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Stereographic Projection References: Dexter Perkins, 2002, Mineralogy, 2nd edition. Prentice Hall, New Jersey, 483 p. Bloss, F.D., 1971, Crystallography and Crystal Chemistry: Holt, Reinhardt, and Winston, New York, 545 p. Klein, C., and Hurlbut, C.S.Jr., 1993, Manual of Mineralogy (after James Dana), 21st edition: John Wiley & Sons, New York, 681 p. Introduction Crystals have a set of 3D geometric relationships among their planar and linear features These include the angle between crystal faces, normal (pole) to these faces, and the line of intersection of these faces Planar features: crystal faces, mirror planes Linear features: pole to crystal faces, zone axis, crystal axes Question: How can we accurately depict all of these planar and linear features on a 2D page and still maintain the correct angular relationships between them? Answer: With equal angle stereographic projection! Stereographic Projection Projection of 3D orientation data and symmetry of a crystal into 2D by preserving all the angular relationships Projection lowers the Euclidian dimension of the object by 1, i.e., planes become lines lines become point In mineralogy, it involves projection of faces, edges, mirror planes, and rotation axes onto a flat equatorial plane of a sphere, in correct angular relationships In mineralogy, in contrast to structural geology, stereograms have no geographic significance, and cannot show shape of crystal faces! Two Types of Stereonet Wulff net (Equal angle) Used in mineralogy & structural geology when angles are meant to be preserved e.g., for crystallography and core analysis Projection is done onto both the upper and lower hemispheres Schmidt net (Equal area) Used in structural geology for orientation analysis when area is meant to be preserved for statistical analysis Uses projection onto the lower hemisphere -90 The ρ angle, is between the c crystal axis and the pole to the crystal face, measured downward from the North pole of the sphere -135 0 +135 Wulff net for Minerals +90 = -90 - 20 40 60 80 =0 + Pole to face http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm axis The ρ angle, is between the c crystal axis and the pole to the crystal face, measured downward from the North pole of the sphere axis a axis The φ angle is measured in the horizontal equatorial plane. Wulff Stereonet (equal angle net) Shows the projection of great circles and small circles Great circle: Line of intersection of a plane, that passes through the center of the sphere, with the surface of the sphere (like lines of longitude on Earth) NOTE: Angular relationships between points can only be measured on great circles (not along small circles)! Small circle: Loci of all positions of a point on the surface of the sphere when rotated about an axis such as the North pole (like lines of latitude on Earth) Why need projection? Projection of all crystal faces of a crystal leads to many great circles or poles to these great circles These great circles and poles allow one to determine the exact angular relationship, and symmetry relationships for a crystal For example, the angle between crystal faces and rotation axes, or between axes and mirror planes To understand these, we first give an introduction to stereographic projection! Stereographic projection of a line • Each line (e.g., rotation axis, pole to a mirror plane) goes through the center of the stereonet (i.e., the thumb tack) • The line intersects the sphere along the ‘spherical projection of the line’, which is a point • A ray, originating from this point, to the eyes of a viewer located vertically above the center of the net (point O), intersect the ‘primitive’ along one point • The point is the stereographic projection of the line. • A vertical line plots at the center of the net • A horizontal line plots on the primitive Special cases of lines Vertical lines (e.g., rotation axes, edges) plot at the center of the equatorial plane Horizontal lines plot on the primitive Inclined lines plot between the primitive and the center Projection of Planar Elements Crystals have faces and mirror planes which are planes, so they intersect the surface of the sphere along lines These elements can be represented either as: Planes, which become great circle after projection Poles (normals) to the planes, which become points after projection Stereographic projection of a plane: • Each plane (e.g., mirror plane) goes through the center of the stereonet (i.e., the thumbtack) • The plane intersects the sphere along the ‘spherical projection of the plane’, which is a series of points • Rays, originating from these points, to the eyes of a viewer located vertically above the center of the net (point O), intersect the ‘primitive’ along a great circle • The great circle is the stereographic projection of the plane. The great circle for a: • vertical plane goes through the center • horizontal plane parallels the primitive Special cases of planes Stereographic projection of a horizontal face or mirror plane is along the primitive (perimeter) of the equatorial plane Stereographic projection of a vertical face or mirror plane is along the straight diameters of the equatorial plane They pass through the center They are straight ‘great circles’ Inclined faces and mirror planes plot along curved great circles that do not pass through the center http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html 3 Face 1: vertical Face 2: inclined Face 3: horizontal 2 1 Face 3 pole to the crystal face Preparing to plot Mark N of the net as -90, E as 0, S as =+90 )(for ). Mount the stereonet on a cardboard. Laminate it. Pass a thumbtack through the center from behind the board Secure the thumbtack with a masking tape from behind the cardboard Place a sheet of tracing paper on the stereonet Put a scotch tape at the center, from both sides of the tracing paper; pierce the paper through the pin Tracing paper can now rotate around the thumbtack without enlarging the hole (because of scotch tape) -90 -135 00 +135 +90 