# How Spherical Projections are Used

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Unlike mapping, we are not interested in projecting patterns on the sphere itself. Rather, we are interested in using the sphere to analyze angular relations between lines and planes.

Consider a two-dimensional analogue: a protractor. To measure the angle between two lines, we move the center of the circle to the intersection of the lines, note where the lines cross the circumference of the circle, and read off the angles. Note that:

• We cannot use the protractor to measure distance, only angle. (Many protractors are combined with rulers as a convenience, but the ruler itself plays no role in measuring angles.)
• The only point of interest is where each line intersects the circle. We don't care how far beyond the protractor the lines extend, or if portions of the lines are missing. And it doesn't matter how precisely the lines are drawn if they don't extend out to the circle. We have to extend them before we can measure the angle.
• In some cases the lines need not even actually meet; we might measure the map azimuths of two roads and determine the angle between them, even if the roads don't actually intersect. This turns out to be very common in three-dimensional angular problems.

Think of a spherical projection as a three-dimensional protractor.

• We are only interested in angles, not distances.
• We are only interested in objects that pass through the center of the sphere.
• A point on the sphere actually represents a line passing through the center of the sphere.
• A great circle on the sphere actually represents a plane passing through the center of the sphere.
• A small circle on the sphere is the hardest thing to visualize. Think of it as representing an infinite number of lines, each passing through the center of the sphere and some point on the circle. All together, the lines sweep out a cone. Shatter cones and conical folds are about the only geological structures that are conical, although borehole problems may involve small circles as well.
• Most spherical projections are drawn with the coordinate (latitude and longitude) axis parallel to the projection plane. This approach has several advantages:
• Every possible great circle is shown, and small circles of every possible radius are shown.
• The meridians represent planes of every possible orientation, so they can be used as templates for solving angle problems.
• The small circles are used mostly for measuring angles along the great circles. They are also useful for performing rotations.

Stereonet constructions are typically accurate to within 2 degrees because of tearing of the overlay at the pivot hole, parallax and plotting errors because the overlay is separated from the net, distortions in printing the net, and hand-eye coordination errors in plotting points and circles. With extra care most of these factors can be minimized to bring the accuracy to within a degree or so. The stereonet works because geologic structures can rarely be measured more precisely than within a couple of degrees or so. Like the slide rule, another effective graphic device, the stereonet is just accurate enough to serve its purpose.

If you attempt to mix stereonet results with those derived by more accurate methods, like calculation, the end result will be only as good as the stereonet accuracy.

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