The mercator and stereographic projections, and many in. Lecture notes mineralogy stereographic projections. In stereographic projection crystal directions are projected. 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. Lecture notes mineralogy stereographic projections the stereographic projection is a device use by mineralogists and structural geologists to represent 3dimensional information in two dimensions. 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. 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. This notion finds utility in projective geometry and complex analysis. Stereographic projection techniques for geologists and civil. Mathematics subject classification 2010 eisbn 9781461401957. Stereographic projection the basics geological digressions. In the rest of the book, the calculus of complex numbers will be built on the propeties that.
To see this, one must recall that the definition of the pre. Pdf history of the stereographic projection and its. The last part of the book is dedicated to geotechnical applications such as analysis of rock slope stability. In order to fully appreciate this book, the reader should have a basic understanding of elementary mathematics especially complex numbers and basic set theory. Stereographic projection techniques for geologists and civil engineers. Stereographic projection an overview sciencedirect topics. Geometrically, it is clear that stereographic projection sends lines to circles. In numerous research fields, especially the applications of electron and xray diffraction, stereographic projection represents a powerful tool for researchers. In geometry, the stereographic projection is a particular mapping function that projects a. See many books on conformal mapping theory or analytic function.
The term planisphere is still used to refer to such charts. This book provides an introduction to complex analysis for students with. Complex analysiscomplex numbers wikibooks, open books. Sometimes we will call c the open complex plane in order to stress the di. Pdf history of the stereographic projection and its early. 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. In text books of complex function analysis, this sphere is called the. Stereographic projection part 1 introduction youtube. Finding the stereographic projection for complex plane. The use of stereographic projection in structural geology paperback 1971. Matthias beck gerald marchesi dennis pixton lucas sabalka. I did a problem about finding the stereographic projection of a cube inscribed inside the riemann sphere with edges parallel to the coordinate axes.
Find all the books, read about the author, and more. It was originally known as the planisphere projection. In geometry, the stereographic projection is a particular mapping function that projects a sphere onto a plane. In complex number theory stereographic projection is. Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other see figures. In 1881, poincare proposed a di erent mapping where the xed projection point coincides with. Once an appreciation is gained of the nature of the problem, the formal solution using the projection becomes both logical and straightforward.
In order to make plotting of the stereographic projection easier, a device called a stereographic net or stereonet is used. Stereographic projection is a projection of points from the surface of a sphere onto its equatorial. Essays on automorphic forms stereographic projection. 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. A central difficulty of complex analysis from nearly the very beginning has been the notion of. Stereographic projection in crystallography is a helpful and illustrative tool when investigating atomic planes or directions and visualizing various orientation dependent phenomena. Stereographic projection simple english wikipedia, the. The second part includes various more specialized topics as the argument principle, the schwarz lemma and hyperbolic. 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. 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. Pdf stereographic projection techniques for geologists. Department of materials science and metallurgy, university of cambridge, uk. It is conformal, meaning that it preserves angles at which curves meet. I also have learned a great amount from reinhold remmerts complex function theory r91.
The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. In the rest of the book, the calculus of complex numbers will be built on the properties that we develop in. Complex analysis has successfully maintained its place as the standard elementary text on functions of one complex variable. Wulff net the wulff net, or stereographic net, is a stereographic. On the other hand, stereographic projection does not preserve area. Where it is defined, the mapping is smooth and bijective. 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. There are various ways to implement stereographic projections in computer programs. The stereographic projection is a conformal map as well. 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. The history and insight of that book will bring me to say a few dozen things this semester, its a joy to read.
The mercator and stereographic projections, and many in between. The use of stereographic projection in structural geology. In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The riemann sphere and stereographic projection the initial and naive idea of the extended complex plane is that one adjoins to the complex plane ca new point, called 1 and decrees that a sequence znof complex numbers converges to 1if and only if the. It contains 115 pages of wellpresented, clearly explained, generally wellillustrated text in short it is userfriendly. Issn 01726056 isbn 9781441972873 eisbn 9781441972880 doi 10. In complex analysis it is used to represent the extended complex plane see for instance 2, chapter i. When i read this book, it brought to me pleasure and sorrow. Fundamentals of complex analysis with applications to engineering and science, e. Markku tilli, atte haapalinna, in handbook of silicon based mems materials and technologies second edition, 2015. The stereographic projection is the transformation of the complex plane into the sphere that satisfies the following.
