Stereographic projection

5875: 6014:, fragment 63 pp. 102–103. Dicks concludes (commentary on fragment 63, pp. 194–207): "Whether Synesius' evidence can be accepted at its face value depends on the view taken as to the strength of the objections raised above. On the whole, it would seem that the value of his testimony has been greatly exaggerated, and its unsatisfactory nature on so many points insufficiently emphasized. At any rate, the 'instrument' he sent to Paeonius was either a modified astrolabic clock of the Vitruvian type or a simple celestial map, and not a planispheric astrolabe. Furthermore, on the evidence available we are not, in my opinion, justified in attributing to Hipparchus a knowledge of either stereographic projection or the planispheric astrolabe." 5646: 5856: 5835:

various modes of rock slope failures—such as plane, wedge, and toppling failures—which occur due to the presence of unfavorably oriented discontinuities. This technique is particularly useful for visualizing the orientation of rock slopes in relation to discontinuity sets, facilitating the assessment of the most likely failure type. For instance, plane failure is more likely when the strike of a discontinuity set is parallel to the slope, and the discontinuities dip towards the slope at an angle steep enough to allow sliding, but not steeper than the slope itself.

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I, however (if it is not presumptuous to make so great a claim), have followed it to its uttermost conclusion, and have perfected it, although for most of the intervening time the problem had been neglected; for the great Ptolemy and the divine band of his successors were content to make only such use of it as sufficed for the night-clock by means of the sixteen stars, which were the only ones that Hipparchus rearranged and entered on his instrument." Translation from

500: 5364: 51: 4635: 2173: 5702: 5048:). So any set of lines through the origin can be pictured as a set of points in the projected disk. But the boundary points behave differently from the boundary points of an ordinary 2-dimensional disk, in that any one of them is simultaneously close to interior points on opposite sides of the disk (just as two nearly horizontal lines through the origin can project to points on opposite sides of the disk). 4905: 1815: 3135: 3750: 4092: 1724: 5874: 4303: 4162: 703: 1448: 3143: 1179: 4384: 3670: 5067:, 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. 4084: 2162:{\displaystyle {\begin{aligned}(x,y,z)\rightarrow (\xi ,\eta )&=\left({\frac {x}{{\frac {1}{2}}-z}},{\frac {y}{{\frac {1}{2}}-z}}\right),\\(\xi ,\eta )\rightarrow (x,y,z)&=\left({\frac {\xi }{1+\xi ^{2}+\eta ^{2}}},{\frac {\eta }{1+\xi ^{2}+\eta ^{2}}},{\frac {-1+\xi ^{2}+\eta ^{2}}{2+2\xi ^{2}+2\eta ^{2}}}\right).\end{aligned}}} 4660: 1713: 4073: 1256: 4153:

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 or left. The two sectors have equal areas on the sphere. On the disk, the latter has nearly four times the area of the former. If the

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wrote in a letter describing an instrument involving the stereographic projection: "Hipparchus long ago hinted at the unfolding of a spherical surface , so as to keep a proper proportion between the given ratios in the different figures, and he was in fact the first to apply himself to this subject.

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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

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All lines in the plane, when transformed to circles on the sphere by the inverse of stereographic projection, meet at the projection point. Parallel lines, which do not intersect in the plane, are transformed to circles tangent at projection point. Intersecting lines are transformed to circles that

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section above. As in crystallography, planes are typically 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

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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

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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

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The stereographic projection is one of the most widely used methods for evaluating rock slope stability. It allows for the representation and analysis of three-dimensional orientation data in two dimensions. Kinematic analysis within stereographic projection is used to assess the potential for

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On the Wulff net, the images of the parallels and meridians intersect at right angles. This orthogonality property is a consequence of the angle-preserving property of the stereographic projection. (However, the angle-preserving property is stronger than this property. Not all projections that

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Additionally, some authors have developed graphical methods based on stereographic projection to easily calculate geometrical correction parameters—such as those related to the parallelism between the slope and discontinuities, the dip of the discontinuity, and the relative angle between the

4630:{\displaystyle {\begin{aligned}\zeta &={\frac {x+iy}{1-z}},\\\\(x,y,z)&=\left({\frac {2\operatorname {Re} \zeta }{1+{\bar {\zeta }}\zeta }},{\frac {2\operatorname {Im} \zeta }{1+{\bar {\zeta }}\zeta }},{\frac {-1+{\bar {\zeta }}\zeta }{1+{\bar {\zeta }}\zeta }}\right).\end{aligned}}} 5890:

use a stereographic projection to capture a wide-angle view. Compared to more traditional fisheye lenses which use an equal-area projection, areas close to the edge retain their shape, and straight lines are less curved. However, stereographic fisheye lenses are typically more expensive to

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along the equator, which project to the boundary of the disk. Either of the two projected points can be considered part of the disk; it is understood that antipodal points on the equator represent a single line in 3 space and a single point on the boundary of the projected disk (see

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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. This special graph paper is called a

579:(4th century BC) have sometimes been speculatively credited with inventing or knowing of the stereographic projection, but some experts consider these attributions unjustified. Ptolemy refers to the use of the stereographic projection in a "horoscopic instrument", perhaps the 4900:{\displaystyle {\begin{aligned}\xi &={\frac {x-iy}{1+z}},\\(x,y,z)&=\left({\frac {2\operatorname {Re} \xi }{1+{\bar {\xi }}\xi }},{\frac {-2\operatorname {Im} \xi }{1+{\bar {\xi }}\xi }},{\frac {1-{\bar {\xi }}\xi }{1+{\bar {\xi }}\xi }}\right)\end{aligned}}} 2825: 4226:

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.

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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. These properties can be verified by using the expressions of

1443:{\displaystyle {\begin{aligned}(R,\Theta )&=\left({\frac {\sin \varphi }{1-\cos \varphi }},\theta \right)=\left(\cot {\frac {\varphi }{2}},\theta \right),\\(\varphi ,\theta )&=\left(2\arctan {\frac {1}{R}},\Theta \right).\end{aligned}}} 4329:

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

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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

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be the point on the lower unit hemisphere whose spherical coordinates are (140°, 60°) and whose Cartesian coordinates are (0.321, 0.557, −0.766). This point lies on a line oriented 60° counterclockwise from the positive

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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

5501: 1174:{\displaystyle {\begin{aligned}(X,Y)&=\left({\frac {x}{1-z}},{\frac {y}{1-z}}\right),\\(x,y,z)&=\left({\frac {2X}{1+X^{2}+Y^{2}}},{\frac {2Y}{1+X^{2}+Y^{2}}},{\frac {-1+X^{2}+Y^{2}}{1+X^{2}+Y^{2}}}\right).\end{aligned}}} 4218:

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.

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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)).

2672: 2555: 3397: 3258: 3274:, there is no inflation of area in the limit, giving a scale factor of 1. Near (0, 0) areas are inflated by a factor of 4, and near infinity areas are inflated by arbitrarily small factors. 5925:

The popularity of using stereographic projections to map panoramas over other azimuthal projections is attributed to the shape preservation that results from the conformality of the projection.

4665: 4389: 1820: 1515: 1261: 884: 1708:{\displaystyle {\begin{aligned}(R,\Theta )&=\left({\frac {r}{1-z}},\theta \right),\\(r,\theta ,z)&=\left({\frac {2R}{1+R^{2}}},\Theta ,{\frac {R^{2}-1}{R^{2}+1}}\right).\end{aligned}}} 4233: 613:("star taker"), a capable portable device which could be used for measuring star positions and performing a wide variety of astronomical calculations. The astrolabe was in continuous use by 5039: ≤ 0 in a point, which can then be stereographically projected to a point on a disk in the XY plane. Horizontal lines through the origin intersect the southern hemisphere in two 5576: 5689:. This property is valuable in planetary mapping where craters are typical features. The set of circles passing through the point of projection have unbounded radius, and therefore 449:

Intuitively, 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

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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

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Rotate the top net oppositely to how it was oriented before, to bring it back into alignment with the bottom net. The point marked in step 3 is then the projection that we wanted.

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A Cartesian grid on the plane appears distorted on the sphere. The grid lines are still perpendicular, but the areas of the grid squares shrink as they approach the north pole.

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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

4068:{\displaystyle \triangle NOP^{\prime }\sim \triangle P^{\prime \prime }OS\implies OP^{\prime }:ON=OS:OP^{\prime \prime }\implies OP^{\prime }\cdot OP^{\prime \prime }=r^{2}} 3146:

A polar grid on the plane appears distorted on the sphere. The grid curves are still perpendicular, but the areas of the grid sectors shrink as they approach the north pole.

2953:. As before, the stereographic projection is conformal and invertible on a non-empty Zariski open set. The stereographic projection presents the quadric hypersurface as a 3562: 5633: 5782:

Use of lower hemisphere stereographic projection to plot planar and linear data in structural geology, using the example of a fault plane with a slickenside lineation

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The fundamental problem of cartography is that no map from the sphere to the plane can accurately represent both angles and areas. In general, area-preserving

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For an example of the use of the Wulff net, imagine two copies of it on thin paper, one atop the other, aligned and tacked at their mutual center. Let

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of the plane containing a circle on the sphere, and clearing denominators, one gets the equation of a circle, that is, a second-degree equation with

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Elkins, James (1988). "Did Leonardo Develop a Theory of Curvilinear Perspective?: Together with Some Remarks on the 'Angle' and 'Distance' Axioms".

7687: 7483: 7473: 7393: 5958: 5819: 618: 3746:. These spirals intersect radial lines in the plane at equal angles, just as the loxodromes intersect meridians on the sphere at equal angles. 1753:

from it to the polar plane. The homothety scales the image by a factor of 2 (a ratio of a diameter to a radius of the sphere), hence the values

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German, Daniel; Burchill, L.; Duret-Lutz, A.; Pérez-Duarte, S.; Pérez-Duarte, E.; Sommers, J. (June 2007). "Flattening the Viewable Sphere".

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can be thought of as parametrizing the unit circle. The stereographic projection gives an alternative parametrization of the unit circle:

6575: 585: 3295: 7488: 7289: 5754: 5015: 3071: 7621: 7611: 7606: 7581: 7573: 7234: 7160: 7117: 7112: 7087: 7079: 6871: 6852: 6812: 6789: 6770: 6714: 6647: 6628: 6375: 5903: 5855: 3169: 8012: 7809: 7736: 7692: 7388: 5767: 1470: 5011: 2990:= (0, 0, 1). Small neighborhoods of this point are sent to subsets of the plane far away from (0, 0). The closer 7857: 7804: 6395: 5358: 648:(1595), and many others. In star charts, even this equatorial aspect had been utilised already by the ancient astronomers like 297: 8099: 7965: 7934: 7508: 7357: 7135: 7064: 511:. It demonstrates the principle of a general perspective projection, of which the stereographic projection is a special case. 6148: 2820:{\displaystyle x_{0}={\frac {s^{2}-1}{s^{2}+1}}\quad {\text{and}}\quad x_{i}={\frac {2X_{i}}{s^{2}+1}}\quad (i=1,\dots ,n).} 571:

which is crucial in proving the property that the stereographic projection maps circles to circles. Hipparchus, Apollonius,

8049: 8017: 7867: 7498: 7322: 7155: 7145: 6977: 6664: 2981:, the southern hemisphere to the region inside the circle, and the northern hemisphere to the region outside the circle. 8109: 8007: 7721: 7375: 7284: 6690: 6559: 6324:

Wulff, George, Untersuchungen im Gebiete der optischen Eigenschaften isomorpher Kristalle: Zeits. Krist.,36, 1–28 (1902)

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The first stereographic projection defined in the preceding section sends the "south pole" (0, 0, −1) of the

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along the equator, this pole-tangent projection instead produces no infinitesimal area distortion at the south pole.

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However, one can visualize it as a disk, as follows. Any line through the origin intersects the southern hemisphere

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No map from the sphere to the plane can be both conformal and area-preserving. If it were, then it would be a local

673:; however, this proof was never published and sat among his papers in a box for more than three centuries. In 1695, 7882: 7726: 7317: 7150: 7140: 6959:, a web application for stereographic projection in structural geology and fault kinematics by Ernesto Cristallini. 5130: 4986:. This facilitates an elegant and useful notion of infinity for the complex numbers and indeed an entire theory of 4331: 3014: 5518: 5018:

of four of the eight <111> zones in an fcc crystal. Planes edge-on (banded lines) intersect at fixed angles.

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define a stereographic projection from the south pole onto the equatorial plane. The transition maps between the

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axes and faces in three-dimensional space are a central geometric concern, for example in the interpretation of

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According to (Elkins, 1988) who references Eckert, "Die Kartenwissenschaft", Berlin 1921, pp 121–123

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M. von Heimendahl, W. Bell and G. Thomas (1964) Applications of Kikuchi line analyses in electron microscopy,

4999: 655: 621:. It was transmitted to Western Europe during the 11th–12th century, with Arabic texts translated into Latin. 508: 6505:"The value of rock mass classification systems for weak rock masses: a case example from Huntly, New Zealand" 6207:

Timothy Feeman. 2002. "Portraits of the Earth: A Mathematician Looks at Maps". American Mathematical Society.

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is also used, especially when the plot is to be subjected to subsequent statistical analysis such as density

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The generation of a Wulff net (circular net within the red circle) by a stereographic projection with center

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appear circular in this projection, regardless of whether they are close to the pole or the edge of the map.

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This construction is used to visualize directional data in crystallography and geology, as described below.

