SPHERICAL PROJECTIONS (II) Schedule Updates and Reminders: Bring tracing paper &needles for Lab 5

Transcription

1 GG303 ecture 9 9/4/01 1 SPHERICA PRJECTINS (II) Schedule Updates and Reminders: Bring tracing paper &needles for ab 5 I ain Topics A Angles between lines and planes B Determination of fold axes C Equal- angle and equal- area projections D Equal-angle projections of s E Computer programs for spherical projections II Angles between lines and planes (see Fig. 8.3) A Angle between lines is measured in the plane (great ) containing the lines. n a spherical projection, this angle is measured along the cyclographic trace of the unique great representing the plane containing the two lines. The angle is obtained by counting across the small s the cyclographic traces crosses. B The angle between two planes is the angle between the poles to the planes. This angle is measured along the cyclographic trace of the unique great representing the plane containing the poles to the two planes. This also could be done using the dot product or cross product of the poles. III Fold axes of cylindrical folds (see Fig. 8.3) A The fold axis is along the line of intersection of beds (β diagram). B The fold axis is perpendicular to the plane containing the poles to beds (π diagram); this approach works better than a β diagram if many poles to beds are being considered. For two poles, the π diagram procedure is analogous to finding the cross product between the bedding plane poles. IV Types of spherical projections A Equal angle projection (Wulff net) See handout B Equal area projection (Schmidt net) See handout Stephen artel 9-1 University of Hawaii

2 GG303 ecture 9 9/4/01 2 C Comparison of Equal Angle and Equal Area projections (see Fig. 9.1) (From Hobbs, eans, and Williams, 1976, An utline of Structural Geology) Property Equal angle projection Equal area projection Net type Wulff net Schmidt net Projection Angles Areas preserves... Projection does not Areas Angles preserve... A line project as a... Point Point Great projection Circle Fourth-order quadric Small projection Circle Fourth-order quadric Distance from center of Rtan primitive to 4 dip R 2 sin 4 dip cyclographic trace measured in direction of dip Distance from center of Rtan dip R 2 sin dip primitive to pole of plane measured in the direction opposite to that of the dip Distance from center of Rtan primitive to point 4 plunge R 2 sin 4 plunge that represents a plunging line Best use easuring angular relations Contouring orientation data For equal-angle projections alone, the angle between two planes equals the angle between tangent lines where the cyclographic traces of two planes intersect (hence the name of the projection) V Equal-angle projections of s (see Figs ) VI Free computer programs for spherical projections 1 S.J. artel s atlab code for spherical projections 2 "Stereonet" by R. Allmendinger at Cornell University (for the ac) Stephen artel 9-2 University of Hawaii

3 GG303 ecture 9 9/4/01 3 Spherical Projections Equal-angle projection (Sterographic Projection) Fig. 9.1 Horizontal plane R φ X B Horizontal plane through center of sphere is projection plane B is the "original" point X is the projection The shapes of plane shapes on the surface of the sphere are preserved in this projection, but the relative areas are altered. Good for measuring the angles between the cyclographic traces of planes. Equal-area projection Horizontal plane φ R B A AX = AB x B is the "original" point X is the projection The relative areas of plane shapes on the surface of the sphere are preserved in this projection, but the shapes are altered. Good for representing the density of poles. Stephen artel 9-3 University of Hawaii

4 GG303 ecture 9 9/4/ Equal-Angle Projection of a Small Circle 2 Fig. 9.2 Vertical Cross-section through sphere r θ θ 1 is isoceles triangle r Small θ Rotate about V such that -> N and -> NQ 5 Rotation axis/ axis of cone ; cone is not a right circular cone θ 1 Q V N So this plane intersects cone in a too! What is orientation of plane? 6 θ 1 θ 1 is isoceles triangle NQ AC 3 Small θ 1 ' V' ' θ 1 (θ2 θ1)(θ2)(θ1)()=180 θ2=90 Q V 1 Small N 2 Small Stephen artel 9-4 University of Hawaii

5 GG303 ecture 9 9/4/01 5 Equal-Angle Projection of a Small Circle (II) Fig. 9.3 Vertical Cross-section through sphere "" φ "V" φ φ "" ' V' ' V V = borehole = pole to bedding = pole to bedding View down of primitive V' is not at the center of ''! The center is at C. ' r ' [ ] [ ] V' C' φ φ ' Projection of small r ' = tan 90 - (φ ) r ' = R tan 90 - (φ ) φ = 90-2( tan -1 [ r ' / R ] ) R 2 2 [ ] [ ] This line marks the trend of the borehole V is where the borehole plots has the trend of V' but plunges at φ has the trend of V' but plunges at φ C is midway between ' and ' and is the center of small '' r ' = tan 90 - (φ ) r ' = R tan 90 - (φ ) φ = 90-2(tan -1 [ r ' / R ]) R 2 2 r C = r ' r ' radius of proj. small = r ' - r ' 2 2 Stephen artel 9-5 University of Hawaii

6 GG303 ecture 9 9/4/01 6 Points utside a Primitive Circle in Equal-angle Projections Fig 9.4 View down onto stereonet The point of intersection of three s is outside the primitive. What would this mean? Point of intersection A' Primitive Circle Point is the center of the primitive et's return to how the projection is done to answer the question Cross section view along A Aup An upward pointing line projects outside the primitive! So our "outside" point A' is really A'up. Projection Plane Adown A'down A'up Whereas Aup-Adown is a diameter, then angle Adown--Aup must be a right angle. Also, is perpendicular to A'up. View down onto stereonet P A'up To plot the downward-pointing pole corresponding to A'up, we turn the equal-angle projection method "on its side": 1 Draw a line from A'up through A'down 2 Draw line P perpendicular to line A'up 3 Draw line A'down perpendicular to A'up. Points A'down,, and A'up lie on one line Stephen artel 9-6 University of Hawaii