## Faulty Reasoning

How to apply Descriptive Geometry to the description of faults

This article explains how to use descriptive geometry to visualize faults in three dimensions. These methods can be used for graphical calculations that are fast and easy, but the real reason for understanding them is the three-dimensional insight they provide. The basic method is explained in Monge's Procedure, so if you are not familiar with multiview orthographic projection, it would be a good idea to study this page first.

The reader is encouraged to make sketches while reading this article. All you really need is a pair of small 30-60 triangles and a millimeter scale, and a protractor to lay off and measure angles. Dividers or a compass to transfer distances is a great convenience, but not necessary. By sliding the two triangles together on their hypotenuses, as shown in the figure, parallel and perpendicular lines can easily be constructed. Draw a short line and practice drawing lines parallel and perpendicular to it in different locations. You will find this a very useful skill, if you are not already familiar with it. It is all you need for doing descriptive geometry. Sketches can be made on plain or quadrille ruled paper, as you prefer. I generally use what is called an "engineering pad."

For clarity and ease, let's consider only plane beds and plane faults. It will be easier to see the relationships in this case, and it is often a good approximation of actuality. We'll treat only the simplest cases. More complicated problems can be found in texts devoted to the subject. We recall that a plane can be completely specified by giving any one point in the plane, as well as its strike and dip. The strike is the bearing of a horizontal line in the plane, the intersection of the plane with the horizontal plane, and is usually specified by the bearing to the east or west of north, from 0° to 90°. The dip is the angle between the trace of the horizontal plane and the trace of the plane under consideration in a plane normal to the strike, a dip plane. It is the maximum inclination of any line in the plane with the horizontal. It is specified by the angle of dip, from 0° (level) to 90° (vertical), and the quadrant in which the plane dips.

At the right, a plane with strike N 45° E and dip 45° SE has been plotted. A and B are two points on the intersection of the plane with the horizontal. Only the strike can be plotted in this plane, the horizontal plane. The folding line 1-1 has been drawn at any convenient place normal to the strike line. Imagine your eye looking straight down B towards A. You are looking perpendicular to the dip plane, where you see the plane as a line inclined at 45°. Points A and B are represented in the dip plane by points A' and B', which are coincident. Now we choose a folding line 2-2 parallel to the line representing the plane in the dip plane, so we are looking normal to the plane, directly down on it. This is the true size view. A and B are represented by A" and B" in this view, and obviously AB = A"B" since in either case we are looking normal to the line and see its true length. However, in view 2 we see every line in the plane in its true length. The distances x and z must be made equal so the views have the proper relation to each other. Note that we measure perpendicular to folding lines, and "skip" a view. When you cross two folding lines, you are looking back along the same direction, since each folding line represents a 90° difference in view. Don't leave this diagram until you understand everything; it contains all we need to know!

To prove to yourself that you know what this drawing represents, construct one yourself. Start by drawing AB at some arbitrary angle and distance; I used 45° because it made the pixel drawing easier. Then draw the folding line 1-1, using the pair of triangles. Project the line to get point A' = B', and then draw the trace of the plane at some arbitrary dip. Now draw folding line 2-2 parallel to this line and project A" and B" at distances x and x + z from it. At all times imagine yourself looking at the actual plane in the field which you are representing.

Now we'll go a little further and locate some point C" on the plane in the other views. Choose this point in the true size view (though you could choose it in any view) to make my explanation easier. Project it normally to the folding line 2-2 to see where it goes in the dip plane, some point C' that lies in the plane. At this point, you have enough information to plot the point on any view whatever! To put it in the horizontal plane, just project normal to the folding line to this plane and lay off the distance y from the true size view (or any adjacent view, for that matter). This gives you point C. If you drilled vertically downward from this point on the horizontal plane, you would hit the plane at point C". The triangle A"B"C" on the plane projects into triangle ABC in the horizontal plan. The same would hold for any figure plotted on the true size view. Alternatively, given any figure represented in the horizontal view, you could find out its actual size and shape in the true size view. Study these things until you appreciate the meaning of choosing a folding line parallel or perpendicular to a line.

Now let's suppose we have a stratum that can be represented, say, by a pair of parallel planes separated by the thickness of the stratum. Or, for simplicity, let's represent it by a single plane. The plane represents a bedding plane or a stratification, or a joint, or cleavage, which we imagine as infinite in extent, and is described by strike and dip. A fault plane can also be represented by a plane intersecting the bedding plane, and we can imagine that the parts of the stratum on either side of the fault plane have been displaced by a certain constant amount. If one point A on the fault plane has been divided into two points A and A' on either side of the fault plane, the vector AA' represents the slip on the fault, where we assume that the motion has been to carry A into A'. Whether we have A→A' or A'→A cannot be told by looking only at the fault; either sense of motion gives the same result. If we are looking at the true size view of the fault plane, the same vector AA' gives the slip at any point. Some line on the fault plane represents a horizontal line, the strike line of the fault. The slip AA' may be resolved into a component along the strike, and perpendicular to it.