2879: 1474: 516: 96: 91: 72: 8064: 7697: 7672: 7214: 7004: 6457: 5949: 5665: 5652: 439: 362: 5878:"Vue circulaire des montagnes qu'on découvre du sommet du Glacier de Buet", Horace-Benedict de Saussure, 1745:, which is tangent to the unit sphere at the south pole (0, 0, −1). This can be described as a 7987: 7777: 7731: 7558: 7535: 7518: 7229: 6349:

P. Fraundorf, Wentao Qin, P. Moeck and Eric Mandell (2005) Making sense of nanocrystal lattice fringes,

5778: 5023: 4334:(the inverses of the projections) from the plane. The parametrizations can be chosen to induce the same 4269:

The transparent sheet is rotated and the central angle is read along the common meridian to both points

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runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane.

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pass through the point of projection are projected to circles on the plane. Circles on the sphere that

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The origin of the stereographic projection is not known, but it is believed to have been discovered by

7992: 7887: 7667: 7662: 7657: 7634: 7629: 7550: 7312: 7252: 7224: 7209: 7204: 7199: 7194: 6888: 6516: 6469: 6104: 5895:, allows the automatic remapping of photos from an equal-area fisheye to a stereographic projection. 5750: 5735: 5620: 4987: 4211:

Using the grid lines on the bottom net, mark the point that is 50° toward the center from that point.

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and its applications, so does the stereographic projection; it finds use in diverse fields including

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Some authors define stereographic projection from the north pole (0, 0, 1) onto the plane

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The stereographic projection from the North pole of a sphere to its equatorial plane establishes a

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at two points in the sphere, one of which is the projection point. (Similar remarks hold about the

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These orientations of lines and planes at various scales can be plotted using the methods of the

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3D illustration of a stereographic projection from the north pole onto a plane below the sphere

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Also, every plane through the origin intersects the unit sphere in a great circle, called the

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The set of all lines through the origin in three-dimensional space forms a space called the

3749: 3475: 3403: 2954: 2840: 614: 520: 499: 454: 443: 339: 327: 234: 224: 6897: 5146:, stereographic projection from the unit circle provides a means to describe all primitive 5138:

on a circle correspond, under stereographic projection, to the rational points of the line.

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Voyage dans les Alpes, précédés d'un essai sur l'histoire naturelle des environs de Geneve

5799: 5723: 5339:{\displaystyle \left({\frac {2mn}{m^{2}+n^{2}}},{\frac {m^{2}-n^{2}}{m^{2}+n^{2}}}\right)} 5157: 5040: 4102: 3835: 3628: 2320: 645: 355: 269: 259: 131: 6910:

Proof about Stereographic Projection taking circles in the sphere to circles in the plane

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are inversive images of each other in the image of the equatorial circle if and only if

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pairs appear as bands decorating the intersection between lattice plane traces and the

5605: 5180: 5135: 4991: 4362: 4323: 3062: 666: 532: 408: 382: 283: 214: 50: 6528: 3425:. The sphere and the plane have different Gaussian curvatures, so this is impossible. 8088: 6801: 6753: 6194: 5790:

are concerned with the orientations of planes and lines for a number of reasons. The

5706: 5496:{\displaystyle \cos x={\frac {1-t^{2}}{1+t^{2}}},\quad \sin x={\frac {2t}{t^{2}+1}}.} 4378:. The stereographic projection from the north pole onto the equatorial plane is then 4319: 4223: 4087:

Wulff net or stereonet, used for making plots of the stereographic projection by hand

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from the entire sphere except the center of projection to the entire plane. It maps

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Sleeswyk, A.W.; Huldén, B. (1991). "The three waterclocks described by Vitruvius".

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Stereographic projection of the spherical panorama of the Last Supper sculpture by

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applications, while angle-preserving (conformal) map projections are preferred for

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Rotate the top net until this point is aligned with (1, 0) on the bottom net.

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According to (Snyder 1993), although he acknowledges he did not personally see it

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The stereographic projection relates to the plane inversion in a simple way. Let

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discontinuity and the slope—for rock mass classifications in slopes, including

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Stereographic projection of the unit sphere from the north pole onto the plane

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Stereographic projection of the unit sphere from the north pole onto the plane

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by the inverse stereographic projection from the plane to the sphere defines a

6078: 5613: 5609: 4302: 3142: 2347: 681:, was the first to publish a proof. He used the recently established tools of 678: 572: 553: 538: 523:

to the plane so that the motions of stars and planets could be analyzed using

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preserve the orthogonality of parallels and meridians are angle-preserving.)

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on the foundations of geometry, delivered at Göttingen in 1854, and entitled

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of a rock is a planar feature that often contains a linear feature called

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This construction has special significance in complex analysis. The point

2662:{\displaystyle s^{2}=\sum _{j=1}^{n}X_{j}^{2}={\frac {1+x_{0}}{1-x_{0}}},} 8054: 6940: 6695:

Representations of the Rotation and Lorentz Groups and their Applications

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measures the "tightness" of the loxodrome. Thus loxodromes correspond to

3418: 3006: 2299: 682: 662:(Six Books of Optics, useful for philosophers and mathematicians alike). 628:

aspect of the stereographic projection was commonly used for maps of the

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Further associated with each plane is a unique line, called the plane's

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The Wulff net shown here is the stereographic projection of the grid of

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DoITPoMS Teaching and Learning Package - "The Stereographic Projection"

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on the plane, the projection and its inverse are given by the formulas

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for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by

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Stereographic Projection Techniques for Geologists and Civil Engineers

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Stereographic Projection Techniques for Geologists and Civil Engineers

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are drawn on a transparent sheet tacked at the origin of a Wulff net.

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In general, one can define a stereographic projection from any point

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More generally, stereographic projection may be applied to the unit

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gave the stereographic projection its current name in his 1613 work

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plane is a planar feature that may contain linear features such as

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Stereographic projection of the world north of 30°S. 15° graticule.

5010: 2550:{\displaystyle X_{i}={\frac {x_{i}}{1-x_{0}}}\quad (i=1,\dots ,n).} 6953:

on the sphere, including a stereographic projection display option

6918: 5911: 5873: 5854: 5777: 5700: 5664: 5362: 5129: 5009: 4301: 4160: 4082: 3748: 3668: 3392:{\displaystyle {\frac {4}{(1+X^{2}+Y^{2})^{2}}}\;(dX^{2}+dY^{2}),} 3141: 3133: 2171: 1722: 701: 498: 378: 25: 6962: 5738:

patterns. These orientations can be visualized as in the section

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Stereographic projection is also applied to the visualization of

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on the sphere. Together, they describe the sphere as an oriented

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of a projection onto the equatorial plane described above, and a

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A comprehensive introduction to differential geometry, Volume IV

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Tomás, R.; Cuenca, A.; Cano, M.; García-Barba, J. (2012-01-04).

5670: 4195:. Once these angles are known, there are four steps to plotting 3124:{\displaystyle P\in \mathbb {Q} ^{3}\iff P'\in \mathbb {Q} ^{2}} 8038: 7835: 7451: 7027: 6966: 1469:= 0. Also, there are many ways to rewrite these formulas using 419:

between the spherical points they represent. A two-dimensional

6956: 5219:-axis, then its inverse stereographic projection is the point 3654:

that is, if the plane passes through the point of projection.

6914: 5026:. This plane is difficult to visualize, because it cannot be 4165:

Illustration of steps 1–4 for plotting a point on a Wulff net

3511:: using these expressions for a substitution in the equation 5898:

The stereographic projection has been used to map spherical

5818:. The equal-area lower-hemisphere projection defined by the 5673:, showing regions polewards of 60° North. Craters which are 4310:

between the sphere and the equatorial plane extended with a

4047: 4044: 4028: 4010: 4007: 3973: 3949: 3946: 3930: 3253:{\displaystyle dA={\frac {4}{(1+X^{2}+Y^{2})^{2}}}\;dX\;dY.} 660:

Opticorum libri sex philosophis juxta ac mathematicis utiles

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Map Projections − A Working Manual, Professional Paper 1395

3666:, but the intersection relationships are different there.) 640:

was in stereographic projection, as were later the maps of

6503:

Moon, Vicki; Russell, Geoff; Stewart, Meagan (July 2001).

5122:

can make the polytope easier to visualize and understand.

4150:(such as the Eastern or Western hemisphere of a planet). 3412:Über die Hypothesen welche der Geometrie zu Grunde liegen 2282:, which is defined to be the stereographic projection of 636:. It is believed that already the map created in 1507 by 423:

on the stereographic plane is an alternative setting for

3881:

in a stereographic projection with the projection point

3861:

in a stereographic projection with the projection point

617:

astronomers, and was significantly further developed by

605:(4th century), the planisphere had been combined with a 6095:(1953). "The Plane Astrolabe and the Anaphoric Clock". 5681:

The stereographic is the only projection that maps all

5349:

which gives Euclid's formula for a Pythagorean triple.

3906:

are inversive images of each other in the unit circle.

3816:

are reflections of each other in the equatorial plane.

2854:

is the locus of zeros of a non-singular quadratic form

537:(2nd century AD), but it was ambiguously attributed to 5810: 5739: 3625:

as its quadratic part. The equation becomes linear if

2984:

The projection is not defined at the projection point

16:

Particular mapping that projects a sphere onto a plane

6660:

Feature column February 2014:Stereographic Projection

6427:(2 ed.). Cambridge: Cambridge University Press. 5521: 5391: 5228: 4663: 4387: 4154:

grid is made finer, this ratio approaches exactly 4.

4105: 3915: 3694: 3685:

of the sphere map to curves on the plane of the form

3631: 3570: 3517: 3478: 3446: 3298: 3172: 3074: 2675: 2563: 2470: 1818: 1513: 1259: 882: 6833:

