Characterizing Structures Using Differential Geometry

1
Characterizing Structures Using Differential
Geometry

• Partial summary of Chapter 3 of Fundamentals of
  Structural Geology by Pollard and Fletcher 2006
• NOTE This Powerpoint presentation is prepared
  by Hassan Babaie for his teaching purposes, and
  does not fully cover the chapter in the
  book!Students are advised to read the entire
  chapter in the book.
• URL http//www.gsu.edu/geohab/pages/geol4013/le
  ctures.htm

2
Why Need Differential Geometry?

• Traditionally, structural data, such as attitude
  of foliation or lineation, are collected at a
  point from scattered exposures
• Data points have locations identified using
  geographic or local coordinates and position
  vector
• Stereographic plots of the attitudes of these
  isolated data lack information about the spatial
  variation of orientation

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• Recall that each plane or line is characterized
  (i.e., measured) by its attitude strike/dip for
  planes, and trend/plunge for lines
• Planar or linear structures measured at a
  specific location (i.e., exposure) represent an
  approximation of the more realistic curviplanar
  or curvilinear structure, whose orientation
  generally varies by position over an extended
  area
• Although such variation can be visualized by
  inspection of geological maps that show spatial
  variation of the structure with symbols, the
  variation cannot be quantified
• We use the differential geometry for the
  quantification and analysis of spatial variation
  

4
Structure contour map of the San Rafael Swell,
Utah. Yellow lines are contours with 300 ft
intervals. Black lines are major faultsNotice
the spatial variation of the attitude
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Position Vector

• Position vectors are used to describe points,
  curves, and surfaces in differential geometry
• Other vector quantities such as tangent vector
  and curvature vector are derived from the
  position vectors

6
Description of Lineation

• Intersection lineation defined by the
  intersection of two, generally curviplanar,
  geological surfaces that separate one volume of
  rock from another
• The intersection of two curviplanar surfaces
  produces a curvilinear lineation
• Examples include contacts, faults, unconformities
  intersecting each other or other surfaces such as
  bedding or foliation (see Fig. 3.1)
• Notice that we measure, e.g., using a compass,
  these surfaces as plane (not as a surface), and
  hence, their intersection turns out to be a
  straight line
• If these intersecting discrete planes remain
  constantly oriented, their intersection, measured
  at different position on different outcrops, plot
  as clusters of points on the stereogram,
  representing discrete lineation
• If the intersection lineation does not permeate
  the rock mass we call it non-penetrative

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Fig. 3.1a-c Two distinct rock volumes separated
across geological surfaces including a) a fault,
F - F b) an igneous contact, I - I and c) an
angular unconformity, U - U. Intersections of
sedimentary or metamorphic layers with these
surfaces define discrete lineations.
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Fig. 3.1d Exposure of a small fault along the
coast of the Bristol Channel, England. The
intersections of the prominent limestone beds
with this fault form discrete lineations as in
Figure 3.1a.
9
Fig. 3.2a-b Discrete lineations defined by
intersections of geological surfaces. a) Straight
lineation at intersection of two planar surfaces.
b) Curved lineation at intersection of curved
surfaces.
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• If the surfaces are not planar, their
  intersection defines a continuous curve in three
  dimensional space (Fig. 3.2b)
• Then measured at different outcrops, the variably
  oriented lineation will plot as a set of points
  along an arc across the stereogram
• Again the stereogram does not depict the spatial
  distribution it only shows the variation in the
  attitude
• To reconstruct the curved intersection lineation
  we need both the orientation and the position of
  each data point

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Non-Penetrative Lineation

• Occurs on the surface of a surface and does not
  permeate the rock mass (e.g., striation on a
  fault plane)

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Fig. 3.3a Superficial lineations on geological
surfaces. a) Slickenlines on fault surface in
granitic rock of the Sierra Nevada, CA (Segall
and Pollard, 1983b).
13
Fig. 3.3a.b Small faults exposed on glaciated
outcrop in the Sierra Nevada, CA. Older aplite
dikes are cut by the faults which offset the
dikes in a left-lateral sense (Segall and
Pollard, 1983b).
14
Fig. 3.3b Superficial lineations on geological
surfaces. b) Lineations on igneous contact in
Henry Mountains, UT (Johnson and Pollard, 1973).
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Penetrative Lineation

