Find Intersection of a Line And a Plane

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

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There are two ways to solve this problem. One is a straightforward application of techniques we have seen before and the other involves a trick. The first method is conceptually simpler but involves more work. The second method involves less work but is conceptually harder.

In both cases we begin by constructing structure contours on the plane and elevation points on the line.

Cross-Section Method

Once we have elevation points and structure contours, we can draw a cross-section along the trend of the line, showing the line and a cross-section through the plane. Where the two meet is the intersection. Measure the depth from the cross-section and project or transfer the intersection back to the map to find its location.

Example

1. Where would we expect the borehole to intersect the dike? 2. Construct structure contours for the plane and elevation points for the line 3. Construct a cross-section along the trend of the line. 4. Find the intersection of the line and plane on the cross-section, project this point onto the map trace of the line, and interpolate its elevation between the contours and elevation points (the two must agree).

Additional Points

Note that for clarity and simplicity we do not show all the structure contours in Step 3.

Imaginary-Plane Method

This method finds the intersection directly using a bit of a trick. Call the given plane in the problem Plane A and call the line L. Now there are an infinite number of pairs of planes that intersect along line L. If we know two such planes, then we have converted this problem into the problem of finding the intersection of three planes, which is easy to do.

We can pick any pair of planes we want. We're only concerned that they intersect along line L. For best results, they should meet the given plane A at large angles. The structure contours of these planes pass through the elevation points of line L. Let's call these planes B and C. Construct two arbitrary sets of structure contours passing through the elevation points on line L. These are the structure contours on planes B and C.

We now have a problem of three intersecting planes, and the intersections A-B, A-C, and B-C meet at a common point. Their common intersection is the intersection of the line L and plane A. The point is located directly on the map by the construction. The only difficulty is determining the elevation of this point, and that can be found by interpolating between structure contours.

Remember, planes B and C are entirely arbitrary. Their only significance is in helping to find the intersection of plane A and line L.

1. Where would we expect the borehole to intersect the dike? 2. Construct structure contours for the plane and elevation points for the line 3. Through the elevation points on the line, draw two sets of arbitrary structure contours. 4. With the intersection located, interpolate between contours to find the elevation

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Created 5 January 1999, Last Update 31 March 1999