A function fz is analytic if it has a complex derivative f0z. Chapter 3 exercise 21, chapter 8 exercises 5 and 15. This notion finds utility in projective geometry and complex analysis. Stereographic projection an overview sciencedirect topics. Especially, fhas only nitely many poles in the plane. Chern, the fundamental objects of study in differential geometry are manifolds. Stuart valhalla, new york, usa abstract mapping points on the sphere by stereographic projection to points on the plane of complex numbers transforms the spherical trigonometry calculations, performed in course of celestial navigation, into arithmetic operations on complex. Chapter 3 problems 1 and 2, chapter 8 exercises 16, 919. Browse other questions tagged complex analysis geometry surfaces or ask your own question. An important limitation of stereographic projections is that they consider only angular relationships between lines and planes, and do not represent the position or size of the feature. Stereographic projection part 1 introduction youtube. The mercator and stereographic projections, and many in.
The group of problems that are approached can usually be synthesized in the following way. The stereographic projection is the transformation of the complex plane into the sphere that satisfies the following. 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. Lecture notes mineralogy stereographic projections. In geometry, the stereographic projection is a particular mapping function that projects a. To perform the projection we connect points on the lower half of our great circle to the topmost point of the sphere or the zenith red lines in fig. This point of view certainly requires more work, but i feel the e.
In text books of complex function analysis, this sphere is called the. Ppt stereographic projection powerpoint presentation. In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. Stereographic projections of noncubic crystals two stereographic projections is required one for the surface normal poles and the other the directions. A circleshaped projection part of a circle then occurs on our horizontal projection plane, and this projection is a stereographic projection of the plane. Applications of complex analysis to celestial navigation. Lecture notes mineralogy stereographic projections the stereographic projection is a device use by mineralogists and structural geologists to represent 3dimensional information in two dimensions. One can also associate a point u,v,w on the unit sphere s u,v,w. The stereographic projection is the transformation of the complex plane into the sphere that. Stereographic projection the basics geological digressions. Under stereographic projection, the points inside the sphere have corresponding imaginary images on the plane. Complex variables for scientists and engineers description this course is an introduction to complex variables accessible to juniors and seniors in engineering, physics and mathematics. Pdf history of the stereographic projection and its. The generalized stereographic projection is a onetoone correspondence between points on a quadric and a twoparameter family of lines in e.
Mineralogists use a wulff stereonet, which is constructed from a simple geometric recipe. I have provided hints and solutions wherever i considered them necessary. 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. Since the riemann sphere is compact, fcan have only nitely many poles, for otherwise a sequence of poles would cluster somewhere, giving a nonisolated singularity. A central difficulty of complex analysis from nearly the very beginning has been the notion of. There are several good software programs and apps to automate projections for large data sets.
The stereographic projection and poincares compacti cations have been used to great advantage in rnin theory and. Stereographic projection maps circles to circles 5. I have provided hints and solutions wherever i considered. Prove that if a root of f0 belongs to the boundary of f, then either f has a repeated. Stereographic projection there are two special projections. Essays on automorphic forms stereographic projection. A spherical projection of a complex hilbert space is conformal iff it is. Mapping points on a sphere by stereographic projection to points on the plane of complex numbers transforms the spherical trigonometry calculations performed in the course of celestial navigation into arithmetic operations on complex numbers. Finally, in lecture 50, we give a brief history of complex numbers. What is the di erence between j j2j j2 1 and j j2 j j2 1. Stereographic projection is conformal let s2 x,y,z. Implementing stereographic projection in sage math 480 simon spicer je beorse kevin lindeman june 2, 2010 figure 1. Where it is defined, the mapping is smooth and bijective.
A spherical projection of a complex hilbert space is. 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. Identify the complex plane c with the x,yplane in r3. For the love of physics walter lewin may 16, 2011 duration. Special cases include the mercator map and the stereographic.
These are problems are meant to be used in a rst course on complex analysis. It can also be interpreted as the spatial analog of inversion. In this paper, we demonstrate how the elementary properties of the stereographic projection can be used for solving a geometric problem. However, a much richer set of conclusions can be drawn about a complex analytic function than is generally true about real di erentiable functions. 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.
A stereographic projection can also be studied more generally. Complex analysis automorphism groups lecture 3 proof. Two and threequbits geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection g. Sample questions, complex analysis fall 2006 chapter 1. Stereographic projection is a transformation in space, used frequently in complex analysis. A spatial problem solved with stereographic projection canadian. Two and threequbits geometry, quaternionic and octonionic. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Show that using these relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskew. Inversion in higherdimensional euclidean space in euclidean space rn of arbitrary dimension, inversion i with respect to the sphere of radius r centered at the origin is the map taking. The last part of the book is dedicated to geotechnical applications such as analysis of rock slope stability. A spherical projection of a complex hilbert space 39 even though these projections have been known for approximately two thousand years, new applications have been found in the previous century and in this new milenia. Stereographic projection encyclopedia of mathematics.
Mirzaei a z adepartment of physics, university of mohaghegh ardabili, p. Pdf stereographic projection techniques for geologists. I despise projections with a passion and i am not entirely sure. History of the stereographic projection and its early use in geology. The mercator and stereographic projections, and many in between daniel daners revised version, july 29, 2011 abstract we consider a family of conformal angle preserving projections of the sphere onto the plane. In general, the rules for computing derivatives will be familiar to you from single variable calculus. The intersection made by the line or plane with the spheres. Complex analysis has successfully maintained its place as the standard elementary text on functions of one complex variable. Finding the stereographic projection for complex plane. A spatial problem solved with stereographic projection 2000. Let the poles occur at the points z 1 through z n with. Stereographic projection from the sphere to the plane. For example, this projection sends the equator to the circle of radius 2 centered at the origin.
Improved solutions to such problems using a stereographic projection of. The family is referred to as the lambert conic conformal projections. Stereographic projection can be used in landslide hazardslope failure studies earthquake studies structural geological analysis. In complex number theory stereographic projection is. Applications of complex analysis to celestial navigation robin g. Browse other questions tagged complex analysis complex numbers or ask your own question. Problems in complex analysis these problems are not in any particular order. Introducing curves, paths and contours, contour integrals and their. 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. A stereographic projection, or more simply a stereonet, is a powerful method for displaying and manipulating the 3dimensional geometry of lines and planes davis and reynolds 1996. However, there is another projection, less known to students, a projection from a circle to the x.
Riemann sphere and the mapping is known as the stereographic projection. The projection is defined on the entire sphere, except at one point. S2 \ n, let tp denote the tangent plane to s2 at p, and let tn denote the tangent plane to s2 at n. Undeformed and deformed strain marker reference used for derivation of formulae. 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.
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