# Find Angle Between Two Planes

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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## Conformality

Find the angle between two planes, one striking 128 and dipping 40 SW, the other striking 034 and dipping 50 NW.

 1. Here the two planes are plotted with their poles. Now the stereographic projection is conformal; that is, it preserves angles but at the expense of large radial distortion. If you look closely, every great circle and every small circle on a stereonet intersect at right angles, just like on the sphere. Therefore, the angle between the planes is simply the angle at which their great circles intersect on the stereonet. 2. To measure the angle precisely, retrace the great circles as exactly as possible near the intersection and construct tangents to the arcs. Measure the angles between the tangents. This method is not quite as accurate as the more preferred method shown below (the error range is typically 3-4 degrees) but it is much more accurate than one might at first guess. And it's a nice illustration of the conformality property. Note: this approach takes advantage of a special property of the stereographic projection, and only works with that projection. Some geologists prefer to work with the equal area projection. This method will not work in that case, although the pole method below works for any projection.

## Pole Method

 To measure the angle between planes, we have to make a cross-section looking down their intersection. The cross-section is perpendicular to both planes, therefore it contains the poles. It's easy to see that the angle between the planes is equal to the angle between the poles. Proof:  Angle Plane1-Plane2 = A and angle Plane1-Pole1 = 90 degrees.  Therefore angle Plane2-Pole1 = 90-A.  Since angle Plane2-Pole2 = 90 degrees and angle Plane2-Pole1 = 90-A, angle Pole2-Pole1 must be A.

The same problem as above is shown below, only using the poles of the planes. Find the angle between two planes, one striking 128 and dipping 40 SW, the other striking 034 and dipping 50 NW.

 1. Plot the pole to the first plane (the stereonet is rotated so strike 034 is vertical. The pole is in blue and the plane in light blue). 2. Plot the pole to the first plane (the stereonet is rotated so strike 128 is vertical. The pole is in green and the plane in light green). 3. Rotate the overlay so the poles lie on a common great circle (purple). Count off the angle between the poles. 4. Return the overlay to its original position.