• These permeate a large volume of the rock
• They can have constant or variable orientations
• How do we define individual curves (a curvilinear
  penetrative structure) in three-dimensional
  space, from scattered measurements or the
  attitudes and positions
• Assume that lineations measured at different
  outcrops represent segments of the same
  curvilinear lineation over a large area

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Fig. 3.4c Penetrative lineations made by the
intersections of two penetrative planar
foliations as shown schematically in Figure 3.4a
(Weiss, 1972).
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Fig. 3.4d Penetrative lineations made by the
intersections of two penetrative planar
foliations as shown schematically in Figure 3.4a
(Weiss, 1972).
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Fig. 3.4a-b Penetrative lineations. a)
Intersections of two penetrative planar
foliations define straight lineations. b)
Intersections of two penetrative curved
foliations define curved lineations.
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Parametric Representation of Curves

• A curve is a set of points, arranged side by side
  in some orderly and continuous distribution
• The position vectors for points and the curve are
  depicted by the symbol p, and c, respectively
• The spatial continuity of the set of points, that
  compose the curve, is achieved by defining c to
  be a continuous function
• c is a vector quantity, called vector function,
  and is defined in terms of a single real
  variable, t, such that
• c(t)cx(t)ex cy(t)ey cz(t)ez
• where the three scalar functions (cx(t), cy(t),
  cz(t)) are components of the vector function with
  respect to the base vectors (ex, ey, ez)
• C(t) determines the position vectors for all
  points on the curve as t varies smoothly from one
  value to another, the points trace out the curve

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Example Circular Helix (Fig. 3.5)

• c(t)cx(t)ex cy(t)ey cz(t)ez
• c(t)a(cost)ex a(sint)ey btez (circular helix)
• compare the components of the vector function for
  the helix and the first equation
• cx(t)a(cost) cy(t) a(sint) cz(t)bt
• the ranges of the constants, a and b, are
• a gt 0 - ? lt b lt ?
• The points on the curve lie on the right cylinder
  of radius a, with the cylinder axis coincident
  with the z-axis
• As t parameter varies from t0 to t2?, the point
  on the curve advances in the z direction a
  distance of 2?b, and the x and y components
  return to their original values

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Fig. 3.5a-b a) Circular helix, defined by
position vector c, with radius a and pitch b. b)
Special case of a circle where b 0. Arbitrary
parameter is t arc length is s.
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• As t continues to increase the points continue to
  advance in the z-direction encircling the z-axis
• For agt0 and b0, the equation for the circular
  helix reduces to the special case of a circle of
  radius a in the (x,y)-plane (Fig. 3.5b)
• c(t)a(cost)ex a(sint)ey (circle)
• The parameter t, in the case of a circle, is the
  counterclockwise angle, measured in radians, from
  0 at the positive axis

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Cylindrical Fold

• Lineations commonly are found on folded layers
  which are approximated by cylinders (cylindrical
  folds) (Fig. 3.6)
• Cylindrical folds can form by a straight line
  (called fold axis) moved in space parallel to
  itself
• Lineations that are perpendicular to the fold
  axis trace the arc of a circle (small circle on
  stereogram) (Fig. 3.6a)
• Those that are not perpendicular to the fold axis
  may approximate the arc of a circular helix (Fig.
  3.6b)

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Fig. 3.6a-b a) Lineations on cylindrical fold
with circular profile shape lie on arcs of
circles. b) Lineations oblique to fold axis lie
on arcs of a circular helix.
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• In the field we measure lineations at different
  locations
• Assuming that the folds in Fig. 3.6 is
  macroscopic, the short lines on figures 3.6a and
  3.6b can represent outcrops at which we measure
  the lineation
• In reality, we have gaps in our measurements in
  the field due to the lack of continuous exposure

