Sanuja Senanayake The stereographic projection is a methodology used in structural geology and engineering to analyze orientation of lines and planes with respect to each other. The stereonets is a type of standardized mapping system that allows us to represent various angles in 3D space on a 1D paper. They are used for analysis of various field data such as bedding attitudes, planes, hinge lines and numerous other structures. This is a very useful tool because it can reduce the workload by avoiding lengthy calculations. In structural geology, we use the bottom half or hemisphere of the spherical projection.

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Equiangular Fig. Stereographic projection of an ESE-dipping plane. Stereographic projection is about representing planar and linear features in a two-dimensional diagram.

The orientation of a plane is represented by imagining the plane to pass through the centre of a sphere Fig. The line of intersection between the plane and the sphere will then represent a circle, and this circle is formally known as a great circle.

We would like to project the plane onto the horizontal plane that runs through the centre of the sphere. Hence, this plane will be our projection plane, and it will intersect the sphere along a horizontal circle called the primitive circle.

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. A circleshaped projection part of a circle then occurs on our horizontal projection plane, and this projection is a stereographic projection of the plane. If the plane is horizontal it will coincide with the primitive circle, and if vertical it will be represented by a straight line.

Once we understand how the stereographic projection of a plane is done it also becomes obvious how lines are projected, because a line is just a subset of a plane. Lines thus project as points, while planes project as great circles. A great circle as any circle can be considered to consist of points, each of which represents a line withinthe plane. Hence, a line contained in a plane, such as a slickenline or mineral lineation, will therefore appear as a point on the great circle corresponding to that plane.

Stereographic projection of a line, in this case the normal or pole to a plane. In Fig. The projection is found by orienting the line through the center and connecting its intersection with the lower hemisphere with the zenith red line in Fig. The intersection of this red line with the projection plane is the pole to the plane. Hence, planes can be represented in two ways, as great circle projections and as poles.

Note that horizontal lines plot along the primitive circle completely horizontal poles are represented by two opposite symbols and vertical lines plot in the centre. The stereonet. To compare this with longitude and latitudes of our planet, remember that the north—south axis is horizontal in the stereonet, and vertical in common views of the Earth.

For stereographic projections to be practical, we have to establish a grid of known surfaces for reference. We have already equipped the primitive circle with geographical directions north, south, east and west , and we can compare the sphere with a globe with longitudes and latitudes.

In three dimensions this is illustrated in Fig. If we now project the small and great circles onto the horizontal projection plane, typically for every 2 and 10 degree interval, we will get what is called a stereographic net or stereonet.

The longitudes are planes that intersect in a common line the N—S line , and thus appear as great circles in the stereonet. The projections of the latitudes, which are not planes but cones coaxial with the N—S line, are usually referred to as small circles also their projections onto the stereonet.

The net that emerges from the particular projection described above is called the Wulff net. Equal area projection Fig. The equal area net. The equal area projection. A plane is projected onto the projection plane, which in this case has been made tangential to the lower pole of the sphere.

Two hundred and fifty quartz c-axes measured on the U-stage and plotted in the stereonet. An asymmetrical pattern with respect to the foliation trending E—W in the plot , such as the one shown here, indicates the sense of shear. The Wulff net makes it possible to work with angular relations it preserves angles between planes across the net , which can be useful in some cases, for instance for crystallographic purposes.

However, for most structural purposes it is more useful to preserve area, so that the densities of projections in one part of the plot can be directly compared to those of another.

The method of plotting is the same, but because the projection is not stereographic but equal area Fig. The net is called a Schmidt net or simply an equal area net Fig. Multiple data plotted in an equal area net can be contoured with respect to density, which can be useful when evaluating concentrations of structural data around certain geographic directions. Contouring is typically done for crystallographic axes such as quartz c-axes Fig.

Contouring is easily done by means of one of the many computer programs available for personal computers. Plotting planes Planes can be represented in a stereonet in two different ways: by means of great circles or poles Fig 2. Plotting the plane N E, 30 NE in the stereonet equal area projection. A plane striking or N 30 E and dipping 30 to the SE is plotted as an example. Tracing paper is placed over a pre-made net an equal area net was chosen , and the centers are attached by means of a thumb tack.

The primitive circle and north N are marked on the transparent overlay. We then mark off the strike value of our plane, which is Fig. For our example, this involves rotating the overlay 30 anticlockwise. We then count the dip value from the primitive circle inwards, and trace the great circle that it falls on Fig.

