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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


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                                                                                                      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
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                                                                                          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

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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

				
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