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The Unit Tangent Vector

• In the field, we always measure a curvilinear
  lineation as a line which is tangent to the
  curved lineation
• Hence, we use unit tangent vector, t, along a
  curve for our analysis of curvilinear lineations
• The unit tangent vector is defined by considering
  the position vector to be a function of a special
  parameter, s, such that dc/ds 1
• The difference between the position vectors for
  two points on the curve, say s and s?s, is the
  secant to the curve between the two points (Fig.
  3.7)

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Fig. 3.7a-b Diagrams to define unit tangent
vector t. a) Difference between two position
vectors for curve define vector parallel to the
secant line. b) In the limit as the arc length
goes to zero the secant line is parallel to the
tangent line.
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• Normalizing the difference by the arc length, ?s
  , and taking the limit as this length goes to
  zero, we get the derivative of the vector
  function c with respect to the arc length s
• lim c(s?s)c(s)/?s dc/ds t(s)
• ?s?0
• In this case, as ?s goes to zero, the secant
  becomes parallel to the curve, i.e., parallel to
  the tangent, and of the same length as the arc
• Therefore, the derivative is unit tangent vector
  (t(s)) at point c(s)

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Fig. 3.8a-b a) Unit tangent vector, t, and
curvature vector, k, on circular helix. b) Unit
tangent and curvature vectors on a circle.
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Field Measurement

• The attitude of a lineation measured in the
  field, using geographic angles, can be related to
  the unit tangent vector, t
• Lineations are measured with their plunge (?p)
  and plunge direction (i.e., trend, ?p)
• The measured lineation is parallel to the unit
  tangent vector of a 3D curvilinear lineation at
  the point of measurement

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• Since t is a unit vector, the scalar components
  are equivalent to the direction cosines , i.e.
• tx cos?xsin?pcos?p
• ty cos?ycos?pcos?p
• tz cos?z -sin?p
• These equations relate what we measure in the
  field (at scatted exposures on a continuous
  geological lineation) to the components of the
  unit tangent vectors at correlative points on a
  3D curve

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Example

• Given a lineation with trend, ?p 222 and plunge
  ?p 33
• tx sin?pcos?p -0.561
• ty cos?pcos?p -0.623
• tz -sin?p -0.545
• Note that t 1
• Recall that geographic coordinates are east,
  north, and up (Fig. 2.15)
• The negative numbers for the components of the
  tangent vector make sense, given that the plunge
  direction is to the west (?p 222) and plunge is
  measured down both negative

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The Curvature Vector and the Scalar Curvature

• The unit tangent vector is not one of two
  fundamental properties that uniquely determine
  the shape of a curve
• The first of these, the curvature vector, k, is
  the derivative of the unit tangent vector, and is
  used as a natural representation of a curve, c(s)
• lim t(s?s)t(s)/?s dt/ds k(s)
• ?s?0
• This means that the curvature vector is the
  second derivative the vector function c with
  respect to the natural parameter k(s) d2c/ds2
• This requires a continuous second derivative for
  the curve c(s)

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Fig. 3.9a-b Diagrams to define scalar
curvature. a) Two tangent vectors on curve
separated by arc length. b) Change in angle
between two tangent vectors with respect to arc
length defines scalar curvature in limit as arc
length goes to zero.
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• The curvature vector is directed away from the
  curve on its concave side toward the z-axis
• This means that the curvature vector points in
  the direction that the curve is turning
• The curvature vector is orthogonal to the unit
  tangent vector
• The scalar curvature, k(s), is the magnitude of
  the curvature vector, k(s) i.e., k(s) k(s)
• The scalar curvature is equivalent to the spatial
  rate of change of the orientation of the unit
  tangent vector with arc length along the curve
• The curvature is greater where the orientation of
  the unit tangent vector changes more rapidly with
  position along the curve
• A point on the curve where the curvature is zero
  is called the inflection point

