# Find Intersection of Two Non-planar Surfaces

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Most geologic structures are not ideally planar. Nevertheless, we may still have to locate the intersections of, say, a folded bed or an intrusive contact with a curving fault. One particularly important case of two intersecting surfaces is the intersection of any geologic structure with the topography. For now, let us consider the intersection of relatively simple surfaces.

Most of the methods we used in finding the intersections of two planes still apply; after all, planes are surfaces. The structure contours are not straight lines and the intersection is not a straight line, but in the most important respects, the two situations are the same. We find the intersections of matching structure contours and draw a smooth curve through the intersection points.

Since the spacing of contours varies, it may be useful to interpolate intermediate points.

## A Few Cardinal Rules

An intersection line can never cross one contour unless the same elevation contour from the other surface intersects at that point. You must have three lines crossing: the intersection line and two contour lines.

Be extremely careful that you locate the intersections of matching contours. It can be very easy to mismatch contours.

## Example: Simple Surfaces

 1. Find the intersection of these two dikes with their structure contours shown. 2. Locate the intersections of  matching contours.  3. Interpolate where contours vary greatly in spacing.   4. Construct a smooth curve through the intersections.

## More Complex Problems

Problems such as the intersection of three surfaces or the intersection of a line and a surface can be solved in much the same way as the simple straight line and plane problems by using the intersection method described here.

Geologic structures in the subsurface can result in situations that are unfamiliar. Structure contours can appear to cross (structures may have overhangs or layers can be overturned) and they can terminate (at faults). Structure contours only appear to cross because a 500-meter contour that crosses a 400-meter contour is actually 100 meters above it.

## Example: Complex Surfaces

 1. A fold with structure contours. The fold is overturned. It is actually recumbent and cannot be considered either an anticline or a syncline. 2. A small intrusion that intersects the fold. It is shown here by itself. The intrusion is shaped somewhat like an upside-down pear tilted to the north. 3. Here the top surface of the fold is colored to help visualize its form. 4. The top surface of the intrusion is similarly colored.

Note that the contours appear to cross. That is, their projections on the surface cross. The contours themselves are separated vertically by 100 meter intervals.

In a case like this you do whatever you have to do to visualize the structure. It may mean making cross-sections or other auxiliary techniques. You may end up building three-dimensional models if the stakes are high enough. Whatever it takes.

This is a fairly nasty case. A couple of comments are in order. First of all, where the intrusion cuts the fold, obviously that part of the fold no longer exists. As we do the construction we treat the fold and the intrusion as interpenetrating mathematical surfaces, but physically that is not the case. We might assume the structure contours on the fold were generated by extrapolating surface data, perhaps in conjunction with borehole information. The shape of the intrusion might be known from surface outcrop, borehole data, geophysical data, or a combination of all of them. Perhaps the contact between the intrusion and the contoured unit in the fold is mineralized, creating an economic reason to map the intersection. Geologists analyzing such a case might spend weeks on the problem.

### Analyzing Structural Levels

 Here we have shown the relationships at each contour. Yellow shows the part of the intrusion within the fold, magenta shows the portion outside. The other structure contours are shown subdued for reference. At 200 meters the intrusion cuts the northeast (lower) limb of the fold. At 300 meters it just reaches the southwest (upper) limb. At 400 and 500 meters it encloses the entire hinge of the fold. The intrusion does not reach 600 meters at all.

### Cross-Sections

 Here we draw cross-sections. The cross-section lines are shown in red on the map. The section below the map is the E-W section, that to the right is the N-S section. Note the 2x vertical exaggeration. The diagonal reflection line is a common technique in drafting for transferring dimensions through 90 degrees. The left cross section shows the intrusion enclosing the entire hinge of the fold. On the right the intrusion penetrates through the fold.

In the left diagram the fold appears to be an anticline, but in the right the limbs converge gradually downward and it looks like a syncline! (We can see on the map that the cross-section line will eventually cross the hinge to the south). Paradoxes like this are common with steeply-plunging recumbent folds.

Based on these two analyses, we expect the intersection will be a saddle-shaped curve wrapping around the hinge of the fold. Picture taking a bite out of a taco shell. Now to construct it.

### Constructing the Intersection

 1. Structure contours on the fold (green) and intrusion (blue). 2. Identify intersections of like contours. Here they are color coded. 3. The line of points from 200 to 500 meters at the rear of the intrusion is pretty straightforward. So are a couple of other pairs of points. The intersections are shown in red.  4. The pluton does not extend down to 100 meters or up to 600, so the 200- and 500- meter points must connect as shown. Although the curve appears to intersect the 600-meter contour, it is actually beneath it.
 5. The intrusion and fold contours just graze at the remaining 300- and 400- meter points, suggesting that the contact just reaches those points and turns 6. The complete contact, but what does it mean? 7. The contours on the intrusion are subdued here. The surface of the fold has been colored to illustrate the area enclosed by the contact: yellow on the top surface of the fold, dark green on the back, light green where they overlap. 8. From 7. we see the construction has a flaw - the contact does not wrap around the hinge of the fold (dark gray). Modifying it as shown remedies the problem.

You can see that there is no single recipe for doing this. The only way to do it is to know as much geology as possible, visualize the intersecting surfaces as completely as possible, and proceed from the simpler to the more complex parts. Also, there is no best way to portray the solution. A map view might be very confusing. It might need to be supplemented by cross-sections or perspective views. Computer drafting programs can help enormously, especially if they generate views that can be rotated, but they only supplement geological intuition, not substitute for it.