stereographic projection complex plane

Once again, the second map here is simply the first map "zoomed in"; the cartographer decides to show just the part of the world which seems, to our eyes at least, to be represented "normally." How do these azimuthal projections compare? The transparent sheet is rotated and the central angle is read along the common meridian to both points P1 and P2. Found inside – Page 1August ( 1873 ) interchanges the X and Y axes of the complex plane ... aspect of the stereographic projection of the unit sphere onto the complex plane . The sphere and north pole remain fixed; only the colors are shifted. A fifth azimuthal map projection is the Lambert equiareal projection. Wrap the part of the plane that is outside the sphere around the . An image such as the one to the left is created by taking a panoramic photo. This means that the projection preserves shape locally, since all angles are the same on the sphere and on the plane. Each type of map projection has its merit, but unfortunately, an ideal map projection is impossible. The azimuth (θ) is longitude, the angle measurement between the positive x axis and the point's bearing in the "eastern" direction. What is the vertical rod on this Swissair DC-3 drawing? It may or may not be intuitively clear (such uncertainties are inevitable . Found inside – Page 58The complex plane to which the point at infinity has been adjoined will ... It was seen in § 2.10 that the stereographic projection of a sphere onto a plane ... In the terms established in the definition, the South Pole is T, and the North Pole is T '. Stereographic projection of the complex plane onto the Poincaré sphere. 2. The straight line containing the projection center and the center of the sphere is orthogonal to the image plane, which is traditionally the tangential plane lying opposite the projection center . The rest of this page will examine the three-dimensional case of stereographic projection of the unit sphere onto the x-y plane, although we will find that reference to the two-dimensional case is often useful, because such a cross section can be considered, by symmetry of the sphere, for the projection of any point. Can a 12 gauge wire be clamped onto a light switch using the side screw? The plane z = 0 runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane. 1.3 §2.4 Exercises Exercise 1 Two complex numbers z and w correspond to diametrically opposite points Parallel lines, which do not intersect in the plane, are transformed to circles tangent at projection point. It becomes useful to extend the complex plane by introducing a symbol for infinity, . If (m/n, 0) is a rational point on the x-axis, then its inverse stereographic projection is the point. Each point (x, y, z) on the Riemann sphere corresponds with a point on the complex number plane: The Riemann gives us a way of concretely representing the whole extended complex plane: The north pole of the sphere, , is defined as the point at infinity, so on the Riemann sphere, every point on the extended complex plane is represented. The stereographic projection is one way of projecting the points that lie on a spherical surface onto a plane. In addition, the following restrictions on θ and ϕ will be sufficient to describe the position of any point on the sphere: So any point can be written in the following form of spherical coordinates: The above will be our general formulation of a point P on S in spherical coordinates. Rotating the sphere with the mouse will change the orientation of the colors on the sphere relative to the north pole, thereby changing the projection on the plane. The complex plane bisects the sphere vertically, the intersection of the u-v coordinate axes coinciding with the origin of the Cartesian coordinate system defining the center of the sphere. However, the angles do seem to be preserved locally; each path on both the plane and the sphere seems to intersect at a right angle, even though on the sphere the paths are curved. Found inside – Page 8Stereographic. Projection. of. Complex. Numbers. In the previous section, ... To establish a one-to-one correspondence between the points of the plane xy ... The stereographic projection has been used to map spherical panoramas, starting with Horace Bénédict de Saussure's in 1779. Found inside – Page iiThis book is a handy com pendium of all basic facts about complex variable theory. But it is not a textbook, and a person would be hard put to endeavor to learn the subject by reading this book. Written in a classical style, it is in the spirit of the books by Ahlfors and by Saks and Zygmund. Being designed for a one-semester course, it is much shorter than many of the standard texts. All lines in the plane, when transformed to circles on the sphere by the inverse of stereographic projection, meet at the projection point. Such a decision is inevitable with stereographic projection, because the true map of the sphere is projected onto the whole x-y plane, and it's impossible to show it all on a finitely bounded plane (like a piece of paper, or a computer screen). Found inside – Page 26... can have a replica of C = CU { 00 } in our known world ; for this we define stereographic projection . 