The purpose of the stereographic projection in crystallography the stereographic projection is a projection of points from the surface of a sphere on to its equatorial plane. Stereographic projection and circular inversion iiser pune. Stereographic projection simple english wikipedia, the. Implementing stereographic projection in sage math 480 simon spicer je beorse kevin lindeman june 2, 2010 figure 1. For more on stereographic projection, see any standard complex analysis text. The stereographic projection has the property that all circles on the sphere are mapped onto circles or straight lines on the plane, and therefore it is easy to map astronomical.
First the lines are plotted and then then the overlay is rotated until these two points lie on the same great circle of the stereonet and the angle between the two lines is determined by counting the small circle divisions between the points along the great. Stereographic projection there are two special projections. Applications of complex analysis to precession, nutation and. There is, never theless, need for a new edition, partly because of changes in current mathe matical terminology, partly because of differences in student preparedness and aims.
The intersection made by the line or plane with the spheres. Think of the complex plane as being embedded in r3 as the plane z 0. Special cases include the mercator map and the stereographic projection. Complex analysis has successfully maintained its place as the standard elementary text on functions of one complex variable. Points on the surface of a sphere can be mapped by stereographic projection to points on the plane of complex numbers. The intersection of the line and the sphere is the corresponding projection. The mercator and stereographic projections, and many in between. The riemann sphere can be visualized as the complex number plane wrapped around a sphere by some form of stereographic projection details are given below. However, there is another projection, less known to students, a projection from a circle to the x.
Stereographic projection is a powerful method, not just to solve relatively simple but important problems of dip and strike, but as an analytical tool for more complex structural geology. The stereographic projection is a powerful tool for representing the orientation of discontinuities through rock in a simple, visual way and for allowing potential failure mechanisms to be recognized. As defined in our projection, the n and s poles would plot directly above and below the center of the stereonet. It will be nice to have some useful way to put images related with stereographic projections in our documents. Cauchyriemann equations sufficient condition pdf download. Stereographic projection the basics geological digressions. The equatorial projection became an essential tool. The family is referred to as the lambert conic conformal projections. Riemann sphere, projective space november 22, 2014 2.
In complex analysis it is used to represent the extended complex plane see for instance 2, chapter i. The equalarea lowerhemisphere projection defined by the lambert azimuthal equalarea projection is also used, especially when the plot is to be subjected to subsequent statistical analysis such as density contouring. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection. The circles are cut from the sphere by two planes, red and green respectively, through the lines in the zplane and through. The orientations of lines and planes can be plotted relative to the center of a sphere, called the projection sphere, as shown at the top of fig. Stereographic projection an overview sciencedirect topics. A spherical projection of a complex hilbert space is conformal i it is the stereographic projection harry gingold, yotam gingold and salah hamad abstract. The stereographic projection is a powerful tool for representing the orientation of discontinuities through rock in a simple, visual way and for allowing potential failure mechanisms to. Geometrical properties of stereographic projection continued. If you looked at the last page of the notes in order to do one of the counting exercises in the last section, you have already seen a picture of the poincar e disc model 6. Mobius transformations and stereographic projection. Mapping points on the sphere by stereographic projection to points on the plane of complex numbers transforms the spherical trigonometry calculations, performed. To visualize this compactification of the complex numbers transformation of a topological space into a compact space, one can perform a stereographic projection of the unit sphere onto the complex plane as follows.
Bpass a plane parallel to the original one through the line pp. However, distances and angles can be very badly distorted look what happened to greenland in the picture. In complex number theory stereographic projection is. The origin of the polar and equatorial stereographic projections can be traced back to hipparchos d.
We consider a family of nonlinear projections that map a complex hilbert space onto a bounded \bowl shaped subset of. Stereographic projection gis wiki the gis encyclopedia. I was working out gamelins complex analysis and read through the part where he finds an expression for the chordal distance on the riemann sphere corresponding to. This gives a 11 correspondence between a point on the sphere and a point in the complex plane. Greens theorem with a twist of complex analysis pdf download. Moreover, though some great circles are mapped to straight lines under cylindrical projection exercise which ones. In geometry, the stereographic projection is a particular mapping function that projects a. Stereographic projection maps circles to circles 5. The operation of stereographic projection is depicted in fig. Stereographic projections tex latex stack exchange. Stereographic projection part 1 introduction geology by dr. Chern, the fundamental objects of study in differential geometry are manifolds. This is to be read in parallel with gamelins complex analysis. Stereographic projection maps the intersection z of two lines, one red and one green, in the complex zplane to the intersection z of two circles, one red and one green, in the riemann zsphere.
Stereographic projection is conformal let s2 x,y,z. I was working out gamelins complex analysis and read through the part where he finds an expression for the chordal distance on the riemann sphere corresponding to the stereographic projection w. The riemann sphere is a geometric representation of the extended complex plane the complex numbers with the added point at infinity. It is easy to see show that the formula for stereographic. Finding the stereographic projection for complex plane. Stereographic projection definition is a map projection of a hemisphere showing the earths lines of latitude and longitude projected onto a tangent plane by radials from a point on the surface of the sphere opposite to the point of tangency. Essays on automorphic forms stereographic projection. The program can process this type of data in a variety of formats, and it can analyze the data with a variety of statistical methods. We consider a family of nonlinear projections that map a complex hilbert space onto a bounded \bowl shaped subset of a sphere. Pdf history of the stereographic projection and its early. Stereographic analysis application netprog capabilities the netprog program is designed to plot geologic structure data on a stereographic projection graph or stereonet.
Math 215 complex analysis lenya ryzhik copy pasting from others. August 20, 2009 6 c we want to show that the section of the cone by this plane is a circle. Pdf history of the stereographic projection and its. Intuitively, the stereographic projection is a way of picturing a sphere as a plane, with some inevitable compromises. A geometric construction known as stereographic projection gives rise to a onetoone correspondence between the complement of a chosen point a on the sphere and the points of the plane z.
Geometrically, we join a point z on the complex plane and the north pole of the riemann sphere with a straight line. Stereographic projection from the sphere to the plane. Then the generalized stereographic projection of s 2 is the composition of the hyperbolic projection and the ordinary stereographic projection centered at the north pole of s 2. The stereographic projection is a conformal map as well. The stereographic projection is a bijective smooth map which allows us to think the sphere as the. A central difficulty of complex analysis from nearly the very beginning has been the notion of. We consider a family of conformal angle preserving projections of the sphere onto the plane. If the points on the sphere are identified with the directions of incoming light rays, then the effect of a lorentz transformation, a rotation plus a boost, is represented by a bilinear or mobius transformation applied to points on the complex plane. The complex projective line cp1 for purposes of complex analysis, a better description of a onepoint compacti cation of c is an instance of the complex projective space cpn, a compact space containing cn, described as follows.
In mathematics, the riemann sphere, named after bernhard riemann, is a model of the extended complex plane, the complex plane plus a point at infinity. C can be represented as point x, y in the plane r2. Stereographic projection of crystal faces page 3 of 6 972010. A sphere of unit diameter is tangent to the complex plane at its south pole. Complex integration notes vector calc interpretation pdf download. Stereographic projection part 1 introduction youtube. Stereographic projection let a sphere in threedimensional euclidean space be given. Stereographic projection can be used to measure the angle between any two lines. In this context the stereographic projection is often referred to as the equalangle lowerhemisphere projection. Math 215 complex analysis lenya ryzhik copy pasting from others november 25, 20.
Stereographic projection of a complex number a onto a point. The riemann sphere as a stereographic projection wolfram. In cartesian coordinates x, y, z and spherical coordinates. It intersects the bowl at a single point, as shown in the view from above. Abstract the stereographic projection is a 11 mapping from the plane to the unit sphere and back again which has the special property of being conformal, or angle preserving.
The equatorial projection became an essential tool in astronomy and from the 2nd to. Given the coordinates x,y of a point in the plane, get the stereographic image on that point in the sphere, i. It follows that you probably should read gamelin to begin then read this. On occasion, a section in this guide may have the complete thought on a given topic, but, usually it is merely a summary or comment on what is in gamelin.
Applications of complex analysis to celestial navigation. Thus, all point on a projecting line are mapped to the same point on s 2. The stereographic projection is the transformation of the complex plane into the sphere that satisfies the following. A stereonet is essentially the view of the bowl from above. The mercator and stereographic projections, and many in.
Stereographic projection definition of stereographic. We also develop the cauchyriemannequations, which provide an easier test to verify the analyticity of a function. Show that using these relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskew. The mercator and stereographic projections, and many in between daniel daners abstract.
The gure below shows a sphere whose equator is the unit circle in the complex plane. Set of complex no, stereographic projection duration. A spherical projection of a complex hilbert space is. There are several good software programs and apps to automate projections for large data sets. This notion finds utility in projective geometry and complex analysis. It can be verified that the hyperbolic projection maps all points on a projecting line to the same point on plane p 3 0. Sep 12, 2018 stereographic projection part 1 introduction geology by dr. Identify the complex plane c with the x,yplane in r3. M obius transformations and stereographic projection. However, when plotting directional data in structural geology, they do represent the north and south geographic directions. The stereographic projection of a line is simply a point, so plotting the representation of the point will be pretty easy.
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