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. 1a). 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. Except for the ﬁeld of crystallography, where upper-hemisphere projection is used, geologists use the lower part of the hemisphere for stereographic projections, as shown in Fig. 1b. 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. 1c). 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. Stereographic projections of planes are formally called cyclographic traces, but are almost always referred to as great circles because of their close connection with great circles as deﬁned above.
Once we understand how the stereographic projection of a plane is finished it also will become obvious how lines are projected, because a line is only a subset of a plane. Lines for that reason mission as points, even as planes assignment as super circles. A exceptional circle (as any circle) can be considered to include factors, each of which represents a line withinthe plane. Hence, a line contained in a plane, inclusive of a slickenline or mineral lineation, will therefore seem as a factor on the amazing circle corresponding to that plane.
In Fig. 2 we have additionally projected the road this is normal to a given plane, represented by way of the pole to the aircraft. The projection is determined by means of orienting the line thru the middle and connecting its intersection with the decrease hemisphere with the zenith (pink line in Fig. 2a). The intersection of this (red) line with the projection plane is the pole to the aircraft. Hence, planes may be represented in two approaches, as notable circle projections and as poles. Note that horizontal strains plot along the primitive circle (completely horizontal poles are represented by means of opposite symbols) and vertical lines plot in the centre.
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. 3, looking from the south pole toward the north pole: longitudes and latitudes are the lines of intersection between great circles (the original meaning) and so-called small circles. 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.
|Fig. 5. The equal area projection. A plane is projected onto the projection aircraft, which in this case has been made tangential to the decrease pole of the sphere. The projection is illustrated in 3-D (a) and alongside a seasoned?Le thru C and A (b).|
|Fig. 6. Two hundred and fifty quartz c-axes measured at the U-level and plotted inside the stereonet. An asymmetrical pattern with recognize to the foliation (trending E?W in the plot), consisting of the only 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. 4), the positions of planes and lines in the plot become somewhat different. The net is called a Schmidt net or simply an equal area net (Fig. 5 shows the equal area projection). 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. 6), and the contour values exhibit the number of points as the percentage within a given 1% area of the stereonet. Contouring is easily done by means of one of the many computer programs available for personal computers.
Planes can be represented in a stereonet in two different ways: by means of great circles or poles (Fig 2). Fig. 5 gives a demonstration of how to plot both by hand, and we will start with a great circle representation.
|Fig. 7. Plotting the plane N 030 E, 30 NE in the stereonet (equal area projection).|
A aircraft placing 030 (or N 30 E) and dipping 30 to the SE is plotted as an instance. Tracing paper is located over a pre-made internet (an equal vicinity internet was chosen), and the facilities are attached by way of a thumb tack. The primitive circle and north (N) are marked on the transparent overlay. We then mark off the strike price of our plane, that's 030 (Fig. 7a), and then rotate the overlay so that this mark happens above the N direction of the underlying stereo plot (Fig. 7b). For our instance, this includes rotating the overlay 30 anticlockwise. We then count the dip fee from the primitive circle inwards, and hint the superb circle that it falls on (Fig. 7b). When N on the tracing paper is circled returned to its unique orientation (Fig. 7c) we have a splendid circle that represents our aircraft. The shallower its dip, the nearer it comes to the primitive circle, which itself represents a horizontal aircraft.
The technique is pretty similar if we need to devise poles. All we do otherwise is to matter the dip from the middle of the plot inside the course opposite to that of the dip, which in our example is to the left. When performed efficaciously, there can be ninety between the tremendous circle and the pole of the same aircraft (see Fig. 7b). The pole for this reason falls on the other aspect of the diagram from that of the corresponding superb circle. Poles are commonly favored in structural analyses that involve big amounts of orientation information, and specially if grouping of structural orientations is an problem (which commonly is the case).
|Fig. Eight. Plotting the line plunging 40 tiers toward N 030 E inside the stereonet (equal location projection).|
Plotting a line orientation is similar (but distinctive) to plotting a aircraft orientation. For instance, a line plunging 40 ranges closer to 030 (NE) is considered. As for the plane, we mark off the trend (030) (Fig. 8a), rotate the overlay both 30 levels anticlockwise, as for the plane (Fig. 8b), or until it reaches the E course (a clockwise rotation of 60 stages for our instance). Now remember the plunge fee along the directly line towards the middle, beginning at the primitive circle, and mark off the pole (Fig. 8b). Back-rotate the overlay, and the assignment is completed (Fig. 8c).