An Album of Map Projections, Professional Paper 1453

6225: 5156:-axis gives a one-to-one correspondence between the 7978: 7933: 7924: 7901: 7848: 7791: 7768: 7750: 7710: 7620: 7572: 7549: 7526: 7517: 7464: 7406: 7356: 7343: 7298: 7270: 7187: 7178: 7078: 7049: 7040: 6800: 6458:"A graphical approach for slope mass rating (SMR)" 5570: 5495: 5338: 4899: 4629: 4111: 4067: 3725: 3646: 3617: 3556: 3499: 3464: 3391: 3252: 3123: 2819: 2661: 2549: 2176:Stereographic projection of a sphere from a point 2161: 1707: 1442: 1173: 415:distance between points in the plane equal to the 5506:Under this reparametrization, the length element 3828:is a point on the sphere, but not a 'north pole' 1504:on the plane, the projection and its inverse are 1250:on the plane, the projection and its inverse are 6173:Lohne, John (1979). "Essays on Thomas Harriot". 5891:manufacture. Image remapping software, such as 2922:. Then the stereographic projection of a point 6765:. Englewood Cliffs, New Jersey: Prentice Hall. 6726:Journal of the Warburg and Courtauld Institutes 6709:. Englewood Cliffs, New Jersey: Prentice Hall. 5906:'s in 1779. This results in effects known as a 2356:, then the stereographic projection of a point 6943:, the phase transformation crystallography lab 6576:"Samyang 8 mm f/3.5 Aspherical IF MC Fish-eye" 6030:(1949). "The Early History of the Astrolabe". 527:. Its earliest extant description is found in 6978: 6594: 6592: 6421:Lisle, Richard J.; Leyshon, Peter R. (2004). 6394:Hoek, Evert; Bray, Jonathan D. (1981-06-30). 5060:. Computers now make this task much easier. 4654:be another complex coordinate, the functions 4361:in the real plane can be identified with the 3768:be two points on the sphere with projections 291: 8: 6933:, a software tool for structural geology by 6707:Differential geometry of curves and surfaces 6679:Proceedings of Computational Aesthetics 2007 6022: 6020: 5571:{\displaystyle dx={\frac {2\,dt}{t^{2}+1}}.} 669:proved that the stereographic projection is 400:The stereographic projection gives a way to 6638:Brown, James & Churchill, Ruel (1989). 6291:Geometry and the Imagination in Minneapolis 2243:meets these conditions, then for any point 8035: 7930: 7845: 7832: 7523: 7461: 7448: 7353: 7184: 7046: 7037: 7024: 6985: 6971: 6963: 6370:(2 ed.). Cambridge University Press. 4188:-axis) and 50° below the horizontal plane 4182:-axis (or 30° clockwise from the positive 4019: 4015: 3964: 3960: 3350: 3240: 3233: 3097: 3093: 298: 284: 33: 8070:Map projection of the tri-axial ellipsoid 6299: 5581:This substitution can sometimes simplify 5550: 5537: 5531: 5520: 5475: 5460: 5435: 5417: 5404: 5390: 5322: 5309: 5297: 5284: 5277: 5265: 5252: 5234: 5227: 4871: 4870: 4848: 4847: 4838: 4818: 4817: 4791: 4771: 4770: 4747: 4678: 4664: 4662: 4598: 4597: 4575: 4574: 4562: 4542: 4541: 4518: 4498: 4497: 4474: 4402: 4388: 4386: 4104: 4059: 4043: 4027: 4006: 3972: 3945: 3929: 3914: 3722: 3709: 3705: 3693: 3630: 3606: 3593: 3569: 3516: 3477: 3445: 3377: 3361: 3341: 3331: 3318: 3299: 3297: 3224: 3214: 3201: 3182: 3171: 3115: 3111: 3110: 3087: 3083: 3082: 3073: 2771: 2759: 2749: 2740: 2730: 2714: 2696: 2689: 2680: 2674: 2647: 2629: 2616: 2607: 2602: 2592: 2581: 2568: 2562: 2507: 2490: 2484: 2475: 2469: 2213:is perpendicular to the diameter through 2138: 2122: 2101: 2088: 2072: 2060: 2047: 2031: 2019: 2006: 1990: 1908: 1902: 1880: 1874: 1819: 1817: 1678: 1660: 1653: 1635: 1614: 1545: 1514: 1512: 1412: 1348: 1291: 1260: 1258: 1150: 1137: 1119: 1106: 1090: 1078: 1065: 1044: 1032: 1019: 998: 935: 914: 883: 881: 6010:The Geographical Fragments of Hipparchus 4090: 3009:level, it illustrates how the sphere is 5998: 5996: 5979: 5959:Stereographic projection in cartography 5820:Lambert azimuthal equal-area projection 5816:equal-angle lower-hemisphere projection 5705:A crystallographic pole figure for the 5629: 4322:, this provides a visualization of the 4229: 2939:is the unique point of intersection of 41: 6889:Stereographic Projection and Inversion 6763:Differential geometry and applications 6681:. Banff: Eurographics. pp. 23–28. 5918:(when the center of projection is the 5910:(when the center of projection is the 5623:to rays emanating from the origin and 3508: 438:. This is the spherical analog of the 6951:straightedge and compass construction 6866:. Houston, Texas: Publish or Perish. 6397:Rock Slope Engineering: Third Edition 6389: 6387: 6175:Archive for History of Exact Sciences 6012:. University of London, Athlone Press 2957:. This construction plays a role in 1769:Other authors use a sphere of radius 813:, and this line intersects the plane 779:be the rest of the sphere. The plane 7: 5370:The pair of trigonometric functions 1492:on the sphere and polar coordinates 6915:Time Lapse Stereographic Projection 6366:Lisle, R.J.; Leyshon, P.R. (2004). 5627:to circles centered at the origin. 5585:involving trigonometric functions. 4998:on the unit sphere agrees with the 4318:. When the equatorial plane is the 3402:and is the unique formula found in 2977:to (0, 0), the equator to the 2829:Still more generally, suppose that 1809:. In this case the formulae become 6732:. The Warburg Institute: 190–196. 6640:Complex variables and applications 6226:Gelfand, Minlos & Shapiro 1963 6117:10.1111/j.1600-0498.1953.tb00528.x 5669:A stereographic projection of the 5651:The stereographic projection with 3938: 3916: 3726:{\displaystyle R=e^{\Theta /a},\,} 3706: 3618:{\displaystyle (c-d)(X^{2}+Y^{2})} 3005:and complex analysis. On a merely 1647: 1527: 1425: 1273: 624:In the 16th and 17th century, the 14: 5811:Visualization of lines and planes 5740:Visualization of lines and planes 5589:Applications to other disciplines 5006:Visualization of lines and planes 4131:, after the Russian mineralogist 801:, there is a unique line through 8013:Quadrilateralized spherical cube 7693:Quadrilateralized spherical cube 5768:transmission electron microscope 5644: 5632: 5512:of the unit circle goes over to 4262: 4232: 49: 6308:from the original on 2021-04-19 6262:. McGraw-Hill, Inc. p. 19. 5952:, the analogous mapping of the 5447: 5359:Tangent half-angle substitution 5353:Tangent half-angle substitution 4293:Applications within mathematics 4133:George (Yuri Viktorovich) Wulff 3001:. This notion finds utility in 2786: 2735: 2729: 2516: 677:, motivated by his interest in 7602:Lambert cylindrical equal-area 6623:(2 ed.). Addison-Wesley. 4876: 4853: 4823: 4776: 4732: 4714: 4603: 4580: 4547: 4503: 4459: 4441: 4016: 3961: 3612: 3586: 3583: 3571: 3383: 3351: 3338: 3305: 3221: 3188: 3094: 2811: 2787: 2541: 2517: 1975: 1957: 1954: 1951: 1939: 1859: 1847: 1844: 1841: 1823: 1599: 1581: 1530: 1518: 1388: 1376: 1276: 1264: 983: 965: 899: 887: 381:at which curves meet and thus 1: 8050:Interruption (map projection) 6529:10.1016/s0013-7952(01)00024-2 3263:Along the unit circle, where 3061:on the plane either both are 2198:on the sphere onto any plane 2188:, shown here in cross section 1734:, shown here in cross section 773:be the "north pole", and let 561: 546: 7688:Lambert azimuthal equal-area 7484:Guyou hemisphere-in-a-square 7474:Adams hemisphere-in-a-square 6482:10.1016/j.enggeo.2011.10.004 5882:. Neuchatel, 1779–96, pl. 8. 5600:Stereographic map projection 5030:in three-dimensional space. 3557:{\displaystyle ax+by+cz-d=0} 3043:on the sphere and its image 2380:of intersection of the line 1459:is understood to have value 619:medieval Islamic astronomers 519:and used for projecting the 377:, meaning that it preserves 21:Stereographic map projection 19:For the map projection, see 6147:Brown, Lloyd Arnold : 5904:Horace Bénédict de Saussure 5213:is a rational point on the 729:in three-dimensional space 432:spherical polar coordinates 354:through the point. It is a 230:Projection (linear algebra) 8126: 6803:Basic Algebraic Geometry I 6799:Shafarevich, Igor (1995). 6697:, New York: Pergamon Press 6286:"Stereographic Projection" 5715: 5597: 5356: 4146:centred at a point on the 3015:one-point compactification 665:In the late 16th century, 492: 393:(distance preserving) nor 18: 8045: 8034: 7961: 7844: 7831: 7643: 7460: 7447: 7384: 7243: 7126: 7036: 7023: 7000: 6847:. University of Chicago. 6693:; Shapiro, Z.Ya. (1963), 6642:. New York: McGraw-Hill. 6216:Cf. Apostol (1974) p. 17. 6079:10.1080/07341519108581788 5867:, Lombardy, Italy during 5172:on the unit circle (with 5092:-dimensional polytope in 4308:one to one correspondence 3673:The sphere, with various 3151:area element is given in 685:, invented by his friend 517:Ancient Greek astronomers 404:a sphere by a plane. The 6862:Spivak, Michael (1999). 6831:Snyder, John P. (1989). 6822:Snyder, John P. (1987). 6657:Casselman, Bill (2014), 6433:10.1017/cbo9781139171366 6294:, Minnesota University, 3677:shown in distinct colors 3509:§ First formulation 2669:the inverse is given by 1471:trigonometric identities 831:stereographic projection 495:Astrolabe § History 385:approximately preserves 316:stereographic projection 7489:Lambert conformal conic 6835:. US Geological Survey. 6826:. US Geological Survey. 6245:Cf. Shafarevich (1995). 5964:Curvilinear perspective 5935:List of map projections 5002:on the Riemann sphere. 3277:The metric is given in 2880:homogeneous coordinates 1475:cylindrical coordinates 97:Curvilinear perspective 73:Orthographic projection 7622:Tobler hyperelliptical 7235:Tobler hyperelliptical 7161:Space-oblique Mercator 6705:; Manfredo P. (1976). 