Three-Point Problem by Simultaneous Linear Equations
Those not familiar with the 3-point problem will learn how to determine the attitude of a (planar) bed from given elevations at three locations.
All students will learn:
- to use simultaneous linear equations to provide a quick and accurate solution of the 3-Point problem
- to use mathematical models to approximate realistic geologic phenomena
- the value of a Linear Algebraic approach to solving Systems of Linear Equations
The exercise can also serve as an intuitive springboard for subsequent learning in data fitting, parameter estimation, experiment design, data resolution, and factor analysis.
Context for Use
The activity can be used in courses on structural geology, field methods, and stratigraphy where determining the attitude of planar features is important, or courses on quantitative methods in the geosciences as an introduction to linear algebra. Computational tools involved can be either the Excel spreadsheet, a downloadable linear-equation solver called LINEQ, or the basic MatLab package. An Excel simulator is provided to generate sample problems.
The concepts of simultaneous linear equations and matrix representation are assumed, though it has been found useful to review them, along with matrix multiplication and inverses, as part of the lesson. It is helpful, though not required, for the students to have a visual understanding of the 3-point problem in advance.
Description and Teaching Materials
Notes about the Analytic Geometry (Acrobat (PDF) 154kB Aug5 04) involved in this problem.
Versions of the exercise for different calculation methods:
- Activity Description/Assignment (MatLab based) (Acrobat (PDF) 140kB Jul20 04)
- Activity Description/Assignment (Excel based) (Acrobat (PDF) 138kB Jul20 04)
- Activity Description/Assignment (LINEQ based) (Acrobat (PDF) 158kB Jul20 04)
- Excel spreadsheet to generate raw data (Excel 16kB Jun30 04)
- LINEQ simultaneous equation solver for PCs ( 59kB Jul16 04)
- Typical problem set (Acrobat (PDF) 58kB Aug4 04)
- Solutions to Typical Problem Set (Excel 16kB Aug5 04)
- Exam questions that can also be used for a problem set (Acrobat (PDF) 9kB Jun30 04)
- Example of a spreadsheet solution: Excel solution of 3-Point problem (Excel 18kB Jul15 04)
Preparatory material: A prior assignment on Analytic Geometry (Acrobat (PDF) 316kB Jun30 04)
Teaching Notes and Tips
This approach to the 3-point problem has been presented several times to students who have already seen more traditional techniques of solution; they have generally accepted it as more convenient and useful than their previous experience. Frankly, I believe that their impressions rest largely on their already having the geometry well in mind!
Though most students these days have previously had glimpses of matrix solutions to linear equations, they often need a refresher. I have found it easy to do in classroom discussion, reminding them of the rules of matrix multiplication and pointing out the parallel to the linear equations that we've just written for depths to a plane at three different locations.
A major benefit of this approach has been that the concepts of simultaneous linear equations and linear algebra (i.e., matrix methods) can be recognized as the easy way to do a problem which students already understand. The practical problem brings their prior knowledge to the fore, and helps them seek ways to apply those skills elsewhere.
Productive directions for extension include:
- factor analysis of geochemical data (see example exam problem)
- least squares fitting through development of the overdetermined problem as u = [ATA]-1 AT z
- discussion of data resolution and experiment design by considering the placement of the drillholes in the problem
Those using the Excel version may find it handy to refer students to the Excel Help for Students
References and Resources
Alternate approaches to the 3-Point problem may be found in most structural geology textbooks.
Vacher, H.L. (2000). Computational Geology 12: Cramer's rule and the three-point problem. J. Geosci. Ed., 48(4), 522-532.
Dr. Vacher provides a solution to this problem using linear algebra.