|Fig. 9. A constructed, however realistic, state of affairs wherein diverse structural factors in a deformed rock series are represented in stereonets. Plotting bedding orientations exhibits the b-axis (local orientation of the hinge line). The attitude between the fracture sets can be observed by way of counting levels alongside the super circle that ?Ts both statistics units. Fault facts are plotted one after the other, showing the fault aircraft as a amazing circle and lineations as dots on that exquisite circle.|
When doing fault analyses it's far useful to plot each the slip plane and its lineation(s) in the equal plot. In this example the lineation will lie at the first-rate circle that is representing the slip plane. The attitude among the horizontal direction and the lineation is referred to as the rake or pitch, and is plotted by using rotating the extraordinary circle of the plane to a N?S orientation after which counting the wide variety of ranges from the horizontal (N or S), i.E. The pitch cost measured within the ?Eld (Fig. 9). Users of the proper hand rule will always degree the pitch clockwise from the strike value, in order that the attitude might be up to 180 ranges . The proper-hand rule has been used in Fig. Nine. Others degree the acute attitude and count from the ideal strike course, wherein case the pitch will not exceed ninety levels.
Fitting a plane to traces
If or greater strains are known to lie in a not unusual plane, the aircraft is observed by plotting the traces in a internet. The lines are then turned around until they fall on a commonplace incredible circle, which represents the aircraft we are seeking out.
Line of intersection
The line of intersection among two planes is possibly most easily seen by means of plotting the awesome circles of the 2 planes, in which case the road of intersection is represented by the point wherein the 2 first rate circles go. When plotting poles to planes, the road of intersection is the pole to the excellent circle that ?Ts (includes) the two poles.
Angle among 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. 9, where the angle of two sets of fractures is considered). The angle between two lines is found in a similar manner, where the two lines are ﬁtted to a great circle (the plane containing both lines) and the distance between them (in degrees) represents the angle (Fig. 9, lineations).
Orientation from apparent dips
Finding the orientation of a planar structure from observations of apparent dips can sometimes be useful. If two or more apparent dips are measured on two arbitrarily oriented planes, and each of those two planes is represented by a great circle and a point representing the apparent dip (measured at the outcrop), then the great circle that best ﬁts the points represents the plane we want to ﬁnd. An inverse problem would be to determine apparent dips of a known planar fabric or structure as exposed on selected surfaces. We then plot the planar fabric as a great circle, and the apparent dip will be deﬁned by the point of intersection between the planar fabric and the surface of interest.
Rotation of planes and contours
Rotation of planar and linear structures may be done via shifting them along a amazing circle, the pole of which represents the rotation axis. Rotation about a horizontal axis is simple: simply rotate the tracing paper so that the rotation axis falls along the N?S path, and then rotate poles via counting levels along the small circles.
Rotating alongside an inclined axis is a chunk greater bulky. It involves rotating the whole lot so that the axis of rotation will become horizontal, then appearing the rotation as above, and ?Nally, again-rotating so that the axis of rotation achieves its authentic orientation.
|Fig. 10. The plots display the versions within every subarea, portrayed via poles, rose diagrams, and an arrow indicating the common orientation. The number of records within every subarea is indicated with the aid of ?N?.|
Sometimes best the strike issue of planes is measurable or of interest, wherein case the facts may be represented inside the form of a rose diagram. A rose diagram is the predominant circle subdivided into sectors, wherein the quantity of measurements recorded inside every sector is represented by using the length of the respective petal. This is a visually attractive manner of representing the orientation of fractures and lineaments as they appear at the surface of the Earth, and can also be used to symbolize the trend distribution of linear structures (Fig. 10).
All of these operations and more may be achieved extra quick with the aid of stereographic plotting applications, inclusive of the one generously made available to the structural community with the aid of Richard Allmendinger (1998). However, expertise the underlying standards is the important thing to success when the use of such packages. Several plotting packages additionally have statistical add-ons which are quite beneficial.
Credits: Haakon Fossen (Structural Geology)