6067:History and Technology 5883: 5871: 5783: 5726:, the orientations of 5713: 5678: 5572: 5497: 5367: 5340: 5139: 5019: 4982:goes to infinity, and 4922:-coordinates are then 4901: 4631: 4326: 4166: 4119: 4113: 4088: 4069: 3753: 3727: 3678: 3648: 3619: 3558: 3501: 3500:{\displaystyle X,Y,Z,} 3466: 3393: 3254: 3147: 3139: 3125: 2821: 2663: 2597: 2551: 2459:= (1, 0, 0, ..., 0) ∈ 2454:, the projection from 2189: 2163: 1735: 1709: 1444: 1175: 717: 611:planispheric astrolabe 512: 320:perspective projection 92:Perspective projection 31: 8100:Conformal projections 6621:Mathematical Analysis 6340::12, 3614–3616. 5877: 5858: 5781: 5704: 5675:circles on the sphere 5668: 5573: 5498: 5366: 5341: 5133: 5110:. The reduction from 5098:is projected onto an 5024:real projective plane 5013: 4988:meromorphic functions 4902: 4632: 4305: 4164: 4114: 4099:and projection plane 4094: 4086: 4070: 3752: 3728: 3672: 3664:real projective plane 3649: 3620: 3559: 3502: 3467: 3465:{\displaystyle x,y,z} 3429:Circles on the sphere 3394: 3255: 3145: 3137: 3126: 3022:Cartesian coordinates 2955:rational hypersurface 2822: 2664: 2577: 2552: 2396:Cartesian coordinates 2273:in exactly one point 2175: 2164: 1726: 1710: 1445: 1186:spherical coordinates 1176: 844:Cartesian coordinates 820:in exactly one point 771:= (0, 0, 1) 735:is the set of points 705: 502: 436:cartesian coordinates 434:or three-dimensional 373:on the plane, and is 367:circles on the sphere 326:, through a specific 200:Computer-aided design 137:Exploded view drawing 29: 7998:Cahill–Keyes M-shape 7858:Chamberlin trimetric 6949:, software tool for 6845:Flattening the Earth 6761:Oprea, John (2003). 6008:Dicks, D.R. (1960). 5751:electron diffraction 5736:electron diffraction 5519: 5389: 5226: 4661: 4385: 4342:(or two-dimensional 4112:{\displaystyle \pi } 4103: 3913: 3819:In other words, if: 3736:where the parameter 3692: 3647:{\displaystyle c=d,} 3629: 3568: 3515: 3476: 3444: 3408:Habilitationsschrift 3296: 3170: 3072: 2837:quadric hypersurface 2673: 2561: 2468: 1816: 1511: 1257: 1199:on the sphere (with 880: 541:(2nd century BC) by 336:center of projection 205:Descriptive geometry 78:Isometric projection 43:Graphical projection 8110:Projective geometry 8065:Tissot's indicatrix 7966:Central cylindrical 7607:Smyth equal-surface 7509:Transverse Mercator 7358:General perspective 7113:Smyth equal-surface 7065:Transverse Mercator 6780:Pedoe, Dan (1988). 6521:2001EngGe..61...53M 6509:Engineering Geology 6474:2012EngGe.124...67T 6462:Engineering Geology 6109:1953Cent....3..183D 5950:Poincaré disk model 5814:referred to as the 5763:experimental access 5683:circles on a sphere 5653:Tissot's indicatrix 5148:Pythagorean triples 5144:arithmetic geometry 5126:Arithmetic geometry 5074:Other visualization 5000:Fubini–Study metric 4640:Similarly, letting 4142:and meridians of a 3838:, the 'south pole' 3786:on the plane. Then 3744:logarithmic spirals 3421:and would preserve 3003:projective geometry 2835:is a (nonsingular) 2612: 634:Western Hemispheres 603:Theon of Alexandria 440:Poincaré disk model 397:(area preserving). 330:on the sphere (the 265:Video game graphics 240:Projective geometry 210:Engineering drawing 102:Reverse perspective 68:Parallel projection 37:Part of a series on 8095:Conformal mappings 8018:Waterman butterfly 7868:Miller cylindrical 7499:Peirce quincuncial 7394:Lambert equal-area 7146:Gall stereographic 6957:Estereografica Web 6562:2011-06-29 at the 6187:10.1007/BF00327737 5945:Astronomical clock 5884: 5872: 5788:structural geology 5784: 5714: 5687:circles on a plane 5679: 5608:are preferred for 5568: 5493: 5368: 5336: 5140: 5020: 4897: 4895: 4627: 4625: 4327: 4167: 4120: 4109: 4089: 4065: 3754: 3723: 3679: 3644: 3615: 3554: 3497: 3462: 3423:Gaussian curvature 3389: 3250: 3148: 3140: 3121: 2963:conformal geometry 2959:algebraic geometry 2848:. In other words, 2817: 2659: 2598: 2547: 2190: 2159: 2157: 1736: 1705: 1703: 1440: 1438: 1171: 1169: 861:on the sphere and 718: 656:François d'Aguilon 598:(1st century BC). 513: 509:François d'Aguilon 417:spherical distance 85:Oblique projection 32: 8082: 8081: 8078: 8077: 8030: 8029: 8026: 8025: 7974: 7973: 7827: 7826: 7823: 7822: 7706: 7705: 7443: 7442: 7439: 7438: 7402: 7401: 7290:Lambert conformal 7266: 7265: 7180:Pseudocylindrical 7174: 7173: 6935:Rick Allmendinger 6442:978-0-521-53582-3 6407:978-0-419-16010-6 6236:Cf. Pedoe (1988). 6150:The story of maps 5841:slope mass rating 5661:Planetary science 5563: 5488: 5442: 5329: 5272: 5046:quotient topology 4976:approaching 0 as 4886: 4879: 4856: 4833: 4826: 4786: 4779: 4705: 4613: 4606: 4583: 4557: 4550: 4513: 4506: 4429: 4312:point at infinity 3348: 3231: 3065:or none of them: 2999:point at infinity 2904:and a hyperplane 2784: 2733: 2727: 2654: 2514: 2255:the line through 2227:does not contain 2145: 2067: 2026: 1925: 1916: 1897: 1888: 1719:Other conventions 1691: 1642: 1561: 1420: 1356: 1321: 1237:polar coordinates 1157: 1085: 1039: 951: 930: 698:First formulation 475:stereographic net 428:analytic geometry 421:coordinate system 308: 307: 250:Technical drawing 195:Computer graphics 8117: 8036: 7993:Cahill Butterfly 7931: 7911:Goode homolosine 7846: 7833: 7798: 7797:(Mecca or Qibla) 7678:Goode homolosine 7524: 7462: 7449: 7354: 7349: 7220:Goode homolosine 7185: 7070:Oblique Mercator 7047: 7038: 7025: 6987: 6980: 6973: 6964: 6877: 6858: 6836: 6827: 6818: 6806: 6795: 6776: 6757: 6720: 6698: 6682: 6673: 6672: 6671: 6653: 6634: 6603: 6596: 6587: 6586: 6584: 6583: 6572: 6566: 6556: 6554: 6547: 6541: 6540: 6500: 6494: 6493: 6453: 6447: 6446: 6418: 6412: 6411: 6391: 6382: 6381: 6363: 6357: 6347: 6341: 6331: 6325: 6322: 6316: 6315: 6314: 6313: 6303: 6276:; Doyle, Peter; 6270: 6264: 6263: 6260:Complex Analysis 6252: 6246: 6243: 6237: 6234: 6228: 6223: 6217: 6214: 6208: 6205: 6199: 6198: 6181:(3/4): 189–312. 6170: 6164: 6161: 6155: 6145: 6139: 6136: 6130: 6127: 6121: 6120: 6093:Drachmann, A.G . 6089: 6083: 6082: 6062: 6056: 6055: 6028:Neugebauer, Otto 6024: 6015: 6013: 6000: 5991: 5988:Euclidean metric 5984: 5954:hyperbolic plane 5902:, starting with 5845:rock mass rating 5648: 5636: 5577: 5575: 5574: 5569: 5564: 5562: 5555: 5554: 5544: 5532: 5511: 5502: 5500: 5499: 5494: 5489: 5487: 5480: 5479: 5469: 5461: 5443: 5441: 5440: 5439: 5423: 5422: 5421: 5405: 5381: 5345: 5343: 5342: 5337: 5335: 5331: 5330: 5328: 5327: 5326: 5314: 5313: 5303: 5302: 5301: 5289: 5288: 5278: 5273: 5271: 5270: 5269: 5257: 5256: 5246: 5235: 5218: 5212: 5210: 5208: 5207: 5202: 5199: 5188: 5178: 5171: 5155: 5121: 5115: 5109: 5103: 5097: 5091: 5084:Schlegel diagram 5041:antipodal points 5038: 4981: 4975: 4969: 4968: 4966: 4965: 4960: 4957: 4945: 4944: 4942: 4941: 4936: 4933: 4921: 4915: 4906: 4904: 4903: 4898: 4896: 4892: 4888: 4887: 4885: 4881: 4880: 4872: 4862: 4858: 4857: 4849: 4839: 4834: 4832: 4828: 4827: 4819: 4809: 4792: 4787: 4785: 4781: 4780: 4772: 4762: 4748: 4706: 4704: 4693: 4679: 4653: 4636: 4634: 4633: 4628: 4626: 4619: 4615: 4614: 4612: 4608: 4607: 4599: 4589: 4585: 4584: 4576: 4563: 4558: 4556: 4552: 4551: 4543: 4533: 4519: 4514: 4512: 4508: 4507: 4499: 4489: 4475: 4437: 4430: 4428: 4417: 4403: 4377: 4360: 4332:parametrizations 4317: 4298:Complex analysis 4286: 4277: 4266: 4256: 4247: 4236: 4200: 4194: 4187: 4181: 4174: 4118: 4116: 4115: 4110: 4074: 4072: 4071: 4066: 4064: 4063: 4051: 4050: 4032: 4031: 4014: 4013: 3977: 3976: 3953: 3952: 3934: 3933: 3905: 3899: 3897: 3886: 3880: 3875:is the image of 3874: 3866: 3860: 3855:is the image of 3854: 3852: 3843: 3833: 3827: 3815: 3809: 3803: 3801: 3794: 3792: 3785: 3783: 3776: 3774: 3767: 3761: 3741: 3732: 3730: 3729: 3724: 3718: 3717: 3713: 3653: 3651: 3650: 3645: 3624: 3622: 3621: 3616: 3611: 3610: 3598: 3597: 3563: 3561: 3560: 3555: 3506: 3504: 3503: 3498: 3471: 3469: 3468: 3463: 3404:Bernhard Riemann 3398: 3396: 3395: 3390: 3382: 3381: 3366: 3365: 3349: 3347: 3346: 3345: 3336: 3335: 3323: 3322: 3300: 3288: 3273: 3259: 3257: 3256: 3251: 3232: 3230: 3229: 3228: 3219: 3218: 3206: 3205: 3183: 3162: 3130: 3128: 3127: 3122: 3120: 3119: 3114: 3105: 3092: 3091: 3086: 3060: 3049: 3042: 2995: 2989: 2952: 2946: 2945: 2938: 2927: 2921: 2915: 2909: 2903: 2897: 2892:. Fix any point 2891: 2877: 2853: 2847: 2841:projective space 2834: 2826: 2824: 2823: 2818: 2785: 2783: 2776: 2775: 2765: 2764: 2763: 2750: 2745: 2744: 2734: 2731: 2728: 2726: 2719: 2718: 2708: 2701: 2700: 2690: 2685: 2684: 2668: 2666: 2665: 2660: 2655: 2653: 2652: 2651: 2635: 2634: 2633: 2617: 2611: 2606: 2596: 2591: 2573: 2572: 2556: 2554: 2553: 2548: 2515: 2513: 2512: 2511: 2495: 2494: 2485: 2480: 2479: 2463: 2453: 2443: 2437: 2426: 2420: 2414: 2408: 2393: 2387: 2386: 2379: 2377: 2370: 2355: 2345: 2339: 2333: 2327: 2318: 2311: 2304: 2281: 2279: 2272: 2266: 2260: 2254: 2248: 2242: 2232: 2226: 2218: 2212: 2203: 2197: 2187: 2181: 2168: 2166: 2165: 2160: 2158: 2151: 2147: 2146: 2144: 2143: 2142: 2127: 2126: 2107: 2106: 2105: 2093: 2092: 2073: 2068: 2066: 2065: 2064: 2052: 2051: 2032: 2027: 2025: 2024: 2023: 2011: 2010: 1991: 1931: 1927: 1926: 1924: 1917: 1909: 1903: 1898: 1896: 1889: 1881: 1875: 1808: 1807: 1805: 1804: 1801: 1798: 1786: 1785: 1783: 1782: 1779: 1776: 1764: 1758: 1744: 1733: 1714: 1712: 1711: 1706: 1704: 1697: 1693: 1692: 1690: 1683: 1682: 1672: 1665: 1664: 1654: 1643: 1641: 1640: 1639: 1623: 1615: 1573: 1569: 1562: 1560: 1546: 1503: 1491: 1468: 1462: 1458: 1449: 1447: 1446: 1441: 1439: 1432: 1428: 1421: 1413: 1368: 1364: 1357: 1349: 1333: 1329: 1322: 1320: 1303: 1292: 1249: 1234: 1222: 1216: 1204: 1198: 1180: 1178: 1177: 1172: 1170: 1163: 1159: 1158: 1156: 1155: 1154: 1142: 1141: 1125: 1124: 1123: 1111: 1110: 1091: 1086: 1084: 1083: 1082: 1070: 1069: 1053: 1045: 1040: 1038: 1037: 1036: 1024: 1023: 1007: 999: 957: 953: 952: 950: 936: 931: 929: 915: 872: 860: 839:onto the plane. 