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Downloaded from geology.gsapubs.org on April 13, 2013 Geology Structural uses of tangent diagrams C. A. Bengtson Geology 1980;8;599-602 doi: 10.1130/0091-7613(1980)8<599:SUOTD>2.0.CO;2 Email alerting services click www.gsapubs.org/cgi/alerts to receive free e-mail alerts when new articles cite this article Subscribe click www.gsapubs.org/subscriptions/ to subscribe to Geology Permission request click http://www.geosociety.org/pubs/copyrt.htm#gsa to contact GSA Copyright not claimed on content prepared wholly by U.S. government employees within scope of their employment. Individual scientists are hereby granted permission, without fees or further requests to GSA, to use a single figure, a single table, and/or a brief paragraph of text in subsequent works and to make unlimited copies of items in GSA's journals for noncommercial use in classrooms to further education and science. This file may not be posted to any Web site, but authors may post the abstracts only of their articles on their own or their organization's Web site providing the posting includes a reference to the article's full citation. GSA provides this and other forums for the presentation of diverse opinions and positions by scientists worldwide, regardless of their race, citizenship, gender, religion, or political viewpoint. Opinions presented in this publication do not reflect official positions of the Society. Notes Geological Society of America Downloaded from geology.gsapubs.org on April 13, 2013 C. A. Bengtson Structural uses of tangent diagrams Chevron U.S.A. Inc. San Francisco, California 94119 ABSTRACT Tangent diagrams are polar coordinate graphs on which the attitude of planes and lines is represented by the end point of vectors, proportional in length to the tangent of the angle of dip. They provide convenient and easily visualized vectorial solutions for such problems as finding apparent dip from true dip, true dip from two apparent dips, and the line of intersection of two planes. In addition, they have proved to be especially useful for orienting cylindrical and conical folds by graphic analysis of dip data and distinguishing cylindrical folds from the two possible kinds of conical folds. Dip measure- ments at random locations on cylindrical folds define straight-line "statistical" patterns on tangent diagrams. Dip data for conical folds, however, define two kinds of curved lines corresponding, respectively, to the two possible kinds of conical folds. Lines concave toward the center identify conical anticlines that narrow up-plunge (or synclines that narrow downplunge) and lines concave away from the center define conical anticlines that narrow downplunge (or conical synclines that narrow up-plunge). Although the first kind of conical fold is more common than the second, only the second kind has been treated in the literature. VECTOR OPERATIONS ON TANGENT DIAGRAMS Tangent diagrams, such as the example shown in Figure 1, are special polar coor- dinate graphs that provide convenient graphic solutions for many , problems of structural geology. Direction of dip is read at the circumference, and angle of dip is read from the concentric circles. Notice, however, that the radius of each circle is proportional to the tangent of the angle of dip. High dips, therefore, plot proportionally farther from the center than low dips. Figure 1 accommodates dips from 0° to 65°, and the auxiliary scale at the bottom extends the range to 80°, beyond which the exaggeration of the dip scale becomes excessive. The distinc- tive feature of this method of display is that planes can be represented by true vectors. Although tangent diagrams are more easily applied than stereonets to many problems, their structural uses are apparently not mentioned in the literature. Figure 1. Polar tangent diagram for dips from I I I ! j i L 0° to 65°. Auxiliary scale extends range to 80 GEOLOGY, v. 8, p. 599-602, DECEMBER 1980 599 Downloaded from geology.gsapubs.org on April 13, 2013 N ^-CRESTAL LlNE A ^ J l ^ 360° tan 02 = tan 8, cos a N N Figure 4. (A) Block diagram of nonplunging cylindrical fold, showing generating lines. V 2 = V, c o s a (B) Tangent plot of dip data. Figure 2. (A) Block diagram showing trigono- metric relation between apparent and true dip. (B) Tangent method of finding apparent dip from true dip. shows how this principle is used to find Figure 3. (A) Finding true dip from two the line of intersection of two planes. apparent dips. (B) Finding line of intersection 1. Plot Vt and V2, the true dip vectors of two planes. for the two planes. 