Intersection of a Line with Topography

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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This is not a very common problem. It's hard to envision a case where you'd want to find the intersection of a line intersecting the topography from above. You might conceivably want to find where a linear feature on one side of a canyon would project to the other side.

More likely, the line would be projected from below. Maybe you'd like to know where a borehole in a mine might emerge on the surface, or where the intersection of a fault and a bed might be located.

In any case, the method is pretty straightforward. Draw a cross-section and topographic profile along the trend of the line and locate the intersection. Very often, you can find the intersection by inspection once you have elevation points on the line.

Example 1

Problem: A fold axis outcrops at the locations shown in magenta (below). The far side of the valley is wooded and has poor outcrop. Where would we expect to find the fold axis?

We have enough information to construct the trend of the fold axis and elecation points along it (below). Contours are in meters. Note that the 600 meter elevation point on the line is above the topography and the 500 meter point is below it. By inspection, we can see the intersection lies between the 600-meter elevation point and the 600-m contour.

Next, we construct a topographic profile along the trend of the fold axis. Note the diagram is rotated for legibility.

In this case, we have enough elevation data to construct the fold axis without determining the plunge, but we will certainly want to indicate it on our map and in our report. Loacet the intersection point between the fold axis and the surface (rightmost magenta point).

Finally, transfer the intersection point to the original map.

Example 2

An abandoned mine has a horizontal shaft, or drift, shown by a purple dashed line. An inclined shaft branches off from it and slopes upward at 30 degrees as shown. Note that this is a case of something having a negative plunge. We want to see if any place on the surface is in danger of collapse, so we want to know where the shaft would intersect the surface.

The first step is to construct the trend of the line and construct elevation points on it. It's fairly obvious by inspection that the inclined shaft is below the surface at 1100 meters and above it at 1200. If all we need to do is make a surface inspection and look for ground disturbance or a possible old hidden entrance (and old mining areas have plenty of them), this may be enough.

Construct a topographic profile along the trend of the inclined shaft. Here we're using a diagonal fold line to avoid rotating the map too much.

Construct a profile of the inclined shaft.

And plot the intersection point on the map.

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Created 03 February 2012, Last Update 03 February 2012
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