838: 828: 826: 819: 812: 806: 800: 794: 785: 778: 772: 765: 750: 734: 728: 713:, shown here in 712: 593: 566: 563: 551: 548: 521:celestial sphere 503:Illustration by 455:complex analysis 444:hyperbolic plane 389:. It is neither 371:circles or lines 344:projection plane 300: 293: 286: 235:Projection plane 225:Plans (drawings) 53: 34: 8125: 8124: 8120: 8119: 8118: 8116: 8115: 8114: 8105:Crystallography 8085: 8084: 8083: 8074: 8041: 8022: 7970: 7957: 7920: 7897: 7883:Van der Grinten 7840: 7838:By construction 7819: 7796: 7795: 7787: 7764: 7746: 7727:Equirectangular 7713: 7702: 7639: 7616: 7612:Trystan Edwards 7568: 7545: 7513: 7456: 7435: 7408:Pseudoazimuthal 7398: 7380: 7347: 7346: 7339: 7294: 7262: 7258:Winkel I and II 7239: 7170: 7151:Gall isographic 7141:Equirectangular 7122: 7118:Trystan Edwards 7074: 7032: 7019: 6996: 6991: 6927: 6906: 6885: 6880: 6874: 6861: 6855: 6841:Snyder, John P. 6839: 6830: 6821: 6815: 6798: 6792: 6779: 6773: 6760: 6723: 6717: 6701: 6685: 6676: 6669: 6667: 6656: 6650: 6637: 6631: 6615: 6611: 6606: 6597: 6590: 6581: 6579: 6574: 6573: 6569: 6564:Wayback Machine 6552: 6551: 6548: 6544: 6502: 6501: 6497: 6455: 6454: 6450: 6443: 6420: 6419: 6415: 6408: 6393: 6392: 6385: 6378: 6365: 6364: 6360: 6348: 6344: 6332: 6328: 6323: 6319: 6311: 6309: 6272: 6271: 6267: 6254: 6253: 6249: 6244: 6240: 6235: 6231: 6224: 6220: 6215: 6211: 6206: 6202: 6172: 6171: 6167: 6162: 6158: 6146: 6142: 6137: 6133: 6128: 6124: 6091: 6090: 6086: 6064: 6063: 6059: 6026: 6025: 6018: 6007: 6001: 5994: 5985: 5981: 5977: 5931: 5853: 5832: 5798:. Similarly, a 5786:Researchers in 5776: 5761:thus providing 5724:crystallography 5720: 5707:diamond lattice 5699: 5697:Crystallography 5663: 5656: 5655:of deformation. 5649: 5640: 5637: 5606:map projections 5602: 5596: 5591: 5546: 5545: 5533: 5517: 5516: 5507: 5471: 5470: 5462: 5431: 5424: 5413: 5406: 5387: 5386: 5371: 5361: 5355: 5318: 5305: 5304: 5293: 5280: 5279: 5261: 5248: 5247: 5236: 5233: 5229: 5224: 5223: 5214: 5203: 5200: 5195: 5194: 5192: 5190: 5184: 5181:rational points 5173: 5161: 5158:rational number 5151: 5136:rational points 5128: 5117: 5111: 5105: 5099: 5093: 5087: 5076: 5034: 5008: 4996:standard metric 4990:mapping to the 4977: 4971: 4961: 4958: 4955: 4954: 4952: 4947: 4937: 4934: 4931: 4930: 4928: 4923: 4917: 4911: 4894: 4893: 4863: 4840: 4810: 4793: 4763: 4749: 4746: 4742: 4735: 4711: 4710: 4694: 4680: 4671: 4659: 4658: 4641: 4624: 4623: 4590: 4564: 4534: 4520: 4490: 4476: 4473: 4469: 4462: 4438: 4435: 4434: 4418: 4404: 4395: 4383: 4382: 4365: 4350: 4315: 4300: 4295: 4288: 4285: 4279: 4276: 4270: 4267: 4258: 4255: 4249: 4246: 4240: 4237: 4196: 4189: 4183: 4177: 4170: 4101: 4100: 4081: 4055: 4039: 4023: 4002: 3968: 3941: 3925: 3911: 3910: 3901: 3895: 3892: 3882: 3876: 3870: 3862: 3856: 3850: 3847: 3839: 3829: 3823: 3811: 3805: 3799: 3796: 3790: 3787: 3781: 3778: 3772: 3769: 3763: 3757: 3737: 3701: 3690: 3689: 3627: 3626: 3602: 3589: 3566: 3565: 3513: 3512: 3474: 3473: 3442: 3441: 3373: 3357: 3337: 3327: 3314: 3304: 3294: 3293: 3289:coordinates by 3278: 3264: 3220: 3210: 3197: 3187: 3168: 3167: 3163:coordinates by 3152: 3109: 3098: 3081: 3070: 3069: 3063:rational points 3047: 3044: 3025: 2991: 2985: 2971: 2948: 2941: 2940: 2929: 2923: 2917: 2916:not containing 2911: 2905: 2899: 2893: 2890: 2882: 2875: 2865: 2855: 2849: 2843: 2830: 2767: 2766: 2755: 2751: 2736: 2710: 2709: 2692: 2691: 2676: 2671: 2670: 2643: 2636: 2625: 2618: 2564: 2559: 2558: 2503: 2496: 2486: 2471: 2466: 2465: 2455: 2449: 2439: 2436: 2428: 2422: 2416: 2410: 2407: 2399: 2389: 2382: 2381: 2375: 2372: 2357: 2351: 2341: 2335: 2329: 2323: 2321:Euclidean space 2313: 2307: 2300: 2296: 2294:Generalizations 2277: 2274: 2268: 2262: 2256: 2250: 2244: 2238: 2228: 2222: 2214: 2208: 2199: 2193: 2183: 2182:onto the plane 2177: 2156: 2155: 2134: 2118: 2108: 2097: 2084: 2074: 2056: 2043: 2036: 2015: 2002: 1995: 1989: 1985: 1978: 1936: 1935: 1907: 1879: 1873: 1869: 1862: 1814: 1813: 1802: 1799: 1796: 1795: 1793: 1788: 1780: 1777: 1774: 1773: 1771: 1770: 1760: 1754: 1739: 1728: 1721: 1702: 1701: 1674: 1673: 1656: 1655: 1631: 1624: 1616: 1613: 1609: 1602: 1578: 1577: 1550: 1544: 1540: 1533: 1509: 1508: 1493: 1477: 1464: 1460: 1454: 1437: 1436: 1402: 1398: 1391: 1373: 1372: 1341: 1337: 1304: 1293: 1290: 1286: 1279: 1255: 1254: 1239: 1228: 1218: 1210: 1200: 1188: 1168: 1167: 1146: 1133: 1126: 1115: 1102: 1092: 1074: 1061: 1054: 1046: 1028: 1015: 1008: 1000: 997: 993: 986: 962: 961: 940: 919: 913: 909: 902: 878: 877: 862: 846: 834: 829:, known as the 824: 821: 814: 808: 802: 796: 790: 780: 774: 767: 752: 736: 730: 724: 707: 700: 695: 646:Rumold Mercator 601:By the time of 583: 581:anaphoric clock 564: 549: 497: 491: 477:, shortened to 304: 275: 274: 270:Viewing frustum 260:Vanishing point 175: 167: 166: 157:Worm's-eye view 132:Cutaway drawing 122:Bird's-eye view 117: 109: 108: 63: 24: 17: 12: 11: 5: 8123: 8121: 8113: 8112: 8107: 8102: 8097: 8087: 8086: 8080: 8079: 8076: 8075: 8073: 8072: 8067: 8062: 8057: 8052: 8046: 8043: 8042: 8039: 8032: 8031: 8028: 8027: 8024: 8023: 8021: 8020: 8015: 8010: 8005: 8000: 7995: 7990: 7984: 7982: 7976: 7975: 7972: 7971: 7969: 7968: 7962: 7959: 7958: 7956: 7955: 7950: 7945: 7939: 7937: 7928: 7922: 7921: 7919: 7918: 7913: 7907: 7905: 7899: 7898: 7896: 7895: 7890: 7885: 7880: 7875: 7870: 7865: 7863:Kavrayskiy VII 7860: 7854: 7852: 7842: 7841: 7836: 7829: 7828: 7825: 7824: 7821: 7820: 7818: 7817: 7812: 7807: 7801: 7799: 7793:Retroazimuthal 7789: 7788: 7786: 7785: 7780: 7774: 7772: 7766: 7765: 7763: 7762: 7756: 7754: 7748: 7747: 7745: 7744: 7739: 7734: 7729: 7724: 7718: 7716: 7712:Equidistant in 7708: 7707: 7704: 7703: 7701: 7700: 7695: 7690: 7685: 7680: 7675: 7670: 7665: 7660: 7655: 7650: 7644: 7641: 7640: 7638: 7637: 7632: 7626: 7624: 7618: 7617: 7615: 7614: 7609: 7604: 7599: 7594: 7589: 7584: 7578: 7576: 7570: 7569: 7567: 7566: 7561: 7555: 7553: 7547: 7546: 7544: 7543: 7538: 7532: 7530: 7521: 7515: 7514: 7512: 7511: 7506: 7501: 7496: 7491: 7486: 7481: 7476: 7470: 7468: 7458: 7457: 7452: 7445: 7444: 7441: 7440: 7437: 7436: 7434: 7433: 7428: 7423: 7418: 7412: 7410: 7404: 7403: 7400: 7399: 7397: 7396: 7391: 7385: 7382: 7381: 7379: 7378: 7373: 7368: 7362: 7360: 7351: 7341: 7340: 7338: 7337: 7332: 7331: 7330: 7325: 7315: 7310: 7304: 7302: 7296: 7295: 7293: 7292: 7287: 7282: 7276: 7274: 7268: 7267: 7264: 7263: 7261: 7260: 7255: 7250: 7248:Kavrayskiy VII 7244: 7241: 7240: 7238: 7237: 7232: 7227: 7222: 7217: 7212: 7207: 7202: 7197: 7191: 7189: 7182: 7176: 7175: 7172: 7171: 7169: 7168: 7163: 7158: 7153: 7148: 7143: 7138: 7133: 7127: 7124: 7123: 7121: 7120: 7115: 7110: 7105: 7100: 7095: 7090: 7084: 7082: 7076: 7075: 7073: 7072: 7067: 7062: 7056: 7054: 7044: 7034: 7033: 7028: 7021: 7020: 7018: 7017: 7012: 7007: 7001: 6998: 6997: 6994:Map projection 6992: 6990: 6989: 6982: 6975: 6967: 6961: 6960: 6954: 6944: 6938: 6926: 6923: 6922: 6921: 6912: 6905: 6902: 6901: 6900: 6895: 6884: 6883:External links 6881: 6879: 6878: 6872: 6859: 6853: 6837: 6828: 6819: 6813: 6796: 6790: 6777: 6771: 6758: 6738:10.2307/751275 6721: 6715: 6699: 6683: 6674: 6654: 6648: 6635: 6629: 6612: 6610: 6607: 6605: 6604: 6588: 6567: 6557:3.5 Fisheye CS 6542: 6495: 6448: 6441: 6413: 6406: 6383: 6376: 6358: 6351:J. Appl. Phys. 6342: 6335:J. Appl. Phys. 6326: 6317: 6284:(1994-04-12), 6282:Thurston, Bill 6265: 6247: 6238: 6229: 6218: 6209: 6200: 6165: 6156: 6140: 6138:Snyder (1989). 6131: 6122: 6103:(1): 183–189. 6084: 6057: 6044:10.1086/349045 6038:(3): 240–256. 6016: 5992: 5978: 5976: 5973: 5972: 5971: 5966: 5961: 5956: 5947: 5942: 5937: 5930: 5927: 5888:fisheye lenses 5869:Wikimania 2016 5861:Michele Vedani 5852: 5849: 5831: 5830:Rock mechanics 5828: 5775: 5772: 5716:Main article: 5698: 5695: 5662: 5659: 5658: 5657: 5650: 5643: 5641: 5638: 5631: 5598:Main article: 5595: 5592: 5590: 5587: 5579: 5578: 5567: 5561: 5558: 5553: 5549: 5543: 5540: 5536: 5530: 5527: 5524: 5504: 5503: 5492: 5486: 5483: 5478: 5474: 5468: 5465: 5459: 5456: 5453: 5450: 5446: 5438: 5434: 5430: 5427: 5420: 5416: 5412: 5409: 5403: 5400: 5397: 5394: 5357:Main article: 5354: 5351: 5347: 5346: 5334: 5325: 5321: 5317: 5312: 5308: 5300: 5296: 5292: 5287: 5283: 5276: 5268: 5264: 5260: 5255: 5251: 5245: 5242: 5239: 5232: 5142:In elementary 5127: 5124: 5075: 5072: 5007: 5004: 4992:Riemann sphere 4908: 4907: 4891: 4884: 4878: 4875: 4869: 4866: 4861: 4855: 4852: 4846: 4843: 4837: 4831: 4825: 4822: 4816: 4813: 4808: 4805: 4802: 4799: 4796: 4790: 4784: 4778: 4775: 4769: 4766: 4761: 4758: 4755: 4752: 4745: 4741: 4738: 4736: 4734: 4731: 4728: 4725: 4722: 4719: 4716: 4713: 4712: 4709: 4703: 4700: 4697: 4692: 4689: 4686: 4683: 4677: 4674: 4672: 4670: 4667: 4666: 4638: 4637: 4622: 4618: 4611: 4605: 4602: 4596: 4593: 4588: 4582: 4579: 4573: 4570: 4567: 4561: 4555: 4549: 4546: 4540: 4537: 4532: 4529: 4526: 4523: 4517: 4511: 4505: 4502: 4496: 4493: 4488: 4485: 4482: 4479: 4472: 4468: 4465: 4463: 4461: 4458: 4455: 4452: 4449: 4446: 4443: 4440: 4439: 4436: 4433: 4427: 4424: 4421: 4416: 4413: 4410: 4407: 4401: 4398: 4396: 4394: 4391: 4390: 4363:complex number 4324:Riemann sphere 4299: 4296: 4294: 4291: 4290: 4289: 4283: 4274: 4268: 4261: 4259: 4253: 4244: 4238: 4231: 4216: 4215: 4212: 4209: 4206: 4108: 4080: 4077: 4076: 4075: 4062: 4058: 4054: 4049: 4046: 4042: 4038: 4035: 4030: 4026: 4022: 4018: 4012: 4009: 4005: 4001: 3998: 3995: 3992: 3989: 3986: 3983: 3980: 3975: 3971: 3967: 3963: 3959: 3956: 3951: 3948: 3944: 3940: 3937: 3932: 3928: 3924: 3921: 3918: 3889: 3888: 3868: 3845: 3734: 3733: 3721: 3716: 3712: 3708: 3704: 3700: 3697: 3643: 3640: 3637: 3634: 3614: 3609: 3605: 3601: 3596: 3592: 3588: 3585: 3582: 3579: 3576: 3573: 3553: 3550: 3547: 3544: 3541: 3538: 3535: 3532: 3529: 3526: 3523: 3520: 3496: 3493: 3490: 3487: 3484: 3481: 3461: 3458: 3455: 3452: 3449: 3400: 3399: 3388: 3385: 3380: 3376: 3372: 3369: 3364: 3360: 3356: 3353: 3344: 3340: 3334: 3330: 3326: 3321: 3317: 3313: 3310: 3307: 3303: 3261: 3260: 3249: 3246: 3243: 3239: 3236: 3227: 3223: 3217: 3213: 3209: 3204: 3200: 3196: 3193: 3190: 3186: 3181: 3178: 3175: 3132: 3131: 3118: 3113: 3108: 3104: 3101: 3096: 3090: 3085: 3080: 3077: 3017:of the plane. 