How Spherical Projections are Used
Steven Dutch, Professor Emeritus, Natural and Applied Sciences,University
of Wisconsin - Green Bay
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
moreprecisely than within a couple of degrees or so. Like the slide rule, anothereffective
graphic device, the stereonet is just accurate enough to serve itspurpose.
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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Created
28 December 1998, Last Update 16 March 1999