# Constructing an Equatorial Stereonet

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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## Constructing a Stereonet

Most of the time, it's much more convenient to use a printed stereonet. Stereonets are widely reproduced in textbooks on mineralogy and structural geology, and can also be purchased in bulk from numerous suppliers. If you need a specially-created stereonet, for example, for a classroom demonstration, you can probably make one by enlarging a standard stereonet on a copier.

However, it's possible that you may need to plot a stereonet for special purposes, or perhaps construct one that extends beyond 180 degrees. This page describes how to do it.

### Fundamental Formulas for Circles in the Stereographic Projection

If r is the radius of the circle in degrees and a is the angle between the center of the projection and the center of the circle, then, in terms of the primitive circle,

• Distance to center of projected circle = sin a /(cos a + cos r)
• Radius of projected circle = sin r /(cos a + cos r)

For derivation, see Constructing The Stereographic Projection

### Constructing Great Circles

For great circles, the radius is 90 degrees and cos r = 0. Thus the distance and radius formulas for the projected circle become:

• Distance = tan a

If we want to construct a stereonet in the standard orientation with the great circles like meridians of longitude, then the circle that represents longitude w will have its pole at longitude w plus or minus 90. Thus the distance to the pole is -1/tan(w) and the center of the circle lies on the equator. The minus sign in the distance formula simply means that the center of the circle lies on the opposite side of the sphere from the meridian. The radius of the projected circle is 1/cos(w).

 The formulas suggest the simple construction at left. Linear dimensions are in red, angles in purple.It is useful to introduce angle u, since 1/tan w = tan(90-w) = tan u. Measure angle u from the prime meridian as shown and draw SPC. Point C is the center of the great circle. Alternatively, if the primitive circle is marked with degrees, simply measure off angles 2u and 2w as shown and construct SPC.

For the great circles very close to the prime meridian, point C may be so far away it is impossible to plot. It's useful to have some formula that relates distance AO and angle w. (AO is sometimes called the sagitta, from the Latin word for arrow, because arc NAS and line AO somewhat resemble a bow and arrow.) The formula is simple and follows directly from the construction of the projection: AO = tan (w/2).

### Constructing Small Circles

For the small circles of the stereonet, a always equals 90 because the circles are centered on the pole. Thus, a circle of latitude l has radius 90-l. The distance to its center is 1/cos r = 1/sin l. Its radius is tan r = 1/tan l.

 The simple construction at left can be used to plot a small circle of latitude l. Draw OP making angle l with the equator. Draw PC perpendicular to OP. C is the center of the circle.

### Construction Overview

 The diagram at left gives an overview of stereonet construction.  Radii of small circles are constructed using the vertical axis. Radii of great circles are constructed on the horizontal axis. Radii for circles with dips less than 45 degrees would be marked off on the upper half of the circle and the centers of the projected circles would fall inside the primitive circle. To plot great circles, start at the top of the primitive circle with zero, count off twice the dip around the primitive circle. Draw a line from the bottom of the primitive circle through the desired point on the circumference. Where the line (extended if necessary) intersects the horizontal axis is the center of the projected circle.

Created 5 September 2000, Last Update 6 September 2000

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