2. Connect the end points of the two Figure 2A, a block diagram of a sloping 2. Draw a line in the direction of the vectors with a straight line. plane, illustrates the basic principle of the apparent dip. 3. Draw V3, the perpendicular from the tangent diagram. Line B t is a horizontal 3. From the terminus of Vj, draw a origin to the straight line. This vector line in the direction of true dip, and B 2 is line perpendicular to the direction of gives the bearing and plunge of the line another horizontal line making an angle a apparent dip. of intersection of the two planes. with Bji 0 j is the angle of true dip, and 4. Read the apparent dip, V2, from the Figure 3B provides insight for an 9 2 is the angle of apparent dip in the intersection of the two lines. important statistical application of tan- direction of line B2. The trigonometric gent diagrams: The lines of intersection relations entered on this drawing demon- Finding True Dip from Two of planes tangent to the bedding on the strate that the tangent of apparent dip in Apparent Dips same or opposite flanks of an ideal cylin- any direction is equal to the tangent of Figure 3A shows how the tangent drical fold are parallel to the crestal line. the true dip times the cosine of the angle diagram is used to find true dip from two Dip measurements obtained at random between the directions of true dip and apparent dips. locations on such a structure will fall on a apparent dip. The tangent of dip, there- 1. Plot V! and V 2 , the two apparent straight line when plotted on tangent fore, is a true vector that obeys the cosine dips. diagrams, as exemplified by the dashed law of vector addition and resolution. 2. Draw perpendicular lines through line in Figure 3B. their end points. Finding Apparent Dip from True Dip 3. Read the true dip, V3, from the CYLINDRICAL AND CONICAL The problem of finding apparent dip intersection of the perpendicular lines. FOLDS from true dip can be resolved vectorially It has been recognized for a long time on a tangent diagram, as shown in Figure Finding the Line of Intersection that the geometry of most folds is well 2B. of Two Planes approximated by either cylindrical or 1. Plot V1( the true dip, as a vector If two planes intersect, they have equal conical bulk curvature (Dahlstrom, 1954; from the origin with length proportional apparent dips in the vertical plane con- Stockwell, 1950; Tischer, 1962; Wilson, to the tangent: of the angle of dip. taining their line of intersection. Figure 3B 1967). Dip data from random locations 600 DECEMBER 1980 Downloaded from geology.gsapubs.org on April 13, 2013 VERTEX 360° 560° Figure 5. (A) Block diagram showing plunging cylindrical fold. (B) Tangent plot of dip data. Figure 6. (A) Block diagram of type I conical Figure 7. Block diagram of type II conical fold; anticlinal vertex up-plunge. (B) Tangent fold; anticlinal vertex downplunge. plot of dip data. (B) Tangent plot of dip data. on such structures define smooth "statistical" curves when plotted on tan- gent diagrams, provided the data are through the center of the tangent diagram Type I and Type II Conical Folds accumulated from one plunge only. of Figure 4B. Notice that the azimuth of Figure 6A is a block diagram of a Similar patterns are also developed from the crestal line, which is known to be conical anticline with its vertex up-plunge. drag zones of dip-slip faults. north-south, is perpendicular to the line The corresponding conical syncline would A cylindrical fold is defined as a defined by the data points. have its vertex downplunge. Such folds, structure whose bedding surfaces can be If the structure of Figure 4A is tilted to which will be called type I conical folds, represented by a moving straight line that the north, it becomes a plunging cylindrical are much more common than the type II remained parallel to a given fixed line fold, such as shown in Figure 5A. The dip folds described in the next section. The called the b axis. A conical fold is defined data, which show variable strike and dip, straight lines, as before, represent instan- as a structure whose bedding surfaces again define a straight line on the tangent taneous positions of the generating line. can be represented by a moving straight diagram (Fig. 5B) but one that does not Dip data accumulated on conical folds, as line that passed through a fixed point pass through the center. This line is might be expiected, define curved rather called the vertex. Each bedding surface parallel to the nonplunging data line of than straight lines when plotted on tangent has a different vertex, but the crestal line the previous example and therefore diagrams. The data curve for an ideal of each bedding surface remains parallel demonstrates that each dip on a cylin- type I conical fold is a hyperbola (Adams, to a fixed line that could be called a local dricàl fold comprises a constant com- 1919; Deetz and Adams, 1945) that opens b axis. Cylindrical folds, or approximately ponent in the direction of the crestal line to the center, such as shown in Figure 6B. cylindrical folds, are more common than (equal to the angle of plunge) and a The shortest line from the center to the conical folds, but both occur in nature. variable component in the perpendicular