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Fig. 3.10a-b Diagram to define radius of
curvature (Lipschutz, 1969). a) Radius of
curvature defined as radius of best fitting
circle to curve at point s. b) Scalar curvature
and radius of curvature at two points on a
parabola.
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• Another vector, called the unit principal normal
  vector is defined to be parallel to the curvature
  vector, but directed to remain continuous along
  the curve
• n(s) ?k(s)/k(s) Fig. 3.11
• Unit binormal vector, b(s), is the second
  property that uniquely describes curves
• It is normal to the plane containing the unit
  tangent vector, t(s), and the unit principal
  normal vector, n(s)
• b(s) t(s)xn(s) (x is cross product)
• The three unit vectors, t(s), n(s), b(s) (Fig.
  3.12b) for the so called moving trihedron for the
  curve, which travels along the curve with change
  in arc length, s

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Fig. 3.11a-b Diagrams to define unit principal
normal vector, n. a) Curve with curvature
vectors, k(s). b) Same curve with unit principal
normal vectors, n(s).
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Fig. 3.12a-b a) Circular helix with moving
trihedron defined by unit tangent, principal
normal, and binormal vectors (t, n, b) all
functions of the arbitrary parameter t. b)
Derivative of the binormal vector, db/ds, used to
define the scalar torsion.
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Description of Curved Surfaces

• Objects of curved surfaces types are common in
  geology contact, fault, unconformity, axial
  plane, folded layers
• The curvature over a large area of any of these
  objects may carry information as how it formed
• E.g., a faults curvature may represent evolution
  from a series of discrete segments or bending.
  The curvature of the fault may constrain the
  direction and magnitude of slip during earthquake
• Curvature of an igneous dike may be related to
  the stiffness of the surrounding host rock and
  the distribution of the magma pressure
• Curvature of an angular unconformity provides
  information about the sedimentary processes that
  shaped the surface

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Fold Geometry Terminology

• A fold represents a deformation of a geological
  layer into a curved surfaces the fold may be in
  one or more layers
• Fold hinge (hingeline) A physical line
  connecting points of maximum curvature on the
  folded surface
• Hingeline can be measure in the field with an
  compass
• Hinge plane plane containing many hingelines in
  a sequence of folded layers
• Axial plane An approximation of the fold axial
  surface, can be determined in the field or using
  stereonet
• A plane containing one or more axial traces
  (i.e., a great circle containing axial traces)
• A plane containing the fold axis at least one
  other axial trace
• Hinge plane plane containing several hingelines
  on several folded surfaces
• Axial trace Intersection of the axial plane and
  any other plane
• e.g., intersection with an erosional surface or
  intersection with the folded surface (in this
  case, the axial trace is the axis)

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• Fold axis An imaginary line that can generate a
  cylindrical fold by moving it parallel to itself
  in space
• Fold axis can only be determined from a
  stereogram or through mapping it cannot be
  measured in the field directly!
• Fold axis can be measures as follows
• Intersection of fold axial plane and folded
  layers, e.g., bdg (S1xSo)
• The pole to the profile plane (great circle of
  poles to folded surfaces)
• Intersection of an axial planar foliation and the
  folded surfaces
• e.g., SnxSn-1, SnxSn-2
• Cylindrical fold have both fold axis and one
  hingeline these are parallel
• Non-cylindrical folds have no axis they have
  more than one hingeline

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Fig. 3.13a-b Some geometric attributes of
folded geological surfaces. a) Cylindrical fold
with straight fold hinge line axis (H H). b)
Non-cylindrical fold with hinge line that is a
plane curve (there is no fold axis in b!)
(Turner and Weiss, 1963)
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Fig. 3.14a-c More geometric attributes of
folded geological surfaces. Successive hinge
lines define axial (or hinge) surfaces that may
be a) planar b) cylindrical or c)
non-cylindrical. (Turner and Weiss, 1963)
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Why need Differential Geometry

• Such measure, described in the previous two
  slides, do not provide adequate data or measure
  to quantitatively
• Reproduce the surfaces
• Compare the shapes of different surfaces
• Compare the spatial variations in shape of single
  surface
• These can be done with the differential geometry

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Penetrative Planar Fabric

• A systematic arrangement of layers of different
  composition, fracture, similarly oriented planar
  aggregates of minerals, and other planar
  constituents, distributed in the rock mass over a
  range of scales
• Although these components or parts may be
  planar at a local scale, the pole to these planes
  may systematically change orientation from
  exposure to exposure
• It is possible to represent this spatial
  variation in orientation with a set of curved
  surfaces such that a given surface is everywhere
  tangent to the locally planar fabric

