Plotting Objects on the Opposite Side of the Sphere

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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This page is designed to give an overview of the way mineralogy and structural geology deal with the problem of points outside the primitive circle, or in other words, on the opposite side of the projection sphere. The descriptions refer to techniques that are explained in more detail on other pages.

Mineralogy

In mineralogy, data is plotted on the upper half of the projection sphere. The lower half is the "opposite side" of the sphere. In mineralogy, we can't assume that a line that intersects the upper hemisphere also intersects the lower hemisphere. The reason is that lines are often used to represent crystal faces. These are poles to crystal faces, that is, lines perpendicular to the crystal face. It very often happens that a crystal will have a face on one side but no corresponding face diametrically opposite. Thus, in mineralogy, it's essential to distinguish between lines that intersect the top of the sphere and lines that intersect the bottom.

In a nutshell, objects on the lower half of the projection sphere are reflected through the equatorial plane to the upper hemisphere and then plotted.

A line that intersects the bottom of the projection sphere is simply reflected through the equatorial plane to the upper hemisphere and then plotted. Typically, a solid dot is used for upper-hemisphere points (lines), and an open circle for lower. If a lower-hemisphere point plots at the same position as an upper-hemisphere point (common because the equatorial plane is often a mirror symmetry plane), both a dot and circle are plotted.

The lower-hemisphere of a great circle is similarly reflected. To measure an angle that crosses the primitive circle, count out along the upper-hemisphere circle (blue) and in on the lower hemisphere circle (red) as shown.

A small circle that straddles the primitive circle is plotted as shown. The upper-hemisphere portion is plotted (blue) as well as the reflected lower hemisphere portion (red).

Structural Geology

In structural geology, data is plotted on the lower half of the sphere. We simply imagine we are looking into a sphere half-submerged in the earth. It is rarely necessary to imagine that lines or planes extend in one direction only. Therefore, in structural geology, we simply plot whichever end of the line or plane that intersects the lower hemisphere. In those rare cases where we have to deal with upward-pointing structures (attitudes measured by aiming up at an overhanging outcrop, current directions or slickensides that point up, etc.) we can usually plot the lower end of the structure and denote it with a special symbol.

The one situation in structural geology that does require moving between hemispheres is rotation of data, a fairly advanced and rare application.

Usually lines can be plotted by plotting their lower-hemisphere ends. Special symbols can be used to keep track of cases where lines are unidirectional (current directions, etc.)

It is almost never necessary to distinguish between the upper and lower portions of a plane. Angles can always be measured by counting off within the primitive circle. If for some reason you need to measure an angle that straddles the primitive circle, count out to the primitive circle (blue) and then continue in from the opposite end (red). Imagine sweeping the angle out with a line. As one end of the line moves into the upper hemisphere, the other end enters the lower hemisphere on the opposite side.

Small circles represent lines on the surface of a cone, but the lines continue through the center and sweep out a matching small circle on the other side. Plot the lower-hemisphere portions of both circles.

A Possibly Interesting Piece of Stereo-Trivia

a. A straight line passing through the center of a stereonet obviously represents a vertical great circle. What does a straight line not passing through the center represent?

b. Recall that the projection point for the stereographic projection is on the sphere, and points are projected onto the equatorial plane.

c. To project as a line, the points must be in a plane passing through the projection point. Great circles (blue) are planes that pass through the center of the sphere. But other planes (red) are possible. They cut the sphere in small circles.

d. A small circle represents a cone, whose other side also cuts the sphere (blue). Note that, if one side of the cone passes through the projection point, the other side passes through the diametrically opposite point.

e. The complete cone plotted. Whenever a small circle (blue) passes through the center of the stereonet and cuts the primitive circle, the other side of the small circle plots as a straight line (red).


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Created 24 August 2000, Last Update 24 August 2000
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