Various names are used to classify faults. A fault with dip-slip only is called a normal fault if the block towards which the fault plane dips is lowered relative to the other, lengthening the cross-section. If the movement is opposite, so that the cross-section is shortened, it is called a reverse fault. A fault with strike-slip only is called a shear or transcurrent fault. If we stand on one side of the trace of the fault, and the other side has moved to the right relatively, it is a dextral or right-lateral fault. In the opposite case, it is sinistral or left-lateral fault. It is easy to see that it does not matter which side of the fault you are standing on. The general case is an oblique fault, which mixes the types. The San Andreas fault in California and the Alpine fault in New Zealand are right-lateral transcurrent faults. The fault off the Sierra Maestra in Cuba is left-transcurrent, but the fault off the southern side of the Caribbean is right-transcurrent, showing that the Caribbean Plate is holding firm while the North American and South American plates squeeze by it. The Wasatch Front fault in Utah, and the fault bounding the eastern side of the Grand Tetons are normal faults. The fault at the front of the Rockies at Denver is a reverse fault, caused by the expansion of the rocks when pressure was relieved. A series of great reverse faults across Wyoming was caused by dragging of the basement eastward during the Tertiary. Dip-slip faults are often difficult to recognize because erosion soon softens fault scarps. Transcurrent faults have no scarps, but offset features give clues.

In the diagram, a stratum striking due east and dipping 45° S is displaced by a fault striking N 45° E and dipping 45° SE. In the horizontal plane, the stratum is displaced from A to B along the fault trace. Folding lines 1-1 and 2-2 are drawn perpendicular to the fault trace and to the stratum outcrops, respectively, so that both planes are shown as a line. The stratum is represented by two parallel lines representing the relatively displaced parts of the stratum. Points C''' and D''' are chosen at the same depth x to lie simultaneously in the plane of the strata and in the fault plane, so that lines AD and BC will be the intersections of the fault plane with the two parts of the stratum. Since the view beyond folding line 1-1 also shows depth, we place these points on the fault plane at the same depth x, so locating the points C' and D', which coincide in this view. With these points located in two views on either side of the horizontal plane, we can project into the view between them and locate C and D by intersections. Now folding line 3-3 is drawn to get a true size view of the fault plane, and points A, B, C and D are projected onto it. Corresponding distances are marked with small letters. The lines A"D" and B"C", extended indefinitely, are the traces of the intersection of the stratum with the fault plane, one on one side of the fault and the other on the other.

From the given information, we can only find this much. The slip can be determined only if we know its direction, or if we know two corresponding points on the parts of the stratum. If A" and B" are corresponding points, then the slip is A"B" and the fault is strike-slip or transcurrent. We also see that it is right-lateral. If we know that the fault is dip-slip, then the slip is represented by a horizontal line (parallel to folding line 3-3) joining extended A"D" and B"C", which in this case is quite large. However, the direction of the slip must be determined by other evidence. If we know the direction, then we can determine whether the fault is normal or reverse. This solution can be applied to the mining problem of finding the rest of a vein when a vein ends against a fault plane, and the strike, dip and slip of the fault is known. Of course, this will require a modification of the solution. It is solved by finding the trace of the known vein on the fault plane, drawing the displaced vein from the known slip, and projecting back the remainder of the vein.

It can be appreciated that these problems are by no means simple, and require a high order of three-dimensional visualization. Descriptive geometry supplies the tools for aiding this visualization, and it is clear that it is not just a matter of inserting known dimensions.

### References

M. C. Hawk, Theory and Problems of Descriptive Geometry (New York: McGraw-Hill Schaum's Outline Series, 1962).

G. F. Pearce, Engineering Graphics and Descriptive Geometry in 3-D (Toronto: The Macmillan Company of Canada, 1977). The use of red-green anaglyphs is an excellent aid to visualization.

Bailey Willis, Geologic Structures (New York: McGraw-Hill, 1929). Bailey Willis was an outstanding field geologist, and this book is well worth reading if you can find it.

Derek Powell, Interpretation of Geologic Structures Through Maps (New York: John Wiley and Sons, 1992).