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Fig. 3.14d Mesozoic strata of the Colorado
Plateaus upwarped on the margin of the San Rafael
Swell, Utah. Surfaces of these strata form
non-cylindrical folds as depicted schematically
in Figure 3.14c. Photograph by D.D. Pollard.
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Fig. 3.14e Synclinal fold at Rainbow Basin,
Mohave Desert, CA with nearly straight limbs and
planar axial surface as depicted schematically in
Figure 3.14a
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Parametric Representation of Planar Surfaces

• Recall that we defined a curved line as a
  continuous vector function of a single scalar
  variable t, called the arbitrary parameter of the
  curve such that cc(t)
• As t increases in value the heads of successive
  position vectors trace out the curved line

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Fig. 3.14f Outcrop of fold suggesting two
generations of folding in which the planar axial
surface of the first generation fold is itself
folded. Photograph by D.D. Pollard.
51
Fig. 3.14g Ornate (chevron) folds exposed on the
southwest coast of England. Photograph by D.D.
Pollard.
52

• In the case of a plane, close to any point, the
  neighboring points are distributed such that they
  resemble a plane, not a line
• Therefore, a curved surface is a continuous
  vector function of two scalar variables (u, v)
  called the parameters of the surface, such that
  ss(u, v)
• u, and v are the coordinates of points on a
  plane, called the parameter plane, and those
  points map onto the surface according to the
  vector function s(u, v)
• As the two parameters vary, the heads of the
  successive position vectors sweep out the curved
  surface in the 3D space

53
Coordinate System

• The two Cartesian axes (0u, 0v) and associated
  base vectors (eu, ev) define the parameter plane
  that contains the two parameters u, and v
• The (0x, 0y, 0z) Cartesian axes and base vectors
  (ex, ey , ez) comprise the system for the curved
  surface
• The position vector, s, for the curved surface,
  in terms of the two parameters is
• s(u,v) sx(u,v)ex sy(u,v)ey sz(u,v)ez (3.54)
• The three scalar functions sx(u,v), sy(u,v),
  sz(u,v) are the components of the vector
  function s(u,v) w.r.t. the base vectors, which
  together determine the position vectors for all
  points on the curved surface

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• The vector equation (3.54) is called the
  parametric representation of the surface
• Any point on in the parameter plane (3.15a) may
  be defined by a 2D position vector w ueu vev
  w.r.t. the base vectors
• In 3D, the position vectors for a curved surface,
  s, are given by a vector function of the 2D
  vector variable, w, i.e.
• ss(w), or its components ss(u,v)
• An individual point (u0, v0) in the parameter
  plane maps onto the surface with p(u0, v0)
• The coordinate lines u u0 and v v0 in the
  parameter plane map onto the curves c(u0, v) and
  c(u, v0) on the surface

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Fig. 3.15a-b Parametric representation of a
curved surface (Pollard et al., 2004). a) Two
dimensional parameter plane with parameters u and
v. Lines u uo and v vo in the parameter plane
map to v- and u-parameter curves on the surface.
b) Three dimensional surface defined by vector
function of two parameters, s(u, v) (Pollard et
al., 2004)
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Example (Fig. 3.16)

• Given the parametric representation of a curved
  surface is
• s (uv)ex(u-v)ey2(u2v2)ez
• The components of this vector function are equal
  to the coordinates x, y, z in the 3D space
  containing the surface
• x uv yu-v z2(u2v2)
• The last equation is an elliptic paraboloid where
  sections parallel to the (x,y)-plane are circle
• In the geological context, the segment of this
  surface near the origin is similar in shape to
  the surfaces of formations that are deformed into
  a basin-shaped fold

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Fig. 3.16a-b Elliptic paraboloid with circular
sections parallel to the (x, y)-plane. a) The
u-parameter curve, c(u, vo), is a parabola. b)
The v-parameter curve, c(uo, v), also is a
parabola.