When N on the tracing paper is rotated back to its original orientation Fig. The shallower its dip, the closer it comes to the primitive circle, which itself represents a horizontal plane. The procedure is quite similar if we want to plot poles. All we do differently is to count the dip from the center of the plot in the direction opposite to that of the dip, which in our example is to the left.

When done correctly, there will be 90 between the great circle and the pole of the same plane see Fig. The pole thus falls on the opposite side of the diagram from that of the corresponding great circle.

Poles are generally preferred in structural analyses that involve large amounts of orientation data, and particularly if grouping of structural orientations is an issue which commonly is the case. Plotting lines Fig. Plotting the line plunging 40 degrees towards N E in the stereonet equal area projection. Plotting a line orientation is similar but different to plotting a plane orientation.

For example, a line plunging 40 degrees toward NE is considered. As for the plane, we mark off the trend Fig. Now count the plunge value along the straight line toward the center, starting at the primitive circle, and mark off the pole Fig.

Back-rotate the overlay, and the task is completed Fig. Pitch rake Fig. A constructed, but realistic, situation where various structural elements in a deformed rock sequence are represented in stereonets.

Plotting bedding orientations reveals the b-axis local orientation of the hinge line. Fault data are plotted separately, showing the fault plane as a great circle and lineations as dots on that great circle.

When doing fault analyses it is useful to plot both the slip plane and its lineation s in the same plot. In this case the lineation will lie on the great circle that is representing the slip plane. The angle between the horizontal direction and the lineation is called the rake or pitch, and is plotted by rotating the great circle of the plane to a N—S orientation and then counting the number of degrees from the horizontal N or S , i.

Users of the right hand rule will always measure the pitch clockwise from the strike value, so that the angle could be up to degrees.

The right-hand rule has been used in Fig. Others measure the acute angle and count from the appropriate strike direction, in which case the pitch will not exceed 90 degrees. Fitting a plane to lines If two or more lines are known to lie in a common plane, the plane is found by plotting the lines in a net. The lines are then rotated until they fall on a common great circle, which represents the plane we are looking for. Line of intersection The line of intersection between two planes is perhaps most easily seen by plotting the great circles of the two planes, in which case the line of intersection is represented by the point where the two great circles cross.

Angle between planes and lines The angle between two planes is found by plotting the planes as poles and then rotating the tracing paper until the two points fall on a great circle. The angle between the planes is then found by counting the degrees between the two points on the great circle Fig.

Orientation from apparent dips Finding the orientation of a planar structure from observations of apparent dips can sometimes be useful. An inverse problem would be to determine apparent dips of a known planar fabric or structure as exposed on selected surfaces. Rotation of planes and lines Rotation of planar and linear structures can be done by moving them along a great circle, the pole of which represents the rotation axis.

Rotation about a horizontal axis is easy: just rotate the tracing paper so that the rotation axis falls along the N—S direction, and then rotate poles by counting degrees along the small circles. Rotating along an inclined axis is a bit more cumbersome. Rose diagram Fig. The plots show the variations within each subarea, portrayed by means of poles, rose diagrams, and an arrow indicating the average orientation. Sometimes only the strike component of planes is measurable or of interest, in which case the data can be represented in the form of a rose diagram.

A rose diagram is the principal circle subdivided into sectors, where the number of measurements recorded within each sector is represented by the length of the respective petal. This is a visually attractive way of representing the orientation of fractures and lineaments as they appear on the surface of the Earth, and can also be used to represent the trend distribution of linear structures Fig.

Plotting programs All of these operations and more can be done more quickly by means of stereographic plotting programs, such as the one generously made available to the structural community by Richard Allmendinger However, understanding the underlying principles is the key to success when using such programs. Several plotting programs also have statistical add-ons that are quite useful. Credits: Haakon Fossen Structural Geology.

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## Stereographic projection

However, for graphing by hand these formulas are unwieldy. Instead, it is common to use graph paper designed specifically for the task. This special graph paper is called a stereonet or Wulff net, after the Russian mineralogist George Yuri Viktorovich Wulff. In the figure, the area-distorting property of the stereographic projection can be seen by comparing a grid sector near the center of the net with one at the far right or left. The two sectors have equal areas on the sphere. On the disk, the latter has nearly four times the area of the former.

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## Stereographic projection for structural analysis

Equiangular Fig. Stereographic projection of an ESE-dipping plane. Stereographic projection is about representing planar and linear features in a two-dimensional diagram. The orientation of a plane is represented by imagining the plane to pass through the centre of a sphere Fig. The line of intersection between the plane and the sphere will then represent a circle, and this circle is formally known as a great circle.

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