c) Equal-area stereonets are used in structural geology because they present b ) The north pole of the stereonet is the upper point where all lines of longitude. Background information on the use of stereonets in structural analysis The above is an equal area stereonet projection showing great circles as arcuate lines. Page 1. mm. WIDTH. Blunt. TUT. HT. T itillinn.

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Differential geometry and applications. Circles on the sphere that do pass through the point of projection are projected to straight lines on the plane. You can do this by simply rotating the point representing the line on to any great circle, and then count along that great circle 20 degrees in both directions and mark those points which will be two lines 20 degrees either side of the first. This circle maps to a circle under stereographic projection.

Stereographic projection is also applied to the visualization of polytopes.

The open and filled red stars represent two lines solid 58, open 37 and the dashed red great circle represents their common plane with a strike of SE. Stereographic projection plots can be carried out by a computer using the explicit formulas given above. Model Kikuchi maps in reciprocal space, [12] and fringe visibility maps for use with bend contours in direct space, [13] thus act as road maps for exploring orientation space with crystals in the transmission electron microscope.

The numbers in the upper right quadrant represent potential strike line positions from degree, in 10 degree arew see below diagram.

In other words, they provide the best projection for analyzing the direction and the vectors of structural forces. Repeat this on another nearby great circle. The above is an equal area stereonet projection eqqual great circles as arcuate lines connecting the North and South Points and small circles as arcuate lines in a latitudinal type position.

What is the form that results? Once these angles are known, there are four steps to plotting P:.

### Stereographic projection – Wikipedia

That is, crystal axes and poles to crystal planes are intersected with the northern hemisphere and then plotted using stereographic projection. In other words, it is often used to analyze accuracy of data from several different regions of the same area.

If you have understand how 3D vectors work, this should be a no-brainer. The two sectors have equal areas on the sphere.

The loxodromes of the sphere map to curves on the plane of the form. The pair of trigonometric functions sin xcos x can be thought of as parametrizing the unit circle.

## Stereographic projection

However, the angle-preserving property is stronger than this property. The green represents the plane’s orientation when North is rotated back to its standard top-of-the-stereonet position. A circle on the surface of a sphere tsereonet by the intersections of a plane that does not pass through the center of the sphere.

No map from the sphere to the plane can be both conformal and area-preserving. Hence, most educational institutions prefer equal area steronets for their students over the equal angle stereonets. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereographic netshortened to stereonetor Wulff net. Typically university geology and engineering students are expected create stereonets by hand. This is the basic 3D geometry we will start with.

This page was last edited on 21 Decemberat Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. See also Latitude Longitude Tissot’s indicatrix. Equal angle projections preserve angles, which can be seen by all small circles being perfect circles. However, stereographic fisheye lenses are typically more expensive to manufacture. The stereographic projection gives an alternative parametrization of the unit circle:.

They are hemisphere surface paths from one line being rotated about another line the pole of rotationboth passing through the hemisphere center.

So any set of lines through the origin can be pictured, almost perfectly, as a set of points in a disk. These lines are sometimes thought of as circles through the point at infinity, or circles of infinite radius.

Planar Gnomonic Orthographic Stereographic.

## Stereographic projection for structural analysis

Some structural elements whose orientations can be plotted on a stereonet are: Some fisheye lenses use a stereographic projection to capture a wide-angle view. That great circle is the bisecting plane.

Views Read Edit View history. From Wikipedia, the free encyclopedia. The projection used for this kind of plots is the Stereographic Projection with equatorial aspect See: Intersecting lines are transformed to circles that intersect transversally at two points in the sphere, one of which is the projection point. Angles are slightly distorted and make the circles appear as ellipses.

A circle on the surface of the sphere made by the intersection with the spehere of a plane that passes through the center of the sphere. These orientations can be visualized as in the section Visualization of lines and planes above. Parallel lines, which do not intersect in the plane, are transformed to circles tangent at projection point.

### Equal Area (Schmidt) Stereonet

For an example of the use of the Wulff net, imagine two copies of it on thin paper, one atop the other, aligned and tacked at their mutual center.

The stereographic projection has been used to map spherical panoramas. Then measure the angle between them by counting grid lines along that meridian. In this case the formulae become. You will need the following materials in order to proceed with this and the subsequent stereonet exercises: It is helpful to understand the 3-D geometry that is being represented on the 2-D stereonet projection plane.