Plane representation in the c 0 plane or in a plane to c 0 z p p inversion center joint p to z projection p intersection of pz with c0. Stereographic projection maps circles of the unit sphere, which contain the north pole, to euclidean straight lines in the complex plane. Mineralogists use a wulff stereonet, which is constructed from a simple geometric recipe. For many of our students, complex analysis is their rst rigorous analysis if not mathematics class they take, and these notes re ect this very much.
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. However, certain conventions will be used throughout the text, so an introduction is warranted. The mapping z zis called the stereographic projection. One of its most important uses was the representation of celestial charts. Riemann sphere and the mapping is known as the stereographic projection.
Intuitively, the stereographic projection is a way of picturing a sphere as a plane, with some inevitable compromises. History of the stereographic projection and its early use in geology. As more manipulations involving complex numbers were tried, it became apparent that many problems in the theory of realvalued functions could be most easily solved using complex numbers and functions. Applications of complex analysis to celestial navigation. The textbook pages are mentioned as a reference and as a reading guide. 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. History of the stereographic projection and its early use in geology article pdf available in terra nova 86. This is the book that all teachers of stereographic projections have been waiting for. The stereographic projection was known to hipparchus, ptolemy and probably earlier to the egyptians. The point m is called stereographic projection of the complex number z on. Chern, the fundamental objects of study in differential geometry are manifolds.
Depending on the nature of your course and its place in your departments overall. These are the lecture notes of a onesemester undergraduate course which we have taught several times at binghamton university suny and san francisco state university. 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. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. Stereographic projection from the sphere to the plane. Planisphaerium by ptolemy is the oldest surviving document that describes it. To this end, consider the stereographic projection from the unit sphere minus the point 0, 0, 1 onto the plane z 0, which we identify with the complex plane by. Here is a tentative guideline of the course structure, arranged by week. Stereographic projection simple english wikipedia, the free. The material in this book should be more than enough for a typical semesterlong undergraduate course in complex analysis. In text books of complex function analysis, this sphere is called the riemann sphere and the mapping is known as the stereographic projection. The stereographic projection is the transformation of the complex plane into the sphere that. The stereographic projection allows the three dimensional orientation data to be represented and analysed in two dimensions. Jul 17, 2003 the book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school.
The intersection made by the line or plane with the spheres. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection. Spica is a new computer program for stereographic projection in interactive crystallographic analysis, which inherits features from the previous jecpsp and includes more functions for extensive crystallographic analysis. The riemann sphere and stereographic projection the initial and naive idea of the extended complex plane is that one adjoins to the complex plane ca new point, called 1 and decrees that a sequence znof complex numbers converges to 1if and only if the real sequence jznjtends to 1in the usual sense. Here, we would like to refer to some mysterious properties of the point at in. Stereographic projection techniques for geologists and civil engineers 2nd edition by richard j. Stereographic projection for interactive crystallographic. We will assume that the reader had some previous encounters with the complex numbers. For crystallographic analysis in electron diffraction experiments, stereographic. This book sets out to provide a simple introduction to the subject by means of illustrations and exercises, encouraging the student to visualise the problems concerned in three dimensions. One can also associate a point u,v,w on the unit sphere s u,v,w. The point at infinity, stereographic projection, division by zero, division by.
Thinking in the plane as the complex plane, ill refer to its points as ordered pairs x,y. It is also used for estimation of cut slope angle, statistical analysis of joints, for the preparation of hazard maps and estimation of safety factors. Show that using these relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskew. Pdf the origin of the polar and equatorial stereographic projections can be traced back to hipparchos d. The gure below shows a sphere whose equator is the unit circle in the complex plane. Stereographic projection techniques for geologists and. In the 18th century, functions involving complex numberswere found by euler to yield solutions to differential equations. Buy the use of stereographic projection in structural geology on free shipping on qualified orders. S2 \ n, let tp denote the tangent plane to s2 at p, and let tn denote the tangent plane to s2 at n. There are several good software programs and apps to automate projections for large data sets.
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