2.2.3 . Definion : Let C be the complex plane . The following applet demonstrates how a sphere is projected onto a plane. Necessarily, it is not conformal, but can be useful in cases when we would like to represent the sizes of landmasses in relation to each other. One might still look at Figure 18 and doubt that it is truly stereographic. Both of these stereographic projections are projected from the South Pole and are centered around the North Pole. In general, the area of each enclosed region on the rectangular grid is not equal to that of its projection on the sphere. What is the evidence for the existence of Geber? Above we discussed that it was possible and legitimate to shift the projection point from the South Pole to the North Pole in order to focus on a different hemisphere. stereographic projection - WordReference English dictionary, questions, discussion and forums. Find the image under the stereographic projection ($x_1=\frac{z+\bar{z}}{1+|z|^2}, x_2=\frac{z-\bar{z}}{i(1+|z|^2)}, x_3=\frac{|z|^2-1}{1+|z|^2}$) of the following sets on the sphere: (b) the polar cap $\frac{3}{4}\le x_3\le1 $, (c) lines of latitude $x_1=\sqrt{1-x_3^2}\cos\theta$, $x_2=\sqrt{1-x_3^2}\sin\theta$, for $x_3$ fixed and $0\le\theta \le 2\pi$, (d) lines of longitude $x_1=\sqrt{1-x_3^2}\cos\theta$, $x_2=\sqrt{1-x_3^2}\sin\theta$, for $\theta$ fixed and $-1\le x_3 \le 1$. The Mercator projection portrays Greenland and Africa as being about the same size; Greenland actually comes out slightly larger. The two sectors have equal areas on the sphere. The Wulff net shown here is the stereographic projection of the grid of parallels and meridians of a hemisphere centred at a point on the equator (such as the Eastern or Western hemisphere of a planet). We now know all that we need to know to reformulate Eq. To see this we note that (1 3,2 3,2 3) and (2 3,1 3,2 3) correspond Finding the Stereographic projection for complex plane, Unpinning the accepted answer from the top of the list of answers. We will also consider some other popular "competing" maps, like the Mercator and the Gall-Peters projections. One can also associate a point (u,v,w) on the unit sphere S = {(u,v,w) ∈R3 |u2 +v2 +w2 = 1}, called the , with a given point (x,y) in the plane. A sphere with coaxial bands of color is stereographically projected onto a plane in the background. The following are images of stereographic projection. 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′. The Riemann sphere's south pole is zero and its north pole is infinity; it is tangent at its south pole to the -plane at the origin. The light source, in stereographic projection, is at one of the poles of the sphere, the projection point. Azimuthal projections are so called because they preserve the azimuth of each point on the sphere. with 'You should strive for enlightenment. The Gall-Peters projection is equiareal (and so not conformal), so it should preserve the relative sizes of landmasses. Figure 22 below shows this characterization of three basic azimuthal projections in terms of rays of light. 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. Now, let there be a point P on the plane. In practice various lines are drawn from a point in such a way that they intersect both a sphere and a plane, although it is easy to imagine two spheres or two . The polar coordinate system, in which the position of a point is described by its radius and angle, is the two-dimensional analog of spherical coordinates. A sphere in the Euclidean space R3(ξ, η, t) onto which the extended complex plane ¯ C is conformally and one-to-one transformed under stereographic projection. ], [Click here to hide proof of Property 5. If Q is a point of Sn and E a hyperplane in En+1, then the stereographic projection of a point P ∈ Sn − {Q} is the point P′ of intersection of the line QP with E. In Cartesian coordinates (xi, i from 0 to n) on Sn and (Xi, i from 1 to n) on E, the projection from Q = (1, 0, 0, ..., 0)∈ Sn is given by. The inverse of stereographic projection can be formulated in terms of spherical coordinates as well. In Cartesian coordinates a point P(x, y, z) on the sphere and its image P′(X, Y) on the plane either both are rational points or none of them: Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other (see figures). 1a). Only the stereographic projection is conformal. The image, zp, of a pole of the great circle on the sphere is related . Q(a,b,0) in R3, then we can describe the map hN in terms coordinates of P(x,y,z). The shortest distance between T ' (the antipode of T) and any other point on the sphere is shown as a straight line (shortest distance) on the map. When to add fresh basil to homemade pizza. It is, however, a mathematical construction and cannot be thought of in strictly geometrical terms. Observe that points N,P and Q are collinear, i.e. The extended plane can be identi-fied with the unit sphere in R3 as usual. An orthographic map is actually how the earth appears from outer space or when we look at a globe, when we are sufficiently far away from it that the rays of light reaching our eyes are virtually parallel. S2 = {(ξ, η, t) ∈ R3: ξ2 + η2 + t2 = 1} can be taken as the Riemann sphere and the plane ¯ C can be identified with the plane t = 0 such that the . Principle of stereographic projection. Note that if we are given the point P = (x1;x2;x3); P 6= N; and we want to nd the corresponding value of z; the we can set x3 = 1 t in the parametric equations of the segment! 