curve gives the bearing and plunge of the direction ranging from zero at the crest crestal line. Nonplunging and Plunging to a maximum on the flanks. The perpen- If the conical fold of Figure 6A is tilted Cylindrical Folds dicular vector from the center to the data to the north, the angle of plunge will Figure 4A shows a nonplunging line therefore establishes both the direction diminish and finally reverse, and the ver- cylindrical fold. The straight lines on this and amount of plunge of the crestal tex will now be downplunge rather than figure show various positions of the line. Cylindrical fold data are almost up-plunge (Fig. 7A). Anticlines with ver- generating line and also delineate the lines always easier to interpret on tangent plots tices downplunge and synclines with of intersection of neighboring dip-strike than on stereonet pole plots—especially vertices up-plunge will be called type II planes. Except for scatter, all dips are when the data are restricted to one flank, conical folds. The dip data of a type II either due east or due west and therefore a situation that commonly arises when conical fold also define a hyperbola define a "statistical" straight line passing orienting folds from dipmeter data. when plotted on a tangent diagram, but GEOLOGY 601 Downloaded from geology.gsapubs.org on April 13, 2013 one that opens away f r o m the origin, as folds is (1) find the crestal line f r o m the REFERENCES CITED shown in Figure 7B. The curvature of the tangent plot; (2) project the control parallel Adams, O. S., 1919, General theory of poly- conic projections: U.S. Coast and Geodetic data line is the opposite of that for a to this axis. The comparable procedure Survey Special Publication 57. type I conical fold, but the bearing and for conical folds is (1) project data in Dahlstrom, C.D.A., 1954, Statistical analysis plunge of the crestal line again are given crestal positions parallel to the crestal line; of cylindrical folds: Canadian Institute of Mining and Metallurgy Transactions, v. 57, by the shortest line f r o m the center to the (2) project data in flank positions parallel p. 140-145. data line. Conical fold data are usually to appropriate "local projection axes." Deetz, Ch. H., and Adams, O. S., 1945, much easier to interpret on tangent dia- T o find a local projection axis, draw a Elements of map projection with applica- tions to map and chart construction [fifth grams than on stereonet pole plots. This line tangent to the data curve of the tan- edition]: U.S. Coast and Geodetic Survey is especially true when dealing with data gent plot. The perpendicular line f r o m Special Publication 68. f r o m type II conical folds. Rech, W., 1977, Zur Geometrie der geologischen the center to this line is the required Falten: Geologische Rundschau, v. 66, Lines that connect points of equal strike local projection axis. p. 352-373. (and dip) converge up-plunge on contour Stauffer, M. R., 1964, The geometry of conical folds: New Zealand Journal of Geology and maps of type I conical anticlines and CONCLUSIONS Geophysics, v. 7, p. 340-347. downplunge on contour maps of type II Solutions to many structural problems Stockwell, C. H., 1950, The use of the plunge conical anticlines. Because the latter are more easily accomplished on tangent in the construction of cross-sections of folds: Geological Association of Canada behavior is seldom seen, it would appear diagrams than on stereonet pole plots Proceedings, v. 3, p. 97-121. that type II conical folds are quite rare, because (1) there is no need to convert Tischer, G., 1962, Über X-Achsen: Geologische a conclusion that runs counter to the planes to poles and poles back to planes; Rundschau, v. 52, p. 426-447. Wilson, G., 1967, The geometry of cylindrical limited literature on conical folding (2) there is no need to rotate tangent dia- and conical folds: Geological Association (Rech, 1977; Stauffer, 1964; Stockwell, grams, because operations that require [London] Proceedings, v. 78, p. 179-210. 1950; Wilson, 1967), which implies that tracing a great circle on stereonets require all conical anticlines are type II conical only straight-line constructions on tangent ACKNOWLEDGMENTS folds. diagrams; (3) dip data f r o m ideal cylin- Reviewed by John M. Crowell, C Dahlstrom, drical folds define straight-line patterns Nicholas Christie-Blick, and Paul Karl Link. Their suggestions for improvement are greatly PROJECTING STRUCTURAL on tangent diagrams rather, than great appreciated. CONTROL ONTO CROSS SECTIONS circle patterns, as on stereonets; and OF CYLINDRICAL A N D (4) dip data from conical folds, especially MANUSCRIPT RECEIVED MAY 22, 1980 CONICAL FOLDS type II conical folds, define more easily MANUSCRIPT ACCEPTED SEPT. 22, 1980 The procedure for projecting structural recognized patterns on tangent diagrams control onto cross sections of cylindrical than on stereonets. 602 PR I N T E O IN U.S.A. DECEMBER 1980