http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html Symbols used For faces below the equator (when using lower hemisphere), place an open circle symbol () where the ray connecting the spherical projection of the pole to the plane intersects the equatorial plane This is the stereographic projection of the pole to the face For faces above the equator (when using the upper hemisphere), place a solid circle symbol () the ray connecting the spherical projection of the pole to the plane intersects the equatorial plane Use a bull’s-eye symbol () to show a point above the page that coincides with one directly below it (when using both hemispheres) Reorient the stereogram such that lines of symmetry are north- south or east-west Projection of Linear Elements We can show all of the symmetry elements of a crystal and their relative positions stereographically Edges, pole to crystal faces, and rotation or roto-inversion axes are lines When extended through the origin of the sphere, lines intersect the surface of the sphere as points Each of these points, when connected to the upper or lower pole of the sphere (viewer’s eyes), is projected onto the equatorial plane, and depicted as a polygon symbol with the same number of sides as the ‘fold’ of the axis Mirror and Polygon Symbols To plot symmetry axes on the stereonet, use the following symbol conventions: Mirror plane: ― (solid line great circles) Crystal axes (lines): Plotting the rotation axes Vertical axes (normal to page) will only have one polygon symbol Horizontal axes (in the plane of page) intersect the primitive twice, hence they have two polygons Inclined axes will have one polygon symbol An open circle in the middle of the polygon shows there is a center of symmetry Measuring angle between faces This is done using the poles to the faces! Three cases: 1. On the primitive, the angle is read directly on the circumference of the net 2. On a straight diameter, the paper is rotated until the zone is coincident with the vertical diameter (i.e., N-S or E-W) and the angle measured on the diameter 3. On a great circle (an inclined zone), rotate the paper until the zone coincides with a great circle on the net; read the angle along the great circle Going from one hemisphere to another During rotation of the pole to a face by a certain angle, we may reach the primitive before we are finished with the amount (angle) of rotation In this case we are moving from one hemisphere to another Move the pole back away from the primitive along that same small circle you followed out to the primitive, until it has been moved the correct total number of degrees Then note its new position with the point symbol for the new hemisphere How to find reflection of a point Having a symmetry (mirror) plane and a point p, find the reflection of point p (i.e., p’) across the mirror: Align the mirror along a great circle Rotate point p along a small circle to the mirror plane Count an equal angle beyond the mirror plane, on the same small circle, to find point p’ If the primitive is reached before p’, then count inward along the same great circle Crystallographic Angles Interfacial angle: between two crystal faces is the angle between poles to the two faces. The interfacial angle can be measured with a contact goniometer These angles are plotted on the stereonet Making a stereographic projection of a crystal face pole Use a contact goniometer to measure the interfacial angles (also measures poles) Convention By convention (Klein and Hurlbut, p.62), we place the crystal at the center of the sphere such that the: c-axis (normal of face 001) is the vertical axis b-axis (normal of face 010) is east-west a-axis (normal of face 100) is north-south See next slide! http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm Up and down axis Stereonet is the equator a axis N-S axis E-W The ρ and φ angles Generally, it is the angles of the spherical projection, ρ and φ, that are given for each face of a crystal These are measured with goniometer If these are known, then the actual angles between any two faces can easily be obtained through trigonometry, or by the use of the stereonet The ρ angle The ρ angle, is between the c axis and the pole to the crystal face, measured downward from the North pole of the sphere A crystal face has a ρ angle measured in the vertical plane containing the axis of the sphere and the face pole. Note: the (010) face has a ρ angle of 90o (010) face is perpendicular to the b-axis The φ angle is measured in the horizontal equatorial plane. Note: the (010) face has a φ angle of 0o! Plotting ρ and φ Suppose you measured ρ = 60o and φ = 30o for a face with goniometer. Plot the pole to this face on the stereonet. Procedure: Line up the N of the tracing paper with the N of the net. From E, count 30 clockwise, put an x (or a tick mark). Bring x to the E, and then count 60 from the center toward E, along the E-W line. Mark the point with . NOTE: The origin for the φ angle is at E (i.e., φ =0). -φ is counted counterclockwise, horizontally from E to the N on the primitive. +φ is counted clockwise horizontally from E to S (i.e., clockwise) on the primitive. http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html -135 135 http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html These angular measurements are similar to those we use for latitude and longitude to plot positions of points on the Earth's surface For the Earth, longitude is similar to the φ angle, except longitude is measured from the Greenwich Meridian, defined as φ = 0o Latitude is measured in the vertical plane, up from the equator, shown as the angle θ. Thus, the ρ angle is like what is called the colatitude (90o - latitude). http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm Zone plotting Zone: Two or more faces whose edges of intersection are parallel to a specific linear direction in a crystal This direction is called the zone axis. A zone is indicated by a symbol similar to that for the Miller Indices of faces, the generalized expression for a zone is [uvw], e.g., all faces parallel to the c axis in an orthorhombic crystal are said to lie in the [001] zone All faces in a zone lie on a great circle; i.e., a zone is constructed by aligning the poles to these faces on a great circle The zone axis (pole to the zone) is normal (i.e., 90o) to this great circle On the stereogram, the lower hemisphere part of the zone great circle is dashed, while the upper great circle is solid Lower-hemisphere faces are depicted by open circle symbol () Upper hemisphere faces are depicted by filled circle symbol () (111) (100) (111) (011) (100) all coplanar (= zone) Thus all poles in a zone are on the same great circle The following rules are applied: All crystal faces are plotted as poles (lines perpendicular to the crystal face. Thus, angles between crystal faces are really angles between poles to crystal faces The b crystallographic axis is taken as the starting point. Such an axis will be perpendicular to the (010) crystal face in any crystal system. The [010] axis (note the zone symbol) or (010) crystal face will therefore plot at φ = 0o and ρ = 90o Positive φ angles will be measured clockwise on the stereonet, and negative φ angles will be measured counter-clockwise on the stereonet Rules cont’d Crystal faces that are on the top of the crystal (ρ < 90o) will be plotted with the closed circles () symbol, and crystal faces on the bottom of the crystal (ρ > 90o) will be plotted with the "" symbol Place a sheet of tracing paper on the stereonet and trace the outermost great circle. Make a reference mark on the right side of the circle (East) To plot a face, first measure the φ angle along the outermost great circle, and make a mark on your tracing paper. Next rotate the tracing paper so that the mark lies at the end of the E-W axis of the stereonet Rules cont’d Measure the ρ angle out from the center of the stereonet along the E-W axis of the stereonet Note that angles can only be measured along great circles. These include the primitive circle, and the E-W and N-S axis of the stereonet Any two faces on the same great circle are in the same zone. Zones can be shown as lines running through the great circle containing faces in that zone The zone axis can be found by setting two faces in the zone on the same great circle, and counting 90o away from the intersection of the great circle along the E-W axis. That is, the zone axis is the pole to the great circle of the poles to the faces in a zone http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm • As an example, the ρ and φ angles for the (111) crystal face in a crystal model is shown here Note again that the ρ angle is measured in the vertical plane containing the c axis and the pole to the face, and the φ angle is measured in the horizontal plane, clockwise from the b axis. D and E are spherical projections, i.e., where the pole to the faces intersect the inside of the sphere D' and E' are stereographic projections, when DS and ES intersect the equator (when projected to the south pole) Distance GD' = f(ρ) as ρ 90 D’ G as ρ 0 D’ O We are looking along the primitive, and viewer’s eye is at the S pole for the two shown faces. The upper an lower hemisphere shown! Fig 6.3 Example for an isometric crystal a3 axis See the isometric crystal axes in the next slide! NOTE: This is a 3D view! a2 axis a1 axis The stereonet is the equatorial plane of the sphere! Stereographic projection of the isometric crystal in the previous slide Upper hemisphere stereographic projection of the poles to the upper crystal faces are shown by the () symbols Viewer’s eyes are at the south pole Symmetry elements of an isometric crystal. Legend Pole to the upper () and lower () crystal faces Stereographic projection of an isometric crystal NOTE: ρ is measured as the distance (in o) from the center of the projection to the position where the pole to the crystal face plots φ is measured around the circumference of the circle, in a clockwise direction away from the b crystallographic axis (010) (1-00) (1-1-0) (1 10) - (1-01) (1-1-1) - (01-1) (01 0) - (11 1) (1-11) (001) (010) a3 (011) (111) (101) (110) (11-0) (100) a1 axis a2 axis Explanation of Previous Slide • In the previous slide, only the upper faces of an isometric crystal are plotted. These faces belong to forms {100}, {110}, and {111} • Form: set of identical faces related by the rotational symmetry (shown by poles/dots in stereograms) • Faces (111) and (110) both have a φ angle of 45o • The ρ angle for these faces is measured along a line from the center of the stereonet (where the (001) face plots) toward the primitive. For the (111) face the ρ angle is 45o, and for the (110) face the ρ angle is 90o As an example all of the faces, both upper and lower, are drawn for a crystal in the class 4/m 2/m in the forms {100} (hexahedron - 6 faces), {110} (dodecahedron, 12 faces), and {111} (octahedron, 8 faces) in the stereogram to the right Rotation axes are indicated by the symbols as discussed above Mirror planes are shown as solid lines and curves, and the primitive circle represents a mirror plane. Note how the symmetry of the crystal can easily be observed in the stereogram http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm How rotational axes are shown Axes that are parallel to the page are indicated by straight lines with proper polygons at the end Solid line if these are parallel to a mirror plane Dashed otherwise Oblique axes plot as polygons between center and primitive The distance between the polygon and center is proportional to the angle between the axis and pole to the face (ρ angle ) Mirror planes (see Figure 9.20 of Perkins) Horizontal mirror planes (in the plane of the page) plot as solid primitive Vertical mirror planes (i.e., normal to the page) plot as solid straight line through the center Inclined mirror planes (inclined to the page) plot as solid curved great circle 3-D Symmetry Conventions http://www.kean.edu/~csmart/Mineralogy/Lectures http://www.kean.edu/~csmart/Mineralogy/Lectures http://www.kean.edu/~csmart/Mineralogy/Lectures