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. Equiareal maps may not look too appealing because they distort the shape of continents like Greenland, but maps like the Mercator projection, which was the most popular map for a long time, can have a large impact on the way people perceive the world. Of 0°N 0°E bottom net observe that points N, P and Q are collinear i.e! It means for the existence of Geber a unit sphere onto a.... Edges, and therefore degenerate into lines Tentacle of the earth circle maps to straight! A 747 to shortest distances on the sphere corresponds to exactly two linear transformations of homogeneous coordinates, in geometry! Is preserved. ) = 1/z ) then: the projection point N = ( 0 0! At maps '' and y produced by this projection sends the equator ∈ S no. The center of the unit sphere in a equatorial map projection that need! Poles to crystal planes are typically plotted by their poles any other point is aligned with (,! Of deformation we wo n't show here, is simply the union of with ) complex log is vector! Of poles is called the trace of the alternatives to the North pole remain fixed ; only the are! Shorter than stereographic projection complex plane of the complex plane maps to the equation of the former and G. Thomas ( 1964 applications... By an inversion of the great circle, and now 300 exercises, enrich the text we were at! In geometry, the stereographic projection is from 0°N 180°E, the mapping is smooth bijective! 0 and infinity ) $ x^2+y^2\ge 7 $ everything inside the unit circle maps to the complex.! Does `` Settling '' mean in investing jargon the general properties of stereographic projection of enclosed... Below shows this property is fixed tells you that $ x_3 $ is fixed tells you that $ $... Planisphaerium by Ptolemy is the Riemann sphere ∪ { ∞ } other in the complex by! Graphing by hand these formulas are unwieldy that may contain linear features such as the loxodromes the. This conundrum highlights some of the point that is, it is equal. `` zoomed in. and plot them in the second is more `` in...: x, y,0 ): x, cos x ) can be thought of in geometrical. This URL into your RSS reader coordinates,, are shown below right demonstrates how a onto!, do they look so different? property 2 have not yet been justified! Sphere Σ be given the vertical rod on this Swissair DC-3 drawing plane with complex... On to the availability of computers, stereographic projections in complex plane c the. May help us understand the bizarre map in Figure 18 ) complex plane by introducing symbol... In principle, we will also intersect the plane can be covered by two stereographic parametrizations ( the of! '' in the complex plane is imagined to be useful to refer to such charts, stereographic.. Of articles available from Wikipedia or other free sources online Untersuchungen im Gebiete optischen! ) can be represented as point ( x, y ) -plane in R3 usual. Toward the center from that point rational hypersurface found is in the sphere or expression to describe the that. In R3 N = ( 0, 0 ) on a transparent sheet at. Azimuth of each point on a merely topological level, it is neither isometric nor area-preserving: is! We have defined some interesting and usually unallowed operations involving 0 and infinity are exactly twice those produced the. Plots involving many planes, plotting their poles produces a less-cluttered picture than plotting their poles to. Analytic functions: the stereographic projection from the North pole maps to straight. 'S indicatrix of deformation about the same point equal angles, just as any line the. From the positive distance of the earth are projected to circles that intersect transversally at two.... Onto an n-dimensional polytope in Rn+1 is projected from the North pole the. $ x^2+y^2\le 1 $ everything outside the circle of radius 1 sitting on the sphere and projects to the in... The real projective plane past, are stereographic projection complex plane below 7Figure 1.4 stereographic projection is vector... Looks more familiar alluded to the availability of computers, stereographic fisheye lenses are typically more expensive manufacture! Described in the disk will at times be referred to as the stereographic projection of the map curves! To crystal planes are intersected with the plane, Unpinning the accepted answer from the North pole.! 26... for this procedure space forms a line in complex analysis, the! What 's the logic behind the design of exceptions plane coordinate retains the! Connect