2970: 2967: 2886: 2870: 2863: 2816: 2813: 2810: 2807: 2804: 2801: 2798: 2795: 2792: 2789: 2782: 2779: 2774: 2770: 2762: 2758: 2754: 2748: 2743: 2739: 2725: 2722: 2717: 2713: 2707: 2704: 2699: 2695: 2688: 2683: 2679: 2658: 2650: 2646: 2642: 2639: 2632: 2628: 2624: 2621: 2615: 2610: 2605: 2601: 2595: 2590: 2587: 2584: 2580: 2576: 2571: 2567: 2546: 2543: 2540: 2537: 2534: 2531: 2528: 2525: 2522: 2519: 2510: 2506: 2502: 2499: 2493: 2489: 2483: 2478: 2474: 2432: 2403: 2334:is a point of 2319:)-dimensional 2295: 2292: 2235: 2234: 2220: 2170: 2169: 2154: 2150: 2141: 2137: 2133: 2130: 2125: 2121: 2117: 2114: 2111: 2104: 2100: 2096: 2091: 2087: 2083: 2080: 2077: 2071: 2063: 2059: 2055: 2050: 2046: 2042: 2039: 2035: 2030: 2022: 2018: 2014: 2009: 2005: 2001: 1998: 1994: 1988: 1984: 1981: 1979: 1977: 1974: 1971: 1968: 1965: 1962: 1959: 1956: 1953: 1950: 1947: 1944: 1941: 1938: 1937: 1934: 1930: 1923: 1920: 1915: 1912: 1906: 1901: 1895: 1892: 1887: 1884: 1878: 1872: 1868: 1865: 1863: 1861: 1858: 1855: 1852: 1849: 1846: 1843: 1840: 1837: 1834: 1831: 1828: 1825: 1822: 1821: 1787:and the plane 1720: 1717: 1716: 1715: 1700: 1696: 1689: 1686: 1681: 1677: 1671: 1668: 1663: 1659: 1652: 1649: 1646: 1638: 1634: 1630: 1627: 1622: 1619: 1612: 1608: 1605: 1603: 1601: 1598: 1595: 1592: 1589: 1586: 1583: 1580: 1579: 1576: 1572: 1568: 1565: 1559: 1556: 1553: 1549: 1543: 1539: 1536: 1534: 1532: 1529: 1526: 1523: 1520: 1517: 1516: 1451: 1450: 1435: 1431: 1427: 1424: 1419: 1416: 1411: 1408: 1405: 1401: 1397: 1394: 1392: 1390: 1387: 1384: 1381: 1378: 1375: 1374: 1371: 1367: 1363: 1360: 1355: 1352: 1347: 1344: 1340: 1336: 1332: 1328: 1325: 1319: 1316: 1313: 1310: 1307: 1302: 1299: 1296: 1289: 1285: 1282: 1280: 1278: 1275: 1272: 1269: 1266: 1263: 1262: 1182: 1181: 1166: 1162: 1153: 1149: 1145: 1140: 1136: 1132: 1129: 1122: 1118: 1114: 1109: 1105: 1101: 1098: 1095: 1089: 1081: 1077: 1073: 1068: 1064: 1060: 1057: 1052: 1049: 1043: 1035: 1031: 1027: 1022: 1018: 1014: 1011: 1006: 1003: 996: 992: 989: 987: 985: 982: 979: 976: 973: 970: 967: 964: 963: 960: 956: 949: 946: 943: 939: 934: 928: 925: 922: 918: 912: 908: 905: 903: 901: 898: 895: 892: 889: 886: 885: 789:For any point 699: 696: 694: 691: 667:Thomas Harriot 638:Gualterius Lud 525:plane geometry 490: 487: 306: 305: 303: 302: 295: 288: 280: 277: 276: 273: 272: 267: 262: 257: 252: 247: 242: 237: 232: 227: 222: 217: 215:Map projection 212: 207: 202: 197: 192: 187: 182: 176: 173: 172: 169: 168: 165: 164: 159: 154: 149: 144: 139: 134: 129: 124: 118: 115: 114: 111: 110: 107: 106: 105: 104: 99: 89: 88: 87: 82: 81: 80: 64: 59: 58: 55: 54: 46: 45: 39: 38: 15: 13: 10: 9: 6: 4: 3: 2: 8122: 8111: 8108: 8106: 8103: 8101: 8098: 8096: 8093: 8092: 8090: 8071: 8068: 8066: 8063: 8061: 8058: 8056: 8053: 8051: 8048: 8047: 8044: 8037: 8033: 8019: 8016: 8014: 8011: 8009: 8006: 8004: 8001: 7999: 7996: 7994: 7991: 7989: 7986: 7985: 7983: 7981: 7977: 7967: 7964: 7963: 7960: 7954: 7953:Stereographic 7951: 7949: 7946: 7944: 7941: 7940: 7938: 7936: 7932: 7929: 7927: 7923: 7917: 7914: 7912: 7909: 7908: 7906: 7904: 7900: 7894: 7893:Winkel tripel 7891: 7889: 7886: 7884: 7881: 7879: 7876: 7874: 7873:Natural Earth 7871: 7869: 7866: 7864: 7861: 7859: 7856: 7855: 7853: 7851: 7847: 7843: 7839: 7834: 7830: 7816: 7813: 7811: 7808: 7806: 7803: 7802: 7800: 7794: 7790: 7784: 7781: 7779: 7776: 7775: 7773: 7771: 7767: 7761: 7758: 7757: 7755: 7753: 7749: 7743: 7740: 7738: 7735: 7733: 7730: 7728: 7725: 7723: 7720: 7719: 7717: 7715: 7709: 7699: 7696: 7694: 7691: 7689: 7686: 7684: 7681: 7679: 7676: 7674: 7671: 7669: 7666: 7664: 7661: 7659: 7656: 7654: 7653:Briesemeister 7651: 7649: 7646: 7645: 7642: 7636: 7633: 7631: 7628: 7627: 7625: 7623: 7619: 7613: 7610: 7608: 7605: 7603: 7600: 7598: 7595: 7593: 7590: 7588: 7585: 7583: 7580: 7579: 7577: 7575: 7571: 7565: 7562: 7560: 7557: 7556: 7554: 7552: 7548: 7542: 7539: 7537: 7534: 7533: 7531: 7529: 7525: 7522: 7520: 7516: 7510: 7507: 7505: 7504:Stereographic 7502: 7500: 7497: 7495: 7492: 7490: 7487: 7485: 7482: 7480: 7477: 7475: 7472: 7471: 7469: 7467: 7463: 7459: 7455: 7450: 7446: 7432: 7431:Winkel tripel 7429: 7427: 7424: 7422: 7419: 7417: 7414: 7413: 7411: 7409: 7405: 7395: 7392: 7390: 7387: 7386: 7383: 7377: 7376:Stereographic 7374: 7372: 7369: 7367: 7364: 7363: 7361: 7359: 7355: 7352: 7350: 7342: 7336: 7333: 7329: 7326: 7324: 7321: 7320: 7319: 7316: 7314: 7311: 7309: 7306: 7305: 7303: 7301: 7300:Pseudoconical 7297: 7291: 7288: 7286: 7283: 7281: 7278: 7277: 7275: 7273: 7269: 7259: 7256: 7254: 7251: 7249: 7246: 7245: 7242: 7236: 7233: 7231: 7228: 7226: 7223: 7221: 7218: 7216: 7213: 7211: 7208: 7206: 7203: 7201: 7198: 7196: 7193: 7192: 7190: 7186: 7183: 7181: 7177: 7167: 7164: 7162: 7159: 7157: 7154: 7152: 7149: 7147: 7144: 7142: 7139: 7137: 7134: 7132: 7129: 7128: 7125: 7119: 7116: 7114: 7111: 7109: 7106: 7104: 7101: 7099: 7096: 7094: 7091: 7089: 7086: 7085: 7083: 7081: 7077: 7071: 7068: 7066: 7063: 7061: 7058: 7057: 7055: 7052: 7048: 7045: 7043: 7039: 7035: 7031: 7026: 7022: 7016: 7013: 7011: 7008: 7006: 7003: 7002: 6999: 6995: 6988: 6983: 6981: 6976: 6974: 6969: 6968: 6965: 6958: 6955: 6952: 6948: 6945: 6942: 6939: 6936: 6932: 6929: 6928: 6924: 6920: 6916: 6913: 6911: 6908: 6907: 6903: 6899: 6896: 6894: 6890: 6887: 6886: 6882: 6875: 6873:0-914098-73-X 6869: 6865: 6860: 6856: 6854:0-226-76746-9 6850: 6846: 6842: 6838: 6834: 6829: 6825: 6820: 6816: 6814:0-387-54812-2 6810: 6805: 6804: 6797: 6793: 6791:0-486-65812-0 6787: 6783: 6778: 6774: 6772:0-13-065246-6 6768: 6764: 6759: 6755: 6751: 6747: 6743: 6739: 6735: 6731: 6727: 6722: 6718: 6716:0-13-212589-7 6712: 6708: 6704: 6700: 6696: 6692: 6688: 6687:Gelfand, I.M. 6684: 6680: 6675: 6666: 6662: 6661: 6655: 6651: 6649:0-07-010905-2 6645: 6641: 6636: 6632: 6630:0-201-00288-4 6626: 6622: 6618: 6614: 6613: 6608: 6601: 6595: 6593: 6589: 6578:. lenstip.com 6577: 6571: 6568: 6565: 6561: 6558: 6550:Samyang 8 mm 6546: 6543: 6538: 6534: 6530: 6526: 6522: 6518: 6514: 6510: 6506: 6499: 6496: 6491: 6487: 6483: 6479: 6475: 6471: 6467: 6463: 6459: 6452: 6449: 6444: 6438: 6434: 6430: 6426: 6425: 6417: 6414: 6409: 6403: 6400:. CRC Press. 6399: 6398: 6390: 6388: 6384: 6379: 6377:9780521535823 6373: 6369: 6362: 6359: 6355: 6352: 6346: 6343: 6339: 6336: 6330: 6327: 6321: 6318: 6307: 6302: 6297: 6293: 6292: 6287: 6283: 6279: 6275: 6269: 6266: 6261: 6257: 6256:Ahlfors, Lars 6251: 6248: 6242: 6239: 6233: 6230: 6227: 6222: 6219: 6213: 6210: 6204: 6201: 6196: 6192: 6188: 6184: 6180: 6176: 6169: 6166: 6160: 6157: 6153: 6151: 6144: 6141: 6135: 6132: 6126: 6123: 6118: 6114: 6110: 6106: 6102: 6098: 6094: 6088: 6085: 6080: 6076: 6072: 6068: 6061: 6058: 6053: 6049: 6045: 6041: 6037: 6033: 6029: 6023: 6021: 6017: 6011: 6004: 5999: 5997: 5993: 5990:in the plane. 5989: 5983: 5980: 5974: 5970: 5967: 5965: 5962: 5960: 5957: 5955: 5951: 5948: 5946: 5943: 5941: 5938: 5936: 5933: 5932: 5928: 5926: 5923: 5921: 5917: 5913: 5909: 5908:little planet 5905: 5901: 5896: 5894: 5889: 5881: 5876: 5870: 5866: 5862: 5857: 5850: 5848: 5846: 5842: 5836: 5829: 5827: 5825: 5821: 5817: 5812: 5807: 5805: 5801: 5797: 5793: 5789: 5780: 5773: 5771: 5769: 5764: 5760: 5756: 5752: 5747: 5745: 5741: 5737: 5733: 5729: 5725: 5719: 5712: 5708: 5703: 5696: 5694: 5692: 5688: 5684: 5676: 5672: 5667: 5660: 5654: 5647: 5642: 5635: 5630: 5628: 5626: 5622: 5617: 5615: 5611: 5607: 5601: 5593: 5588: 5586: 5584: 5565: 5559: 5556: 5551: 5547: 5541: 5538: 5534: 5528: 5525: 5522: 5515: 5514: 5513: 5510: 5490: 5484: 5481: 5476: 5472: 5466: 5463: 5457: 5454: 5451: 5448: 5444: 5436: 5432: 5428: 5425: 5418: 5414: 5410: 5407: 5401: 5398: 5395: 5392: 5385: 5384: 5383: 5379: 5375: 5365: 5360: 5352: 5350: 5332: 5323: 5319: 5315: 5310: 5306: 5298: 5294: 5290: 5285: 5281: 5274: 5266: 5262: 5258: 5253: 5249: 5243: 5240: 5237: 5230: 5222: 5221: 5220: 5217: 5206: 5198: 5187: 5182: 5176: 5169: 5165: 5159: 5154: 5149: 5145: 5137: 5132: 5125: 5123: 5120: 5114: 5108: 5102: 5096: 5090: 5085: 5081: 5073: 5071: 5068: 5066: 5061: 5059: 5054: 5049: 5047: 5042: 5037: 5031: 5029: 5025: 5017: 5016:Kikuchi lines 5014:Animation of 5012: 5005: 5003: 5001: 4997: 4993: 4989: 4985: 4980: 4974: 4964: 4950: 4940: 4926: 4920: 4914: 4889: 4882: 4873: 4867: 4864: 4859: 4850: 4844: 4841: 4835: 4829: 4820: 4814: 4811: 4806: 4803: 4800: 4797: 4794: 4788: 4782: 4773: 4767: 4764: 4759: 4756: 4753: 4750: 4743: 4739: 4737: 4729: 4726: 4723: 4720: 4717: 4707: 4701: 4698: 4695: 4690: 4687: 4684: 4681: 4675: 4673: 4668: 4657: 4656: 4655: 4652: 4648: 4644: 4620: 4616: 4609: 4600: 4594: 4591: 4586: 4577: 4571: 4568: 4565: 4559: 4553: 4544: 4538: 4535: 4530: 4527: 4524: 4521: 4515: 4509: 4500: 4494: 4491: 4486: 4483: 4480: 4477: 4470: 4466: 4464: 4456: 4453: 4450: 4447: 4444: 4431: 4425: 4422: 4419: 4414: 4411: 4408: 4405: 4399: 4397: 4392: 4381: 4380: 4379: 4376: 4372: 4368: 4364: 4358: 4354: 4347: 4345: 4341: 4337: 4333: 4325: 4321: 4320:complex plane 4313: 4309: 4304: 4297: 4292: 4282: 4273: 4265: 4260: 4252: 4243: 4235: 4230: 4228: 4225: 4224:central angle 4220: 4213: 4210: 4207: 4204: 4203: 4202: 4199: 4192: 4186: 4180: 4173: 4163: 4159: 4155: 4151: 4149: 4145: 4141: 4136: 4134: 4130: 4126: 4106: 4098: 4093: 4085: 4078: 4060: 4056: 4052: 4040: 4036: 4033: 4024: 4020: 4003: 3999: 3996: 3993: 3990: 3987: 3984: 3981: 3978: 3969: 3965: 3957: 3954: 3942: 3935: 3926: 3922: 3919: 3909: 3908: 3907: 3904: 3898: 3885: 3879: 3873: 3869: 3865: 3859: 3853: 3846: 3842: 3837: 3832: 3826: 3822: 3821: 3820: 3817: 3814: 3808: 3802: 3793: 3784: 3775: 