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DOWN PLUNGE CROSS SECTIONS I Main Topics

A Cylindrical folds B Downplunge cross-section views C Apparent dip

II Cylindrical folds A Surface of a cylindrical fold is parallel to a line called the fold axis. B Cylindrical folds maintain their shape for “long” (infinite) distances in the

direction of the fold axis (as opposed to folds bending in the shape of a bowl); they are two-dimensional structures because they do not change in shape along the dimension of the fold axis.

C Planes tangent to cylindrically folded beds intersect in lines parallel to the fold axis.

D Poles to cylindrically folded beds are contained in the plane perpendicular to the fold axis, so taking the cross-product of the poles gives the orientation of the fold axis.

III Down-plunge cross-section views A Down-plunge cross-section views can be obtained directly from a geologic

map by looking obliquely at the map down a fold axis. B Beds appear in true thickness C Graphical technique

1 Find orientation of fold axis 2 Draw a cross-section along a plane parallel to the fold axis. The fold

axis will be contained in this plane and the fold axis will appear "in true length" and its plunge can be measured.

3 Take an adjacent view of the above cross section where the line of sight is parallel to the fold axis. Viewed end-on, the fold axis will appear as a point. All the other lines lying in the surface of a cylindrical fold will also be viewed end-on, so the fold surface will appear as a curve.

D Computer-assisted technique using Matlab 1 Find three-dimensional coordinates of points on the folded units. This

can be done be digitizing a geologic map, for example, by scanning a map and using Matlab’s ginput function: [x,y] = ginput

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2 Transform the coordinates of the digitized points by projecting them onto a new set of right-handed reference axes aligned with the fold axis. a “Manual” procedure

i Define the down-plunge (e.g., X,Y,Z) reference frame in terms of the geographic (e.g., x,y,z) reference frame. For example, let the Y axis be the down-plunge direction, the X axis be horizontal and 90° clockwise from the fold axis trend, and the Z axis be “up” (but not vertical). This is the view one would get if you point you right arm and right index finger down the fold axis, with your thumb pointing to the right. and your middle finger pointing “up”.

ii Transform the coordinates from the x,y,z reference frame to the X,Y,Z reference frame using the matrix transformation equations.

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For one point:
*X
Y
Z
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=
*aX x aX y aXz
aY x aY y aY z
aZx aZy aZz
*

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*x
y
z
*

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(3x1) = (3x3) (3x1)

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For n points:
*X*1 *X*2 ... *Xn
Y*1 *Y*2 ... *Yn
Z*1 *Z*2 ... *Zn
*

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=
*aXx aXy aXz
aYx aYy aYz
aZx aZy aZz
*

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*x*1 *x*2 ... *xn
y*1 *y*2 ... *yn
z*1 *z*2 ... *zn
*

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(3xn) = (3x3) (3xn)

iii Then prepare an (X,Z) plot using Matlab’s plot command: plot(X,Z)

The Y (down-plunge) coordinate is irrelevant for this view. b “Automated Matlab 3-D visualization technique”

i Use the Matlab command plot3(x,y,z)

ii Then use the “view” command to look down the fold axis view(-trend,plunge)

(Here the trend and plunge are in degrees, not radians)

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Lab 7 Exercise 1: Down-plunge views Read each problem before you start to do it. 1A) Graphical solution (18 points total) a Draw in a top view a line OF that represents a fold axis that trends N70°W and plunges

30°. Show a north arrow on your top view. Draw an adjacent view “A” that shows the plunge of the line; you need to visualize how to look at the line so that you see its plunge. Adjacent to view “A” draw another view “B” that allows you to see line OF as viewed down-plunge (not up-plunge). In each view label points O and F, where O is the high point and F is the low point. (2 points for the top view, 2 points for the second view, 1 point for the third view, 1 point for the north arrow, 3 points total for the labels; 9 points total)

Assuming that point O is at the coordinate origin and that x=east, y=north, and z = up, find the direction cosines of line line OF.

b α= _________________ (1 point) c β = _________________ (1 point) d γ = _________________ (1 point) e The ratio of the length of OF to the length of the horizontal projection of OF is (to three

significant figures) _______________ (1 point)

In the third (down-plunge) view, draw and label two right-handed axes z” and x”; y” points into the page, so you can draw it. Draw the axes such that:

f The z”-axis points directly from “line” OF toward the fold line separating views A and B. (1 point)

g The x”-axis points away from “line” OF, parallel to the fold line separating views A and B, and is consistent with the y” axis pointing in the down-plunge direction. (1 point)

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1B) Spherical projection (equal-angle) solution (36 points) Here you will revisit the same line as in 1a, but you will obtain the down-plunge view of the line by successive rotations about the z, x’, and y” axes using an equal-angle (stereonet) projection. Earlier we discussed how the rotations could be done about the x, y’, and z”, but they also could be done about the z, x’, and y” axes. Use one spherical projection for this problem.

a Plot a fold axis that plunges 30° in the direction N70°W. Label the line with an F.

Consider this l ine as being fixed in space. (2 points for plotting the l ine, 1 point for the label; 3 points total) Plot three right-handed axes:

b The x-axis trends due east and plunges 0°. Label it “x”. (2 points for plotting the l ine, 1 point for the label; 3 points total) c The y-axis trends due north and plunges 0°. Label it “y”. (2 points for plotting the l ine, 1 point for the label; 3 points total) d The negative z-axis, where the positive z-axis points up at 90°. Label it “-z””.

You can’t plot the positive z-axis (as defined here) on a lower hemispherical projection, so that is why you are plotting the negative z-axis.

(2 points for plotting the l ine, 1 point for the label; 3 points total)

Now rotate the axes about the positive z-axis to yield a new set of axes (x’, y’, z’) such that the trend of the y’ axis matches trend of the fixed fold axis F.

e The angle of rotation about z is: ______ Pay attention to the sign of the angle! (1 point for the magnitude, 1 point for the sign; 2 point total) Plot the new right-handed axes:

f The x’-axis. Label it “ x’ ”. (2 points for plotting the l ine, 1 point for the label; 3 points total) g The y’-axis. Label it “ y’ ”. (2 points for plotting the l ine, 1 point for the label; 3 points total) h The negative z’-axis. Label it “ -z’ ”. (2 points for plotting the l ine, 1 point for the label; 3 points total) i Draw a curved arrow that shows the angle of rotation connecting y to y’ and label the

curved arrow with the angle of rotation. (1 point)

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