Lecture GG303 8
9/4/01
1
SPHERICAL PROJECTIONS (I) Schedule Updates and Reminders: I Bring tracing paper &needles for Lab 5
Main Topics A What is a spherical projection? B Spherical projection of a line C Spherical projection of a plane II What is a spherical projection? A A 2-D projection for describing the orientation of 3-D features. A spherical projection shows where lines or planes that intersect the surface of a (hemi)sphere, provided that the lines/planes also pass through the center of the (hemi)sphere. B Uses in geology and geophysics 1 Maps 2 Representation of the orientation of planar features (e.g., bedding, fractures, crystal faces) 3 Representation of the orientation of linear features (e.g., fold axes) 4 Representation of first motion data from earthquakes C Great circle: intersection of the surface of a sphere with a plane that passes through the center of the sphere (e.g., lines of longitude) D Small circle: intersection of the surface of a sphere with a plane that does not pass through the center of the sphere (e.g., lines of latitude). A line rotated about an axis traces a small circle too. III Spherical projection of a line A Technique (see handout): 1 A line is at the intersection of two planes: 1) a vertical plane coinciding with the trend of the line and (2) an inclined plane coinciding with the plunge of the line. 2 Trend and plunge: The point representing a line plots away from the center of the spherical plot in the direction of the trend of the line. The trend of a line is measured along a horizontal great circle. The plunge of the line is measured along a vertical great circle by counting down from the horizontal plane. 3 Rake: If the strike and dip of a plane is specified, the rake (pitch) of a line in the plane can be measured along the cyclographic trace of the
Stephen Martel 8 -1 University of Hawaii
GG303 Lecture 8
9/4/01
2
great circle representing that plane. Rake is measured from the direction of strike. B Plane containing two lines: Two intersecting lines uniquely define a plane. The cyclographic trace of the great circle representing that plane will pass through the points representing the lines. IV Spherical projection of a plane A A plane plots as the cyclographic trace of a great circle B Strike and dip: The strike is measured around the perimeter of the primitive circle. The dip of the line is measured along a vertical great circle perpendicular to the line of strike. C Pole to a plane 1 Pole can be plotted directly using its trend and plunge 2 Pole also can be plotted 90 along the cyclographic of trace a great circle that is perpendicular to the plane. D Intersection of two planes 1 Two planes intersect in a line, which projects as a point in a spherical projection. This point is at the intersection of the cyclographic traces of the two planes. 2 The intersection is also 90 from the plane (great circle) containing the poles to the two planes; these 90 angles are measured along the great circles representing the planes containing the poles. This procedure is analogous to finding the cross product between poles.
Stephen Martel
8 -2
University of Hawaii
GG303 Lecture 8
9/4/01
3
Equal-Angle (Sterographic) Projection of a Line
Fig. 8.1
View down of primitive circle R O r V'
= plunge
North = trend
Vertical Cross-section through sphere
Z
Polar coordinates of point V' inside the primitive circle = (r,)
(90 - )/2 R O r R V V'
VOZ is isoceles triangle
r = tan[ (90- )/2 ] R
r = R( tan [ (90- )/2 ] )
= 90 - 2( tan-1 [ r / R ] )
Stephen Martel
8 -3
University of Hawaii
GG303 Lecture 8
9/4/01
4
Stereographic (Equal-angle) Projections (I) A
Horizontal O plane Inclined plane Projection sphere Zenith
FIg. 8.2
B
Cyclographic trace of plane
Projection plane
Primitive Circle
Oblique view of plane intersecting a sphere
Stereographic projection of plane
North Azimuth of strike Cyclographic trace of plane Y Pole
C
- 4 2
- 4 2
D
Trend of pole O
90
R Y O
Inc l Pla ined ne X
X
Azimuth of dip
Pole to plane
Cross section view along strike of inclined plane OX = R tan (/4 - /2)
North Strike directio...