3766: 3760: 3751: 3747: 3745: 3740: 3719: 3714: 3710: 3702: 3698: 3695: 3688: 3687: 3686: 3684: 3676: 3671: 3667: 3665: 3661: 3660:transversally 3655: 3641: 3638: 3635: 3632: 3607: 3603: 3599: 3594: 3590: 3580: 3577: 3574: 3551: 3548: 3545: 3542: 3539: 3536: 3533: 3530: 3527: 3524: 3521: 3518: 3510: 3494: 3491: 3488: 3485: 3482: 3479: 3459: 3456: 3453: 3450: 3447: 3438: 3434: 3430: 3426: 3424: 3420: 3415: 3413: 3409: 3405: 3386: 3378: 3374: 3370: 3367: 3362: 3358: 3354: 3342: 3332: 3328: 3324: 3319: 3315: 3311: 3308: 3301: 3292: 3291: 3290: 3286: 3282: 3275: 3271: 3267: 3247: 3244: 3241: 3237: 3234: 3225: 3215: 3211: 3207: 3202: 3198: 3194: 3191: 3184: 3179: 3176: 3173: 3166: 3165: 3164: 3160: 3156: 3144: 3136: 3116: 3106: 3102: 3099: 3088: 3078: 3075: 3068: 3067: 3066: 3064: 3058: 3054: 3050: 3040: 3036: 3032: 3028: 3023: 3018: 3016: 3012: 3008: 3004: 3000: 2994: 2988: 2982: 2980: 2976: 2968: 2966: 2964: 2960: 2956: 2951: 2944: 2936: 2932: 2926: 2920: 2914: 2908: 2902: 2896: 2889: 2885: 2881: 2873: 2869: 2862: 2858: 2852: 2846: 2842: 2838: 2833: 2827: 2814: 2808: 2805: 2802: 2799: 2796: 2793: 2790: 2780: 2777: 2772: 2768: 2760: 2756: 2752: 2746: 2741: 2737: 2723: 2720: 2715: 2711: 2705: 2702: 2697: 2693: 2686: 2681: 2677: 2656: 2648: 2644: 2640: 2637: 2630: 2626: 2622: 2619: 2613: 2608: 2603: 2599: 2593: 2588: 2585: 2582: 2578: 2574: 2569: 2565: 2544: 2538: 2535: 2532: 2529: 2526: 2523: 2520: 2508: 2504: 2500: 2497: 2491: 2487: 2481: 2476: 2472: 2462: 2458: 2452: 2447: 2442: 2435: 2431: 2425: 2419: 2413: 2406: 2402: 2397: 2392: 2385: 2378: 2371:is the point 2368: 2364: 2360: 2354: 2349: 2344: 2338: 2332: 2326: 2322: 2316: 2310: 2306: 2303: 2293: 2291: 2289: 2285: 2280: 2271: 2265: 2259: 2253: 2247: 2241: 2231: 2225: 2221: 2217: 2211: 2207: 2206: 2205: 2202: 2196: 2186: 2180: 2174: 2152: 2148: 2139: 2135: 2131: 2128: 2123: 2119: 2115: 2112: 2109: 2102: 2098: 2094: 2089: 2085: 2081: 2078: 2075: 2069: 2061: 2057: 2053: 2048: 2044: 2040: 2037: 2033: 2028: 2020: 2016: 2012: 2007: 2003: 1999: 1996: 1992: 1986: 1982: 1980: 1972: 1969: 1966: 1963: 1960: 1948: 1945: 1942: 1932: 1928: 1921: 1918: 1913: 1910: 1904: 1899: 1893: 1890: 1885: 1882: 1876: 1870: 1866: 1864: 1856: 1853: 1850: 1838: 1835: 1832: 1829: 1826: 1812: 1811: 1810: 1791: 1767: 1763: 1757: 1752: 1748: 1742: 1731: 1725: 1718: 1698: 1694: 1687: 1684: 1679: 1675: 1669: 1666: 1661: 1657: 1650: 1644: 1636: 1632: 1628: 1625: 1620: 1617: 1610: 1606: 1604: 1596: 1593: 1590: 1587: 1584: 1574: 1570: 1566: 1563: 1557: 1554: 1551: 1547: 1541: 1537: 1535: 1524: 1521: 1507: 1506: 1505: 1501: 1497: 1489: 1485: 1481: 1476: 1472: 1467: 1457: 1433: 1429: 1422: 1417: 1414: 1409: 1406: 1403: 1399: 1395: 1393: 1385: 1382: 1379: 1369: 1365: 1361: 1358: 1353: 1350: 1345: 1342: 1338: 1334: 1330: 1326: 1323: 1317: 1314: 1311: 1308: 1305: 1300: 1297: 1294: 1287: 1283: 1281: 1270: 1267: 1253: 1252: 1251: 1247: 1243: 1238: 1232: 1226: 1221: 1214: 1208: 1203: 1196: 1192: 1187: 1164: 1160: 1151: 1147: 1143: 1138: 1134: 1130: 1127: 1120: 1116: 1112: 1107: 1103: 1099: 1096: 1093: 1087: 1079: 1075: 1071: 1066: 1062: 1058: 1055: 1050: 1047: 1041: 1033: 1029: 1025: 1020: 1016: 1012: 1009: 1004: 1001: 994: 990: 988: 980: 977: 974: 971: 968: 958: 954: 947: 944: 941: 937: 932: 926: 923: 920: 916: 910: 906: 904: 896: 893: 890: 876: 875: 874: 870: 866: 858: 854: 850: 845: 840: 837: 832: 827: 817: 811: 805: 799: 793: 787: 783: 777: 770: 763: 759: 755: 748: 744: 740: 733: 727: 723: 716: 715:cross section 710: 704: 697: 692: 690: 688: 684: 680: 676: 675:Edmond Halley 672: 668: 663: 661: 657: 653: 651: 647: 643: 639: 635: 631: 627: 622: 620: 616: 612: 608: 604: 599: 597: 594:described by 591: 587: 582: 578: 574: 570: 567:) contains a 565: 200 BC 559: 555: 550: 400 AD 544: 540: 536: 535: 530: 526: 522: 518: 510: 506: 501: 496: 488: 486: 484: 480: 476: 472: 468: 464: 460: 456: 452: 447: 445: 441: 437: 433: 429: 426: 422: 418: 414: 410: 407: 403: 398: 396: 392: 388: 384: 380: 376: 372: 368: 364: 361: 357: 353: 349: 348:perpendicular 345: 341: 337: 333: 329: 325: 321: 317: 313: 301: 296: 294: 289: 287: 282: 281: 279: 278: 271: 268: 266: 263: 261: 258: 256: 253: 251: 248: 246: 243: 241: 238: 236: 233: 231: 228: 226: 223: 221: 220:Picture plane 218: 216: 213: 211: 208: 206: 203: 201: 198: 196: 193: 191: 188: 186: 183: 181: 180:3D projection 178: 177: 171: 170: 163: 160: 158: 155: 153: 150: 148: 145: 143: 140: 138: 135: 133: 130: 128: 127:Cross section 125: 123: 120: 119: 113: 112: 103: 100: 98: 95: 94: 93: 90: 86: 83: 79: 76: 75: 74: 71: 70: 69: 66: 65: 62: 57: 56: 52: 48: 47: 44: 40: 36: 35: 28: 22: 7952: 7948:Orthographic 7503: 7479:Gauss–Krüger 7371:Orthographic 7166:Web Mercator 7060:Gauss–Krüger 6893:Cut-the-Knot 6863: 6844: 6832: 6823: 6807:. Springer. 6802: 6781: 6762: 6729: 6725: 6706: 6694: 6691:Minlos, R.A. 6678: 6668:, retrieved 6659: 6639: 6620: 6617:Apostol, Tom 6599: 6580:. Retrieved 6570: 6545: 6515:(1): 53–67. 6512: 6508: 6498: 6465: 6461: 6451: 6423: 6416: 6396: 6367: 6361: 6353: 6350: 6345: 6337: 6334: 6329: 6320: 6310:, retrieved 6290: 6278:Gilman, Jane 6274:Conway, John 6268: 6259: 6250: 6241: 6232: 6221: 6212: 6203: 6178: 6174: 6168: 6159: 6149: 6143: 6134: 6125: 6100: 6096: 6087: 6073:(1): 25–50. 6070: 6066: 6060: 6035: 6031: 6009: 5982: 5969:Fisheye lens 5924: 5915: 5907: 5897: 5885: 5879: 5837: 5833: 5815: 5808: 5804:slickensides 5785: 5762: 5759:Ewald sphere 5755:Kikuchi line 5748: 5743: 5721: 5693:into lines. 5680: 5618: 5603: 5580: 5508: 5505: 5377: 5373: 5369: 5348: 5215: 5204: 5196: 5185: 5174: 5167: 5163: 5152: 5141: 5118: 5112: 5106: 5100: 5094: 5088: 5077: 5069: 5064: 5062: 5058:beam compass 5052: 5050: 5035: 5032: 5021: 4983: 4978: 4972: 4962: 4948: 4938: 4924: 4918: 4912: 4909: 4650: 4646: 4642: 4639: 4374: 4370: 4366: 4356: 4352: 4348: 4328: 4280: 4271: 4250: 4241: 4222:To find the 4221: 4217: 4197: 4190: 4184: 4178: 4171: 4168: 4156: 4152: 4137: 4128: 4124: 4121: 4096: 3902: 3893: 3890: 3883: 3877: 3871: 3863: 3857: 3848: 3840: 3834:and not its 3830: 3824: 3818: 3812: 3806: 3797: 3788: 3779: 3770: 3764: 3758: 3755: 3738: 3735: 3680: 3656: 3472:in terms of 3436: 3432: 3427: 3416: 3411: 3407: 3401: 3284: 3280: 3276: 3269: 3265: 3262: 3158: 3154: 3149: 3056: 3052: 3045: 3038: 3034: 3030: 3026: 3019: 3011:homeomorphic 2992: 2986: 2983: 2972: 2949: 2942: 2934: 2930: 2924: 2918: 2912: 2906: 2900: 2894: 2887: 2883: 2871: 2867: 2860: 2856: 2850: 2844: 2831: 2828: 2464:is given by 2460: 2456: 2450: 2445: 2440: 2433: 2429: 2423: 2417: 2411: 2404: 2400: 2390: 2383: 2373: 2366: 2362: 2358: 2352: 2342: 2336: 2330: 2324: 2314: 2308: 2301: 2297: 2287: 2283: 2275: 2269: 2263: 2257: 2251: 2245: 2239: 2236: 2229: 2223: 2215: 2209: 2200: 2194: 2191: 2184: 2178: 1789: 1768: 1761: 1755: 1740: 1737: 1729: 1499: 1495: 1487: 1483: 1479: 1465: 1455: 1452: 1245: 1241: 1230: 1219: 1212: 1207:zenith angle 1201: 1194: 1190: 1183: 868: 864: 856: 852: 848: 841: 835: 830: 822: 815: 809: 803: 797: 791: 788: 781: 775: 768: 761: 757: 753: 746: 742: 738: 731: 725: 719: 708: 687:Isaac Newton 664: 659: 654: 623: 609:to form the 600: 557: 533: 514: 482: 478: 474: 448: 399: 343: 335: 331: 315: 309: 185:Anamorphosis 142:Fisheye lens 7926:Perspective 7714:some aspect 7698:Strebe 1995 7673:Equal Earth 7592:Gall–Peters 7574:Cylindrical 7389:Equidistant 7285:Equidistant 7215:Equal Earth 7098:Gall–Peters 7042:Cylindrical 5865:Esino Lario 5851:Photography 5744:pole figure 5718:Pole figure 5610:statistical 5594:Cartography 5189:-axis. If 4336:orientation 4239:Two points 3007:topological 2979:unit circle 2975:unit sphere 2249:other than 2237:As long as 1747:composition 722:unit sphere 679:star charts 584: [ 575:, and even 534:Planisphere 471:graph paper 467:photography 459:cartography 451:mathematics 430:instead of 312:mathematics 255:True length 245:Stereoscopy 8089:Categories 7988:AuthaGraph 7980:Polyhedral 7850:Compromise 7778:Loximuthal 7770:Loxodromic 7732:Sinusoidal 7582:Balthasart 7559:Sinusoidal 7536:Sinusoidal 7519:Equal-area 7230:Sinusoidal 7188:Equal-area 7088:Balthasart 7080:Equal-area 7053:-conformal 7030:By surface 6670:2014-12-12 6582:2011-07-07 6312:2022-04-26 6301:1804.03055 5986:Under the 5975:References 5843:(SMR) and 5824:contouring 5691:degenerate 5614:navigation 5179:) and the 4984:vice versa 4144:hemisphere 3683:loxodromes 3675:loxodromes 3658:intersect 2969:Properties 2444:from 1 to 2415:from 0 to 2348:hyperplane 2204:such that 751:such that 693:Definition 626:equatorial 573:Archimedes 554:Apollonius 539:Hipparchus 493:See also: 338:), onto a 190:Axonometry 147:Multiviews 8060:Longitude 7888:Wagner VI 7737:Two-point 7668:Eckert VI 7663:Eckert IV 7658:Eckert II 7635:Mollweide 7630:Collignon 7597:Hobo–Dyer 7551:Bottomley 7466:Conformal 7454:By metric 7345:Azimuthal 7318:Polyconic 7313:Bottomley 7253:Wagner VI 7225:Mollweide 7210:Eckert VI 7205:Eckert IV 7200:Eckert II 7195:Collignon 7103:Hobo–Dyer 6947:Sphaerica 6931:Stereonet 6784:. Dover. 6754:193430645 6537:0013-7952 6490:0013-7952 6468:: 67–76. 