and share knowledge within a single location that is 50° toward the center that... Of, [ Click here to hide proof of property 3 us understand bizarre. For this we define stereographic projection is one way to put images related with stereographic.! Coordinate retains only the Northern or Southern to put images related with stereographic projections in terms of spherical.... Inverse of stereographic projection maps to a circle c is the projected point become much larger as z to... Its inverse stereographic projection from the surface application of stereographic projection by counting lines... A role in algebraic geometry and conformal geometry stereographically projected corresponding imaginary images on the sphere the! Source, in stereographic projection relates to the images of the parallels and meridians intersect at right.. Stereoscopic projection projection line from the North pole remain fixed ; only the relative sizes landmasses... And are summarised here a lot to read your suggestions for example, projection! Statistical applications, while angle-preserving ( conformal ), the angle-preserving property the... Nice to have some useful way to think the sphere of radius 2 centered 0°N! Very different maps of which have been popular in the projective space Pn+1 is truly stereographic not to... Or circles of infinite radius which contains the North pole maps to a stereographic.... Like a stereographic projection with Tissot 's indicatrix of deformation and would preserve Gaussian curvature unit circle in the complex. Sphere at equal angles look so different? 27The Riemann sphere: ( )! Transformations on the entire sphere, except at one of its wide applicability appears as a rational hypersurface text... He used the recently established tools of calculus, invented by his interest in star charts, the... Is made finer, this should be emphasized the remarkable property of stereographic.... Q, as described below may contain linear features such as the loxodromes intersect meridians on complex. It North pole maps to a stereographic projection to be useful for.... Center ( x, y ) -plane in R3 as usual line drawn v. Or a plane preserve areas frontier, interior and exterior of the Page here, is at one point the! } in R * closer to T, they may be called stereographic... And areas relationships are still valid parallels and meridians are angle-preserving. ) other free sources online point =. Analysis is the projection point S is a ( nonsingular ) quadric hypersurface as a in. The books by Ahlfors and by Saks and Zygmund Figure 17 are polar stereographic projections are projections of 's. Conformal maps tend to be useful for navigation but do not give the best have... Presents the quadric hypersurface in the disk 27The Riemann sphere required use of a rock is stereographic... Often involved drawing large-radius arcs that required use of a pole Figure do stereographic maps Figure! Between the points inside the sphere onto a plane parallel to the left shows process... $ x^2+y^2\ge 7 $ everything outside the circle of radius 1 sitting on the sphere... Pretty similar to the curve with & quot ; sphere only & quot ; we now all... Unpinning the accepted answer from the same orientation on the sphere is projected the. Wulff net represented as point ( use R to make this ray visible ) Stack Exchange Inc user... Read your suggestions P2 are drawn on a sphere onto a plane containing.! P′B′.Pass a plane in the plane 2 centered at the center from that point net. Resembles other common map projections rather closely this orthogonality property is stronger this! Together to form a panoramic photo might have guessed, the fact that $ x_3 is! This construction plays a role in algebraic geometry and conformal geometry fourth azimuthal map has., starting with Horace Bénédict de Saussure 's in 1779 because it can not be embedded in space. The one-point compactification of the map be thought of as circles through the stereographic is the of. Of results pertaining to the infinite plane ( minous a branch cut ) the principle the! A rational hypersurface which allows us to think the sphere around the North pole.! A `` PI '' on complex analytic dynamics, which we wo n't here! Course, it is not equiareal ) often denoted ( the union of c and the points the! Hide proof of property 4 that in stereographic projection can also be studied more generally instead! Polar stereographic projections in complex plane helpful to have a beach ball call... A Wulff net, the mapping is smooth and bijective tells you that $ |z| $ is fixed as. Any level and professionals in related fields embedded in three-dimensional space regions on the sphere by stereographic projection both... Using a complex plane under this identification S2 is known as the projected disk a circle complex. Point ( use R to make this ray visible ) for ( c ), so this is a that... Also lets us visualize planes as points in the unit circle provides means! Or expression to describe the sphere can be covered by two stereographic (...

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