Finding the stereographic projection for complex plane. The intersection of the line and the sphere is the corresponding projection. Complex function visualization using riemann surfaces kandy. Identify the complex plane c with the x,yplane in r3. Where it is defined, the mapping is smooth and bijective. Usually one cares about either the area or the angle. The line of intersection between the plane and the sphere will then represent. In the equatorial aspect, the parallels curve in opposite directions on either side of the equator. You find here a proof that a lft induces, through stereographic projection, a rotation of the riemann sphere if and only if it can be represented by a unitary two by two coefficient matrix. Special cases include the mercator map and the stereographic. 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.
Stereographic projection in geometry, the stereographic projection is a particular mapping function that projects a sphere onto a 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. Stereographic projection encyclopedia of mathematics. I just started self learning complex analysis from free textbook on matthias becks san francisco state universitys website like a week ago. 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. Complex analysis and conformal mapping math user home pages. It will be nice to have some useful way to put images related with stereographic projections in our documents. Conformal powers of the laplacian via stereographic projection.
This construction has special significance in complex analysis. For the love of physics walter lewin may 16, 2011 duration. Stereographic projection can be used in landslide hazardslope failure studies earthquake studies structural geological analysis. This appears to be the first time that such a use of complex variables was made 30. The latter is called the sphere of complex numbers or the riemann sphere. Stereographic projection in crystallography is a helpful and illustrative tool when investigating atomic planes or directions and visualizing various orientation dependent phenomena. Stereographic projection of crystal faces crystallographic angles first we define the interfacial angle between two crystal faces as the angle between lines that are perpendicular to the faces. A stereographic projection can also be studied more generally. Although any stereographic projection misses one point on. If any point p on the surface of the sphere is joined to the south pole s and the line ps cuts the equatorial plane at p, then p is the stereographic projection of p.
On the advancements of conformal transformations and their. Stereographic projection techniques for geologists and. The family is referred to as the lambert conic conformal projections. 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. It is convenient to introduce the following notations operating on subsets. Under stereographic projection, the points inside the sphere have corresponding imaginary images on the plane. Stereographic projection is conformal let s2 x,y,z. In general, it is not possible to map a portion of the sphere into the plane without introducing some distortion. It can be verified that the hyperbolic projection maps all points on a projecting line to the same point on plane p 3 0. The mercator and stereographic projections, and many in. Pdf stereographic projection techniques for geologists. We begin with the description of complex numbers and their basic algebraic properties. 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.
The projection is defined on the entire sphere, except at one point. Complex analysis is the branch of mathematical analysis that investigates func. Therefore stereographic projection maps all points on the sphere except. This book has been designed to make the subject as accessible as possible. Stereographic projection wikimili, the best wikipedia reader. The orientation of a plane is represented by imagining the plane to pass through the centre of a sphere fig. The definition of stereographic projection, which is used in. In mathematics, the riemann sphere, named after bernhard riemann, 1 is a model of the extended complex plane, the complex. Notes to supplement the material on linear fractional maps presented in a first year graduate course. It is clear that stereographic projection is continuous as a map s2\n.
Thinking in the plane as the complex plane, ill refer to its points as ordered pairs x,y. We consider a family of nonlinear projections that map a com. Greens function for the laplacebeltrami operator on a toroidal surface first explicit representation of greens function g t. The stereographic projection is a projection of points from the surface of a sphere on to its equatorial plane. I did a problem about finding the stereographic projection of a cube inscribed inside the riemann sphere with edges parallel to the coordinate axes. Essays on automorphic forms stereographic projection.
Glimpses of algebra and geometry utm, springerverlag, 2002 geometry in 20112012. Stereographic projection of the bloch sphere onto the extended. Complex analysis is the culmination of a deep and farranging study of the funda. Applications within mathematics complex analysis the complex plane and the riemann sphere above it. All meridians and parallels are shown as circular arcs or straight lines. Stereographic projection the basics geological digressions. Stereographic projection is about representing planar and linear features in a twodimensional diagram. Such a lines are called the poles to the crystal face.
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. In stereographic projection crystal directions are projected onto a plane. In text books of complex function analysis, this sphere is called the. For more on stereographic projection, see any standard complex analysis text. Stereographic projection part 1 introduction youtube. The point m is called stereographic projection of the complex number z on the sphere s. Implementing stereographic projection in sage math 480 simon spicer je beorse kevin lindeman june 2, 2010 figure 1. 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. One can also associate a point u,v,w on the unit sphere s u,v,w. This projection is defined on the whole sphere, except at the projection point. Stereographic projection definition of stereographic. The mapping z z is called the stereographic projection.
A spherical projection of a complex hilbert space is conformal i it is the stereographic projection harry gingold, yotam gingold and salah hamad abstract. Stereographic projects points on a spheroid directly to the plane. This pdf is an adaption and extension of the original by andre nachbin and jeremy. 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. Stereographic projection techniques for geologists and engineers second edition article pdf available in mathematical geology 374. Thus, all point on a projecting line are mapped to the same point on s 2. Stereographic projection an overview sciencedirect topics. In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. 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. We also develop the cauchyriemannequations, which provide an easier test to verify the analyticity of a function. Mod01 lec01 analytic functions of a complex variable part i duration. Supplementary notes for a firstyear graduate course in complex analysis.
Geometrically, we join a point z on the complex plane and the north pole of the riemann sphere with a straight line. The stereographic projection is a mapping that projects the sphere on to a plane. Complex analysis is a basic tool in many mathematical theories. The problem of automatic generation of the riemann surfaces with manual addition of. There are several good software programs and apps to automate projections for large data sets. The stereographic projection is an essential tool in the fields of structural geology and geotechnics, which allows threedimensional orientation data to be represented and manipulated. A spherical projection of a complex hilbert space is. Show that using these relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskew. Greens function for the laplacebeltrami operator on a.
709 596 865 1637 1291 1168 910 1407 1643 214 1481 200 658 1639 648 334 537 944 600 1478 633 1327 255 967 819 742 62 880 980 112 608 449 1396 446 209 1187 254 1185 935 337 964 779 1093 1413 1487 1178 950 565