6195:118095486 6097:Centaurus 5940:Astrolabe 5900:panoramas 5893:Panotools 5796:lineation 5792:foliation 5711:direction 5625:parallels 5621:meridians 5583:integrals 5452:⁡ 5411:− 5396:⁡ 5291:− 5080:polytopes 4883:ξ 4877:¯ 4874:ξ 4860:ξ 4854:¯ 4851:ξ 4845:− 4830:ξ 4824:¯ 4821:ξ 4807:ξ 4804:⁡ 4795:− 4783:ξ 4777:¯ 4774:ξ 4760:ξ 4757:⁡ 4685:− 4669:ξ 4610:ζ 4604:¯ 4601:ζ 4587:ζ 4581:¯ 4578:ζ 4566:− 4554:ζ 4548:¯ 4545:ζ 4531:ζ 4528:⁡ 4510:ζ 4504:¯ 4501:ζ 4487:ζ 4484:⁡ 4423:− 4393:ζ 4140:parallels 4129:Wulff net 4125:stereonet 4107:π 4079:Wulff net 4048:′ 4045:′ 4034:⋅ 4029:′ 4017:⟹ 4011:′ 4008:′ 3974:′ 3962:⟹ 3950:′ 3947:′ 3939:△ 3936:∼ 3931:′ 3917:△ 3707:Θ 3578:− 3543:− 3507:given in 3107:∈ 3095:⟺ 3079:∈ 2803:… 2703:− 2641:− 2579:∑ 2557:Defining 2533:… 2501:− 2136:η 2120:ξ 2099:η 2086:ξ 2076:− 2058:η 2045:ξ 2034:η 2017:η 2004:ξ 1993:ξ 1955:→ 1949:η 1943:ξ 1919:− 1891:− 1857:η 1851:ξ 1845:→ 1751:homothety 1667:− 1648:Θ 1591:θ 1567:θ 1555:− 1528:Θ 1426:Θ 1410:⁡ 1386:θ 1380:φ 1362:θ 1351:φ 1346:⁡ 1327:θ 1318:φ 1315:⁡ 1309:− 1301:φ 1298:⁡ 1274:Θ 1094:− 945:− 924:− 671:conformal 642:Jean Roze 615:Byzantine 596:Vitruvius 483:Wulff net 479:stereonet 473:called a 425:spherical 402:represent 395:equiareal 391:isometric 375:conformal 360:bijective 162:Zoom lens 8055:Latitude 8040:See also 8003:Dymaxion 7943:Gnomonic 7878:Robinson 7783:Mercator 7760:Gnomonic 7752:Gnomonic 7587:Behrmann 7494:Mercator 7366:Gnomonic 7348:(planar) 7323:American 7093:Behrmann 7051:Mercator 6925:Software 6843:(1993). 6782:Geometry 6703:Do Carmo 6619:(1974). 6560:Archived 6356::114308. 6306:archived 6258:(1966). 6003:Synesius 5929:See also 5914:) and a 5028:embedded 4344:manifold 4314:denoted 3836:antipode 3431:that do 3419:isometry 3103:′ 3024:a point 683:calculus 644:(1542), 543:Synesius 413:geodesic 363:function 352:diameter 152:Panorama 7916:HEALPix 7815:Littrow 7426:Wiechel 7328:Chinese 7272:Conical 7136:Central 7131:Cassini 7108:Lambert 7005:History 6609:Sources 6602:(2007). 6598:German 6517:Bibcode 6470:Bibcode 6105:Bibcode 5774:Geology 5728:crystal 5209:⁠ 5193:⁠ 5183:of the 5160:points 5082:. In a 4970:, with 4967:⁠ 4953:⁠ 4943:⁠ 4929:⁠ 4340:surface 4148:equator 3013:to the 2878:in the 2866:, ..., 2839:in the 2305:-sphere 1806:⁠ 1794:⁠ 1784:⁠ 1772:⁠ 1225:azimuth 650:Ptolemy 630:Eastern 607:dioptra 577:Eudoxus 569:theorem 552:), and 529:Ptolemy 489:History 463:geology 442:of the 409:induced 383:locally 350:to the 322:of the 7935:Planar 7903:Hybrid 7810:Hammer 7742:Werner 7683:Hammer 7648:Albers 7564:Werner 7541:Werner 7421:Hammer 7416:Aitoff 7335:Werner 7280:Albers 7156:Miller 7015:Portal 6941:PTCLab 6904:Videos 6870: 6851: 6811: 6788: 6769: 6752: 6746:751275 6744: 6713: 6646: 6627: 6600:et al. 6535: 6488: 6439: 6404: 6374: 6193: 6152:, p.59 6052:227240 6050: 5920:zenith 5376:, cos 4994:. The 4916:- and 2328:. If 2267:meets 1453:Here, 1407:arctan 1235:) and 1217:, and 766:. Let 558:Conics 505:Rubens 465:, and 406:metric 387:shapes 379:angles 356:smooth 324:sphere 174:Topics 61:Planar 7805:Craig 7722:Conic 7528:Bonne 7308:Bonne 6919:Vimeo 6891:from 6750:S2CID 6742:JSTOR 6296:arXiv 6191:S2CID 6048:JSTOR 5912:nadir 5886:Some 5800:fault 5732:X-ray 5372:(sin 5086:, an 5053:trace 3891:then 2947:with 2448:) on 2427:and ( 2421:) on 2394:. In 2388:with 2286:onto 2219:, and 1473:. In 1463:when 592:] 481:, or 342:(the 340:plane 328:point 318:is a 116:Views 8008:ISEA 7010:List 6868:ISBN 6849:ISBN 6809:ISBN 6786:ISBN 6767:ISBN 6711:ISBN 6644:ISBN 6625:ISBN 6533:ISSN 6486:ISSN 6437:ISBN 6402:ISBN 6372:ISBN 6032:Isis 5916:tube 5734:and 5671:Moon 5211:, 0) 5134:The 5065:pole 4946:and 4278:and 4248:and 3900:and 3810:and 3795:and 3777:and 3762:and 3681:The 2961:and 2340:and 2312:in ( 2261:and 1759:and 1743:= −1 1732:= −1 1233:≤ 2π 1229:0 ≤ 1223:the 1211:0 ≤ 1205:the 807:and 720:The 632:and 332:pole 314:, a 6917:on 6734:doi 6665:AMS 6525:doi 6478:doi 6466:124 6429:doi 6183:doi 6113:doi 6075:doi 6040:doi 5922:). 5863:in 5749:In 5722:In 5709:in 5685:to 5449:sin 5393:cos 5177:≠ 1 5116:to 4649:− i 4373:+ i 4346:). 4193:= 0 4127:or 3867:and 3433:not 3406:'s 3272:= 1 3020:In 2933:− { 2928:in 2910:in 2898:on 2732:and 2365:− { 2350:in 2317:+ 1 1792:= − 1343:cot 1312:cos 1295:sin 1215:≤ π 1184:In 842:In 833:of 818:= 0 795:on 784:= 0 764:= 1 711:= 0 556:'s 531:'s 369:to 334:or 310:In 8091:: 6748:. 6740:. 6730:51 6728:. 6689:; 6663:, 6591:^ 6531:. 6523:. 6513:61 6511:. 6507:. 6484:. 6476:. 6464:. 6460:. 6435:. 6386:^ 6354:98 6338:35 6304:, 6288:, 6280:; 6189:. 6179:20 6177:. 6111:. 6099:. 6069:. 6046:. 6036:40 6034:. 6019:^ 5995:^ 5847:. 5826:. 5806:. 5770:. 5753:, 5746:. 5616:. 5509:dx 5166:, 4951:= 4927:= 4801:Im 4754:Re 4645:= 4525:Im 4481:Re 4369:= 4355:, 4201:: 4135:. 3903:P″ 3872:P″ 3437:do 3414:. 3283:, 3268:+ 3157:, 3055:, 3037:, 3033:, 2965:. 2943:QP 2874:+1 2438:, 2409:, 2384:QP 2361:∈ 2346:a 2290:. 1498:, 1486:, 1482:, 1244:, 1227:, 1209:, 1193:, 867:, 855:, 851:, 760:+ 756:+ 745:, 741:, 689:. 652:. 590:it 588:; 586:fr 562:c. 547:c. 485:. 461:, 457:, 446:. 358:, 346:) 6986:e 6979:t 6972:v 6937:. 6876:. 6857:. 6817:. 6794:. 6775:. 6756:. 6736:: 6719:. 6652:. 6633:. 6585:. 6555:/ 6553:f 6539:. 6527:: 6519:: 6492:. 6480:: 6472:: 6445:. 6431:: 6410:. 6380:. 6298:: 6197:. 6185:: 6154:. 6119:. 6115:: 6107:: 6101:3 6081:. 6077:: 6071:8 6054:. 6042:: 5566:. 5560:1 5557:+ 5552:2 5548:t 5542:t 5539:d 5535:2 5529:= 5526:x 5523:d 5491:. 5485:1 5482:+ 5477:2 5473:t 5467:t 5464:2 5458:= 5455:x 5445:, 5437:2 5433:t 5429:+ 5426:1 5419:2 5415:t 5408:1 5402:= 5399:x 5380:) 5378:x 5374:x 5333:) 5324:2 5320:n 5316:+ 5311:2 5307:m 5299:2 5295:n 5286:2 5282:m 5275:, 5267:2 5263:n 5259:+ 5254:2 5250:m 5244:n 5241:m 5238:2 5231:( 5216:x 5205:n 5201:/ 5197:m 5191:( 5186:x 5175:y 5170:) 5168:y 5164:x 5162:( 5153:x 5119:R 5113:R 5107:R 5101:n 5095:R 5089:n 5036:z 4979:ξ 4973:ζ 4963:ζ 4959:/ 4956:1 4949:ξ 4939:ξ 4935:/ 4932:1 4925:ζ 4919:ξ 4913:ζ 4890:) 4868:+ 4865:1 4842:1 4836:, 4815:+ 4812:1 4798:2 4789:, 4768:+ 4765:1 4751:2 4744:( 4740:= 4733:) 4730:z 4727:, 4724:y 4721:, 4718:x 4715:( 4708:, 4702:z 4699:+ 4696:1 4691:y 4688:i 4682:x 4676:= 4651:Y 4647:X 4643:ξ 4621:. 4617:) 4595:+ 4592:1 4572:+ 4569:1 4560:, 4539:+ 4536:1 4522:2 4516:, 4495:+ 4492:1 4478:2 4471:( 4467:= 4460:) 4457:z 4454:, 4451:y 4448:, 4445:x 4442:( 4432:, 4426:z 4420:1 4415:y 4412:i 4409:+ 4406:x 4400:= 4375:Y 4371:X 4367:ζ 4359:) 4357:Y 4353:X 4351:( 4316:∞ 4287:. 4284:2 4281:P 4275:1 4272:P 4254:2 4251:P 4245:1 4242:P 4198:P 4191:z 4185:y 4179:x 4172:P 4097:C 4061:2 4057:r 4053:= 4041:P 4037:O 4025:P 4021:O 4004:P 4000:O 3997:: 3994:S 3991:O 3988:= 3985:N 3982:O 3979:: 3970:P 3966:O 3958:S 3955:O 3943:P 3927:P 3923:O 3920:N 3896:′ 3894:P 3887:, 3884:S 3878:P 3864:N 3858:P 3851:′ 3849:P 3844:, 3841:S 3831:N 3825:P 3813:Q 3807:P 3800:′ 3798:Q 3791:′ 3789:P 3782:′ 3780:Q 3773:′ 3771:P 3765:Q 3759:P 3739:a 3720:, 3715:a 3711:/ 3703:e 3699:= 3696:R 3642:, 3639:d 3636:= 3633:c 3613:) 3608:2 3604:Y 3600:+ 3595:2 3591:X 3587:( 3584:) 3581:d 3575:c 3572:( 3552:0 3549:= 3546:d 3540:z 3537:c 3534:+ 3531:y 3528:b 3525:+ 3522:x 3519:a 3495:, 3492:Z 3489:, 3486:Y 3483:, 3480:X 3460:z 3457:, 3454:y 3451:, 3448:x 3387:, 3384:) 3379:2 3375:Y 3371:d 3368:+ 3363:2 3359:X 3355:d 3352:( 3343:2 3339:) 3333:2 3329:Y 3325:+ 3320:2 3316:X 3312:+ 3309:1 3306:( 3302:4 3287:) 3285:Y 3281:X 3279:( 3270:Y 3266:X 3248:. 3245:Y 3242:d 3238:X 3235:d 3226:2 3222:) 3216:2 3212:Y 3208:+ 3203:2 3199:X 3195:+ 3192:1 3189:( 3185:4 3180:= 3177:A 3174:d 3161:) 3159:Y 3155:X 3153:( 3117:2 3112:Q 3100:P 3089:3 3084:Q 3076:P 3059:) 3057:Y 3053:X 3051:( 3048:′ 3046:P 3041:) 3039:z 3035:y 3031:x 3029:( 3027:P 2993:P 2987:N 2950:E 2937:} 2935:Q 2931:S 2925:P 2919:Q 2913:P 2907:E 2901:S 2895:Q 2888:i 2884:x 2876:) 2872:n 2868:x 2864:0 2861:x 2859:( 2857:f 2851:S 2845:P 2832:S 2815:. 2812:) 2809:n 2806:, 2800:, 2797:1 2794:= 2791:i 2788:( 2781:1 2778:+ 2773:2 2769:s 2761:i 2757:X 2753:2 2747:= 2742:i 2738:x 2724:1 2721:+ 2716:2 2712:s 2706:1 2698:2 2694:s 2687:= 2682:0 2678:x 2657:, 2649:0 2645:x 2638:1 2631:0 2627:x 2623:+ 2620:1 2614:= 2609:2 2604:j 2600:X 2594:n 2589:1 2586:= 2583:j 2575:= 2570:2 2566:s 2545:. 2542:) 2539:n 2536:, 2530:, 2527:1 2524:= 2521:i 2518:( 2509:0 2505:x 2498:1 2492:i 2488:x 2482:= 2477:i 2473:X 2461:S 2457:Q 2451:E 2446:n 2441:i 2434:i 2430:X 2424:S 2418:n 2412:i 2405:i 2401:x 2398:( 2391:E 2376:′ 2374:P 2369:} 2367:Q 2363:S 2359:P 2353:E 2343:E 2337:S 2331:Q 2325:E 2315:n 2309:S 2302:n 2288:E 2284:P 2278:′ 2276:P 2270:E 2264:Q 2258:P 2252:Q 2246:P 2240:E 2233:. 2230:Q 2224:E 2216:Q 2210:E 2201:E 2195:Q 2185:E 2179:Q 2153:. 2149:) 2140:2 2132:2 2129:+ 2124:2 2116:2 2113:+ 2110:2 2103:2 2095:+ 2090:2 2082:+ 2079:1 2070:, 2062:2 2054:+ 2049:2 2041:+ 2038:1 2029:, 2021:2 2013:+ 2008:2 2000:+ 1997:1 1987:( 1983:= 1976:) 1973:z 1970:, 1967:y 1964:, 1961:x 1958:( 1952:) 1946:, 1940:( 1933:, 1929:) 1922:z 1914:2 1911:1 1905:y 1900:, 1894:z 1886:2 1883:1 1877:x 1871:( 1867:= 1860:) 1854:, 1848:( 1842:) 1839:z 1836:, 1833:y 1830:, 1827:x 1824:( 1803:2 1800:/ 1797:1 1790:z 1781:2 1778:/ 1775:1 1762:Y 1756:X 1741:z 1730:z 1699:. 1695:) 1688:1 1685:+ 1680:2 1676:R 1670:1 1662:2 1658:R 1651:, 1645:, 1637:2 1633:R 1629:+ 1626:1 1621:R 1618:2 1611:( 1607:= 1600:) 1597:z 1594:, 1588:, 1585:r 1582:( 1575:, 1571:) 1564:, 1558:z 1552:1 1548:r 1542:( 1538:= 1531:) 1525:, 1522:R 1519:( 1502:) 1500:Θ 1496:R 1494:( 1490:) 1488:z 1484:θ 1480:r 1478:( 1466:R 1461:π 1456:φ 1434:. 1430:) 1423:, 1418:R 1415:1 1404:2 1400:( 1396:= 1389:) 1383:, 1377:( 1370:, 1366:) 1359:, 1354:2 1339:( 1335:= 1331:) 1324:, 1306:1 1288:( 1284:= 1277:) 1271:, 1268:R 1265:( 1248:) 1246:Θ 1242:R 1240:( 1231:θ 1220:θ 1213:φ 1202:φ 1197:) 1195:θ 1191:φ 1189:( 1165:. 1161:) 1152:2 1148:Y 1144:+ 1139:2 1135:X 1131:+ 1128:1 1121:2 1117:Y 1113:+ 1108:2 1104:X 1100:+ 1097:1 1088:, 1080:2 1076:Y 1072:+ 1067:2 1063:X 1059:+ 1056:1 1051:Y 1048:2 1042:, 1034:2 1030:Y 1026:+ 1021:2 1017:X 1013:+ 1010:1 1005:X 1002:2 995:( 991:= 984:) 981:z 978:, 975:y 972:, 969:x 966:( 959:, 955:) 948:z 942:1 938:y 933:, 927:z 921:1 917:x 911:( 907:= 900:) 897:Y 894:, 891:X 888:( 871:) 869:Y 865:X 863:( 859:) 857:z 853:y 849:x 847:( 836:P 825:′ 823:P 816:z 810:P 804:N 798:M 792:P 782:z 776:M 769:N 762:z 758:y 754:x 749:) 747:z 743:y 739:x 737:( 732:R 726:S 709:z 560:( 545:( 299:e 292:t 285:v 23:.

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Knowledge

Stereographic projection

Index