic projection is a mapping that projects each point on a sphere onto a tangent p... above. That is, crystal axes and poles to crystal planes are intersected with the northern hemisphere and then plotted using stereographic projection. A plot of poles is called a pole figure. In electron diffractionElectron diffraction Overview Electron diffraction is a technique used to study matter by firing electrons at a sample and observing the resulting interfe... , Kikuchi lineKikuchi line Kikuchi lines pair up to form bands in electron diffraction from single crystal specimens, there to serve as "roads in orient... pairs appear as bands decorating the intersection between lattice plane traces and the Ewald sphere thus providing experimental access to a crystal's stereographic projection. Model Kikuchi maps in reciprocal space, and fringe visibility maps for use with bend contours in direct space, thus act as road maps for exploring orientation space with crystals in the transmission electron microscope. Geology Researchers in structural geologyStructural geology Structural geology is the study of the three dimensional distribution of rock bodies and their planar or folded surfaces... are concerned with the orientations of planes and lines for a number of reasons. The foliationFoliation (geology) Foliation is any penetrative planar fabric present in rocks.... of a rock is a planar feature that often contains a linear feature called lineation. Similarly, a fault plane is a planar feature that may contain linear features such as slickensideSlickenside slickenside, in pedology, is a term describing the surfaces of the cracks produced in soils containing a high proportion of ... s which indicate the direction of the fault's movement. These orientations of lines and planes at various scales can be plotted using the methods of the Visualization of lines and planesStereographic projection In cartography and geometry, the stereographic projection is a mapping that projects each point on a sphere onto a tangent p... section above. As in crystallography, planes are plotted by their poles. Unlike crystallography, the southern hemisphere is used instead of the northern one (because the geological features in question lie below the Earth's surface). In this context the stereographic projection is often referred to as the equal-angle lower-hemisphere projection. The equal-area lower-hemisphere projection defined by the Lambert azimuthal equal-area projectionLambert azimuthal equal-area projection The Lambert azimuthal equal-area projection, or Lambert azimuthal projection, is an... is also used, especially when the plot is to be subjected to subsequent statistical analysis such as density contourCONTOUR The Comet Nucleus Tour was a NASA Discovery-class space probe that failed shortly after launch.... ing. Photography Some fisheye lensFisheye lens In photography, a fisheye lens is a wide-angle lens that takes in an extremely wide, hemispherical image.... es use a stereographic projection to capture a wide angle view. These lenses are usually preferred to more traditional fisheye lenses, which use an equal-area projection. This is probably a result of the conformal property of the stereographic: even areas close to the edge retain their shape, and straight lines are less curved. Unfortunately stereographic fisheye lenses are expensive to manufacture (none are currently being produced). Image remapping software, such as Panotools, allows the automatic remapping of photos from an equal-area fisheye to a stereographic projection. The stereographic projection has been used to map spherical panoramas. This results in interesting effects: the area close to the point opposite to the center of projection becomes significantly enlarged, resulting in an effect known as little planet (when the center of projection is the nadirNadir Overview The nadir is the astronomical term for the point in the sky directly below the observer, or more precisely, the point in the... ) and tube (when the center of projection is the zenithZenith In broad terms, the zenith is the direction pointing directly above a particular location .... ). Compared to other azimuthal projections, the stereographic projection tends to produce especially visually pleasing panoramas; this is due to the excellent shape preservation that is a result of the conformality of the projection. See also AstrolabeAstrolabe The astrolabe is a historical astronomical instrument used by classical astronomers and astrologers.... Astronomical clockAstronomical clock An astronomical clock is a clock with special mechanisms and dials to display astronomical information, such as the relative... External links , from radicalcartography.net et")