Find Trend And Plunge, Given Pitch

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
First-time Visitors: Please visit Site Map and Disclaimer. Use "Back" to return here.

It is sometimes useful to characterize the orientation of a line by referring to its direction in a dipping plane. For example, you may have ripple marks on a bed or slickensides on a fault, and it may be difficult to determine their trend and plunge accurately. On the other hand, it is very important that these lines are contained within some particular plane (the bed and the fault, respectively). In cases like these, the pitch measurement is sometimes used.

Pitch is defined as the angle between some line in a plane and a horizontal line, measured in the plane. A line can have the same pitch in two directions, so it is important to define directions precisely.

Here's how to do this using descriptive geometry. In practice, problems like this are generally done with a stereonet. There are also special diagrams that allow the solution to be read off directly.

We can solve the problem easily if we note that structure contours on a map are foreshortened views of the real thing. If the plane dips with an angle D, the mapped contours are compressed by a factor of cos D

You can determine the true spacing of the contours by calculation or by measuring distance AB in the cross-section. Note: a pitching line can never have a plunge greater than the dip of the plane!

Example

 1. Given the fault shown and slickensides with the observed pitch, find their trend and plunge. 2. Construct structure contours for the plane. Find the true spacing by cross-section or trigonometry and construct a second set of contours with true spacing. 3. On the true set of contours, construct the pitching line as it appears on the dipping plane. Project distances along the contours back to the map view. 4. On the map view, construct the map trace of the line. Find its trend and find the plunge by trigonometry or drawing a cross-section.

Mathematical Note

We know the pitch and the dip. What we want to find is the trend and plunge. The trend is just the strike of the bed (given by the structure contours) plus or minus angle XAB in the top diagram. It will be plus or minus depending on which way the pitch is measured relative to the strike.

We have:

• Tan XAB = XB/AX
• Tan(Pitch) = XB'/AX. Thus:
• Tan XAB/Tan(Pitch) = XB/XB' = S/(S/Cos D) = Cos D
• Therefore, Tan XAB = Tan(Pitch)Cos D

The plunge (P) is found from the diagrams below. Note: we use B to denote location in the vertical cross section, and B' to denote the same point viewed in the dipping plane.

We have:

• U Tan(Pitch) = H/Sin(Dip) or U = H/(Tan(Pitch)Sin(Dip))
• U Tan(Plunge) = H or Tan(Plunge) = H/U
• Substituting U from the first equation gives us Tan(Plunge) =
Tan P = Tan(Pitch)Sin D

Final Result

If X is the angle between the trend and the strike, P is the plunge, D is the dip of the layer, then

• Tan X = Tan(Pitch)Cos D
• Tan P = Tan(Pitch)Sin D

Special Cases

• If Pitch = 0, trend=strike, plunge = 0
• If Pitch = 90, trend=strike plus or minus 90, plunge = dip

Created 5 January 1999, Last Updated 26 January 2012
Not an official UW Green Bay site