In order to make plotting of the stereographic projection easier, a device called a stereographic net or stereonet is used. In mathematics, a projection is a mapping of a set or other mathematical structure into a subset or substructure, which is equal to its square for mapping composition or, in other words, which is idempotent. The mercator and stereographic projections, and many in between. Jul 17, 2003 the book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection. The stereographic projection is a conformal map as well. Buy theory of functions, parts i and ii dover books on mathematics. In the previous lecture we introduced functions holomorphic on domains in \ \mathbbc \. History of the stereographic projection and its early use in geology. Stereographic projection from the sphere to the plane. Singularity at infinity, infinity as a value, compact spaces of meromorphic functions for the spherical metric and spherical derivative, local analysis of n video course course outline this is the second part of a series of lectures on advanced topics in complex analysis. Thanks for contributing an answer to mathematics stack exchange. Maths stereographic projection riemann sphere martin. The purpose of this course is to introduce the main ideas of complex analysis to an undergraduate audience.
Maths stereographic projection riemann sphere martin baker. Jul 23, 2014 stereographic projection of riemann sphere. Introductory complex analysis dover books on mathematics. We can represent any point on a sphere by using a complex number. Intuitively, the stereographic projection is a way of picturing a sphere as a plane, with some inevitable compromises. Geometrical properties of stereographic projection continued. Complex analysis has successfully maintained its place as the standard elementary text on functions of one complex variable. 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. Introduction to complex analysis gives advanced students an introduction to the theory of functions of a complex variable, a fundamental area of mathematics. Although the north n and south s poles are shown on the stereonet, these do not correspond to the north and south poles as defined in the projection above. Topics include complex numbers and their properties, analytic functions and the cauchyriemann equations, the logarithm and other elementary functions of a complex variable, integration of.
Visual complex analysis if you already know the basics of complex numbers but want to get. Stereographic projection there are two special projections. Essays on automorphic forms stereographic projection. 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. 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. Please practice handwashing and social distancing, and. Riemann sphere and the mapping is known as the stereographic projection. On the other hand, stereographic projection does not preserve area.
Stereographic projection of a complex number a onto a point. The stereographic projection is a bijective smooth map which allows us to think the sphere as the. Students should have a background in real analysis as in the course real analysis i, including the ability to write a simple proof in an analysis. The projection is defined on the entire sphere, except at one point. In text books of complex function analysis, this sphere is called the. In complex analysis it is used to represent the extended complex plane see for instance 2, chapter i. An everyday example of a projection is the casting of shadows onto. Theory of functions, parts i and ii dover books on. Theorem stereographic projection circles into circle on. The restriction to a subspace of a projection is also called a projection, even if the idempotence property is lost.
Richard silverman, the editor and translator of the original, has prepared this shorter version expressly to meet the needs of a oneyear graduate or undergraduate course in complex analysis. Introductory complex analysis is a scaleddown version of a. This page overlaps with the page here, i need to combine them. Complex analysis analytic function complex function limit continuity differentiability of a complex function. Complex analysis objectives of the course this is an introductory course in complex analysis, giving the basics of the theory along with applications, with an emphasis on applications of complex analysis and especially conformal mappings. It is clear that stereographic projection is continuous as a map s2\n.
All the points on s 2 are images of a twoparameter family of lines in e 3, called projecting lines, under the generalized stereographic projection. Introduction to complex analysis johns hopkins center for. Compare the angle between l1 and l2 with the angle of. If we identify, via stereographic projection, points in the complex plane with points in s2\n and further identify. 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. However, there is another projection, less known to students, a projection from a circle to the x. Pdf history of the stereographic projection and its. Mathematics subject classification 2010 eisbn 9781461401957 doi 10. Finding the stereographic projection for complex plane. Let us make some explorations with the priemannz, tool about the action of on concentric circles.
