# Plot a Cone (Small Circle) on a Stereonet

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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## Cones and Small Circles

Conical structures are relatively rare in geology and the need to plot small circles on the stereonet is fairly uncommon, but it's good training in understanding the stereographic projection. Shatter cones and conical folds are the most important structures requiring knowledge of cones and small circles. Small circles also crop up in borehole problems.

In general terms, a cone is any surface that is swept out by a line that always passes through one fixed point. Thus cones need not be circular although most geological conical structures are at least approximately circular. A circular cone with its center at the center of a sphere cuts the sphere in a small circle. We can define the cone in terms of the trend and plunge of its axis and the angular radius or apical half-angle of the cone.

## Example

Plot a small circle (cone) whose axis trends 290 degrees, plunges 50 degrees, and has an angular radius of 25 degrees.

### Plotting Without A Stereonet

Strictly speaking, you can do any stereonet construction graphically. This is one of the simpler constructions to do without a net.

 1. Mark the trend of the cone axis. 2. Rotate the overlay until the axis trend is horizontal. Mark off the plunge angle and the angular radius angles as shown. Points A and B are the endpoints of the diameter of the small circle. 3. Construct a circle through the end points of the diameter. Note that its center does not coincide with the projection of the cone axis. 4. Rotate the overlay to its original position.

### Plotting On A Stereonet

 1. Mark the trend of the cone axis. 2. Rotate the axis trend until it is lies along a vertical great circle. Count off the plunge angle and the angular radius angles as shown. 3. Construct a circle through the end points of the diameter. Note that its center does not coincide with the projection of the cone axis. 4. Rotate the overlay to its original position.

### Small Circles That Cross The Primitive Circle

It's perfectly possible for small circles to cross the primitive circle. Geometrically, that means that part of the cone surface is above the horizontal. However, unless you're using an extended stereonet, there is no way to measure or plot points on the exterior portion of the circle. We have to find some way to bring that part of the small circle inside the stereonet. We usually imagine that the line that sweeps out the cone extends indefinitely far in both directions, so that when the cone exits the projection hemisphere on one side it enters on the opposite side.

The easiest way to plot such circles is the graphical approach. You could locate the cone axis, use the great circles to measure off a number of points with the desired radius, then construct a circle through the points, but the graphical method is just as fast.

 1. Mark off the trend of the cone axis.2. Rotate the overlay until the axis trend is horizontal. Mark off the plunge angle and the angular radius angles as shown. Note that one of the endpoints of the small circle diameter is outside the primitive circle. 3. Construct a circle through the end points of the diameter. Note that its center does not coincide with the projection of the cone axis. The farther the center is from the center of the net, the greater the discrepancy gets. 4. The other side of the small circle re-enters the net diametrically opposite the point where it exits. 5. Now we have to locate a third point on the far side of the small circle. One way to do it is simply count off the radius until we reach the primitive circle, then continue counting on the opposite side of the stereonet. (So far, to avoid clutter, we haven't shown the stereonet)6. We can also continue graphically. Since the small circle has an axis plunging 30 degrees and a radius of 45 degrees, the opposing side of the small circle plunges (45-30) = 15 degrees. 7. Either way, once we have a third point on the arc, we can construct the arc. 8. Final result, with the overlay rotated back to the upright position.

The opposing part of the small circle can do one of three things:

• If the main portion of the small circle does not extend to the center of the net, the opposing segment is concave inward. Its curvature is in the opposite direction from the primitive circle.
• If the main portion of the small circle passes through the center of the net, the opposing segment is a straight line. In such a case there is no need to plot a third point; the two points on the primitive circle are enough.
• If the main portion of the small circle extends beyond the center of the net, the opposing segment is concave outward. Its curvature is in the same direction as the primitive circle.