In hana kitasei and tim reckarts animated video, summer vacation, angela says that stereographic projection preserves angles. The overflow blog socializing with coworkers while social distancing. In complex analysis it is used to represent the extended complex plane see for instance 2, chapter i. The circles are cut from the sphere by two planes, red and green respectively, through the lines in the zplane and through. Stereographic projection is conformal let s2 x,y,z. Show that using these relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskew.
Polar stereographic projection and grid national snow. First, lets recall what stereographic projection is. Set of complex no, stereographic projection youtube. Oct 11, 2011 in hana kitasei and tim reckarts animated video, summer vacation, angela says that stereographic projection preserves angles. However, there is another projection, less known to students, a projection from a circle to the x. Set of complex no, stereographic projection ch18 mathematics, physics, metallurgy subjects. It is necessary to emphasize, however, that all constructions on the stereographic projection can be performed by simple methods of vector analysis. The mercator and stereographic projections, and many in between.
Planar perspective projection, where one pole is viewed from the other pole. The operation of stereographic projection is depicted in fig. Stereographic polar zenithal projection practical excercise 2. 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. Like the utm coordinate system, the ups coordinate system uses a metricbased cartesian grid laid out on a conformally projected surface. Here we will allow domains to include the infinity and look at the socalled conformal maps on such extended domains. It is conformal, being free from angular distortion. Mineralogists use a wulff stereonet, which is constructed from a simple geometric recipe. A light source at the north pole projects the continents onto a flat screen tangent to the. This planar grid is designed so that the grid cells at 70 degrees latitude are exactly the nominal grid resolution. Stereographic projection can be used to measure the angle between any two lines.
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. We will assume that the reader had some previous encounters with the complex numbers. Applications of complex analysis to celestial navigation. The universal polar stereographic ups coordinate system is used in conjunction with the universal transverse mercator utm coordinate system to locate positions on the surface of the earth. The stereographic projection is the transformation of the complex plane into the sphere that satisfies the following. The most obvious symmetry element is the fourfold rotational symmetry. Pdf history of the stereographic projection and its. The stereographic projection is an essential tool of the structural geologist which allows threedimensional orientation data to be both represented and manipulated. Please note the image in this listing is a stock photo and may not match the covers of the actual item. Zj is constant, but az and az make equal angles with nz, so the triangle azp is isoceles and ja. One of its most important uses was the representation of celestial charts. The polar stereographic map projection, as defined by isoiec 18026.
Bpass a plane parallel to the original one through the line pp. Stereographic projection is a technique for displaying the angular properties of a plane faced object on a single drawing or diagram. August 20, 2009 6 c we want to show that the section of the cone by this plane is a circle. Stereographic polar zenithal projection draw a stereographic polar zenithal projection for northern hemisphere on a scale 1. This defines a boundary around the stereographic projection. Math 215 complex analysis lenya ryzhik copy pasting from others november 25, 20. Polar stereographic map projection computation of from k. Aug 20, 2012 stereographic polar zenithal projection 1. The polar stereographic map projection, as defined by isoiec 18026 and other places, allows a factor to adjust the scale of the map projection. Describe a conformal map projection that transforms circles on a globe sphere into circles of the map. It was originally known as the planisphere projection. 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. In any book on differentiable manifolds, the stereographic projection map p from the nsphere to the n1plane is discussed as part of an example of how one might cover a sphere with an atlas. Projected points may fall inside or outside of the primitive circle, depending on which pole is used as a projection point, as shown here.
Polar stereographic projection and grid national snow and. This is a circle on the projection plane, which is located where the sphere of projection intersects the projection plane. Readers will realize that the situation with the south polar stereographic projection is symmetric. Threedimensional backanalysis of saturated rock slopes in. Angela uses stereographic projection to make a map of the earth. Doitpoms tlp library the stereographic projection basic. The stereographic projection was known to hipparchus, ptolemy and probably earlier to the egyptians. Finding the stereographic projection for complex plane. Stereographic projection an overview sciencedirect topics. This is usually followed by a comment such as it is obvious or it can be shown that the inverse projection p1 is given by such and such. Other articles where stereographic projection is discussed. Compare the angle between l1 and l2 with the angle of the arcs at n and the image z of z under the projection.
The stereographic projection is a projection of points from the surface of a sphere on to its equatorial plane. The polar stereographic projection is used for all regions not included in the utm coordinate system, regions north of 84 n and south of 80 s. Identify the complex plane c with the x,yplane in r3. This is an exlibrary book and may have the usual libraryusedbook markings inside. Riemann sphere is a plane, but it has the word sphere in it because the essence of this plane is really a sphere, as you see in the videos. The stereographic projection is a conformal map as well. It provides a way of graphically displaying the data collected. Stereographic projection, the riemann sphere, and the. A sphere of unit diameter is tangent to the complex plane at its south pole. In this context the stereographic projection is often referred to as the equalangle lowerhemisphere projection. 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.
The center point is a pole in the common polar aspect, but can be any point. 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. In this paper we give a criterion for a compact minimal submanifold of sm to lie in a given great hypersphere in terms of an integral over the stereographic image in em of the submanifold. Q869 polar stereographic calculation help 11012007, 16. Planisphaerium by ptolemy is the oldest surviving document that describes it.
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. In all likelihood the source of this problem is the astronomer. Directions as well as planes may be shown and any desired angle can be measured directly from the projection using a graphical technique. For more on stereographic projection, see any standard complex analysis text. The term planisphere is still used to refer to such charts. 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 intersection of the line and the sphere is the corresponding projection. Pdf history of the stereographic projection and its early. The use of stereographic projection for describing the. Math 215 complex analysis lenya ryzhik copy pasting from others. In the previous lecture we introduced functions holomorphic on domains in \ \mathbbc \. Stereographic projection gis wiki the gis encyclopedia. Pick a circle on s not containing n and let a be the vertex of the cone tangent to s at this circle fig.
The use of stereographic projection in structural geology. 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 projection we are looking for, which is called a stereographic or polar projection is very important in cartography. This number is called the central scale in the above iso document, and is notated k 0 there. Now, this idealized plane, with a point at infinity in the stereographic projection sense, is called riemann sphere. Stereographic projection there are two special projections. This is essential for the recognition and interpretation of patterns of preferred orientation. Polar stereographic projection and grid nsidcs polar stereographic projection specifies a projection plane or grid tangent to the earths surface at 70 degrees northern and southern latitude. Geometrically, we join a point z on the complex plane and the north pole of the riemann sphere with a straight line. Essays on automorphic forms stereographic projection. The mercator and stereographic projections, and many in. The equatorial projection became an essential tool. One can also associate a point u,v,w on the unit sphere s u,v,w. Universal polar stereographic coordinate system wikipedia.
The intersection made by the line or plane with the spheres. When it is 1, the local scale of the map projection at the north pole is 1. The origin of the polar and equatorial stereographic projections can be traced back to hipparchos d. 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. Browse other questions tagged complexanalysis or ask your own question. Quantarctica is a unique visualization and analysis environment for atmospheric data, allowing users to show observation and model data with fully customizable color ramps, vector arrows, and more features provided by qgis. Here we discuss the method used in crystallography, but it is similar to the method used in structural geology. Therefore stereographic projection maps all points on the sphere except. Stereographic projection, the riemann sphere, and the chordal. The explanations to follow will refer to the stereographic projection of the north polar regions. In drawing a map of a sphere, therefore, some compromises must be made. As we shall see, maps of this kind are analogous to onetoone holomorphic functions.
The equatorial projection became an essential tool in astronomy and from the 2nd to. Inverse of the stereographic projection physics forums. Threedimensional backanalysis of saturated rock slopes. 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. The original data are gridded on a geographic latlon grid, so when they are projected to a polar stereographic projection, this creates problems in gridding straight to 5 km resolution, due to the directionality of the points used in the interpolation procedure. While we have seen the basis of projection for a very simple system, we may now more closely examine the production of a stereogram another word for stereographic projection, by producing the stereogram for a cube. This is a perspective projection on a plane tangent at the center point from the point antipodal to the center point. Stereographic projection maps circles of the unit sphere, which contain the north pole, to euclidean straight lines in the complex plane. Incorporation of stereographic projection techniques provides a fully threedimensional graphical presentation of the problem and helps one visualize complex threedimensional entities. Lecture notes mineralogy stereographic projections. 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 a method used in crystallography and structural geology to depict the angular relationships between crystal faces and geologic structures, respectively. Q869 polar stereographic calculation help atp forum. Stereographic polar zenithal projection linkedin slideshare.
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