Context allow us to deduce if we go from the plane to the sphere or the other way around, at least as far as ive seen. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. It is conformal, meaning that it preserves angles at which curves meet. For many of our students, complex analysis is their. Math 215 complex analysis lenya ryzhik copy pasting from others november 25, 20 1 the holomorphic functions we begin with the description of complex numbers and. Mathematics subject classification 2010 eisbn 9781461401957. Where it is defined, the mapping is smooth and bijective. C can be represented as point x, y in the plane r2. A first course in complex analysis was written for a onesemester undergraduate course developed at binghamton university suny and san francisco state university, and has been adopted at several other institutions. Math 215 complex analysis lenya ryzhik copy pasting from others november 25, 20 1 the holomorphic functions we begin with the description of complex numbers and their basic algebraic properties.
The mercator and stereographic projections, and many in. The riemann sphere is a geometric representation of the extended complex plane the complex numbers with the added point at infinity. Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other see figures. Of course, we have seen that stereographic projection is conformal and it is a basic fact that complex analytic functions, such as the logarithm, are. In this edition, the numbers of certain exercises are preceded by an asterisk. Sometimes we will call c the open complex plane in order to stress the di. The riemann sphere as a stereographic projection wolfram. The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. As we shall see, maps of this kind are analogous to. Thinking in the plane as the complex plane, ill refer to its points as ordered pairs x,y. In mathematics, the riemann sphere, named after the 19th century mathematician.
Rationality of meromorphic functions on the riemann sphere. Chern, the fundamental objects of study in differential geometry are manifolds. Compare the angle between l1 and l2 with the angle of the arcs at n and the image z of z under the projection. Its applications are numerous and can be found in many other branches of mathematics, ranging from. The asterisk indicates that the exercise will be referred to later in. Here we will allow domains to include the infinity and look at the socalled conformal maps on such extended domains. To see that mercator is conformal, we will show that it is the composition of stereographic projection with the complex logarithm. The intersection of the line and the sphere is the corresponding projection. A stereographic projection can also be studied more generally. Markushevichs masterly threevolume theory of functions of a complex variable. The second part includes various more specialized topics as the argument principle, the schwarz lemma and hyperbolic. It can be verified that the hyperbolic projection maps all points on a projecting line to the same point on plane p 3 0.
Stereographic projection from the sphere to the plane is accomplished by drawing the secant line from the north pole through a point on the sphere and seeing where it intersects the. 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. A spherical projection of a complex hilbert space is conformal iff it is. Visual complex analysis if you already know the basics of complex numbers but want to get an in depth understanding using an geometric and intuitive approach then this is a very good book. Show that using these relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskew. Stereographic projection simple english wikipedia, the. Jul 23, 2014 jul 23, 2014 stereographic projection of riemann sphere. Stereographic projection part 1 introduction youtube. Stereographic projection encyclopedia of mathematics. Ii complex analysis sin hitotumatu encyclopedia of life support systems eolss complex analysis sin hitotumatu kyoto university, japan key words. Geometrically, we join a point z on the complex plane and the north pole of the riemann sphere with a straight line. Jan 01, 2019 complex analysis analytic function complex function limit continuity differentiability of a complex function. Singularity at infinity, infinity as a value, compact spaces of meromorphic functions for the spherical metric and spherical derivative, local analysis of n video course course outline this is the second part of a series of lectures on advanced topics in.
Introduction to complex analysis johns hopkins center. S2 \ n, let tp denote the tangent plane to s2 at p, and let tn denote the tangent plane to s2 at n. Stereographic projection an overview sciencedirect topics. Stereographic projection of riemann sphere college math. This elegant little book covers the elements of a senior or 1st year graduate course on complex analysis, although a really good mathematics program like at berkeley may look. Action of stereographic projection on the frontier, interior and exterior of the complex unit disc. 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. By the early 19th century, mathematics was beginning to be applied outside the fields of astronomy, physics and mechanics. The book explains how to represent complex transformations such as the mobius transformations. Below we will only discuss the properties of the generalized stereographic projection for a sphere. Stereographic projection gis wiki the gis encyclopedia.
The mapping z zis called the stereographic projection. Sep 12, 2018 introduction about stereographic projection. In complex number theory stereographic projection is. Please practice handwashing and social distancing, and check out our resources for adapting to these times. In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane.